Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B

Percentage Accurate: 96.5% → 99.5%

Time: 16.5s

Alternatives: 28

Speedup: 1.5×

• 41.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right)\\ \mathbf{if}\;t\_1 + a \cdot \left(\log \left(1 - z\right) - b\right) \leq \infty:\\ \;\;\;\;x \cdot e^{t\_1 - a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* y (- (log z) t))))
   (if (<= (+ t_1 (* a (- (log (- 1.0 z)) b))) INFINITY)
     (* x (exp (- t_1 (* a (+ z b)))))
     (* x (* (* 0.5 (* a a)) (* b b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (log(z) - t);
	double tmp;
	if ((t_1 + (a * (log((1.0 - z)) - b))) <= ((double) INFINITY)) {
		tmp = x * exp((t_1 - (a * (z + b))));
	} else {
		tmp = x * ((0.5 * (a * a)) * (b * b));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (Math.log(z) - t);
	double tmp;
	if ((t_1 + (a * (Math.log((1.0 - z)) - b))) <= Double.POSITIVE_INFINITY) {
		tmp = x * Math.exp((t_1 - (a * (z + b))));
	} else {
		tmp = x * ((0.5 * (a * a)) * (b * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y * (math.log(z) - t)
	tmp = 0
	if (t_1 + (a * (math.log((1.0 - z)) - b))) <= math.inf:
		tmp = x * math.exp((t_1 - (a * (z + b))))
	else:
		tmp = x * ((0.5 * (a * a)) * (b * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y * Float64(log(z) - t))
	tmp = 0.0
	if (Float64(t_1 + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= Inf)
		tmp = Float64(x * exp(Float64(t_1 - Float64(a * Float64(z + b)))));
	else
		tmp = Float64(x * Float64(Float64(0.5 * Float64(a * a)) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y * (log(z) - t);
	tmp = 0.0;
	if ((t_1 + (a * (log((1.0 - z)) - b))) <= Inf)
		tmp = x * exp((t_1 - (a * (z + b))));
	else
		tmp = x * ((0.5 * (a * a)) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x * N[Exp[N[(t$95$1 - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < +inf.0

   1. Initial program 97.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \color{blue}{\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)\right)}\right)\right)\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \left(-1 \cdot \left(a \cdot b + a \cdot z\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \left(\mathsf{neg}\left(\left(a \cdot b + a \cdot z\right)\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \mathsf{neg.f64}\left(\left(a \cdot b + a \cdot z\right)\right)\right)\right)\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \mathsf{neg.f64}\left(\left(a \cdot \left(b + z\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(b + z\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, z\right)\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-a \cdot \left(b + z\right)\right)}} \]

   if +inf.0 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 0.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 50.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 50.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left({b}^{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left({b}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto x \cdot \color{blue}{\left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq \infty:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-7}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-94}:\\ \;\;\;\;\frac{x}{e^{a \cdot \left(z + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{y \cdot \left(t - \log z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.15e-7)
   (* x (exp (* y (- (log z) t))))
   (if (<= y 1.16e-94)
     (/ x (exp (* a (+ z b))))
     (/ x (exp (* y (- t (log z))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.15e-7) {
		tmp = x * exp((y * (log(z) - t)));
	} else if (y <= 1.16e-94) {
		tmp = x / exp((a * (z + b)));
	} else {
		tmp = x / exp((y * (t - log(z))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-3.15d-7)) then
        tmp = x * exp((y * (log(z) - t)))
    else if (y <= 1.16d-94) then
        tmp = x / exp((a * (z + b)))
    else
        tmp = x / exp((y * (t - log(z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.15e-7) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else if (y <= 1.16e-94) {
		tmp = x / Math.exp((a * (z + b)));
	} else {
		tmp = x / Math.exp((y * (t - Math.log(z))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.15e-7:
		tmp = x * math.exp((y * (math.log(z) - t)))
	elif y <= 1.16e-94:
		tmp = x / math.exp((a * (z + b)))
	else:
		tmp = x / math.exp((y * (t - math.log(z))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.15e-7)
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	elseif (y <= 1.16e-94)
		tmp = Float64(x / exp(Float64(a * Float64(z + b))));
	else
		tmp = Float64(x / exp(Float64(y * Float64(t - log(z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.15e-7)
		tmp = x * exp((y * (log(z) - t)));
	elseif (y <= 1.16e-94)
		tmp = x / exp((a * (z + b)));
	else
		tmp = x / exp((y * (t - log(z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.15e-7], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.16e-94], N[(x / N[Exp[N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x / N[Exp[N[(y * N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.15000000000000002e-7

   1. Initial program 93.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(e^{y \cdot \left(\log z - t\right)}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \left(\log z - t\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\log z, t\right)\right)\right)\right) \]

      5. log-lowering-log.f64 87.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right)\right)\right) \]

   5. Simplified 87.3%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   if -3.15000000000000002e-7 < y < 1.16000000000000001e-94

   1. Initial program 95.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right)}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(e^{a \cdot \left(b - \log \left(1 - z\right)\right)}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(a \cdot \left(b - \log \left(1 - z\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \left(b - \log \left(1 - z\right)\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(b, \log \left(1 - z\right)\right)\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(b, \mathsf{log.f64}\left(\left(1 - z\right)\right)\right)\right)\right)\right) \]

      6. --lowering--.f64 87.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(b, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, z\right)\right)\right)\right)\right)\right) \]

   7. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right)}}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b + a \cdot z\right)}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(a \cdot \left(b + z\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \left(b + z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \left(z + b\right)\right)\right)\right) \]

      4. +-lowering-+.f64 92.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(z, b\right)\right)\right)\right) \]

   10. Simplified 92.2%

       \[\leadsto \frac{x}{e^{\color{blue}{a \cdot \left(z + b\right)}}} \]

   if 1.16000000000000001e-94 < y

   1. Initial program 99.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot \left(t - \log z\right)}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(e^{y \cdot \left(t - \log z\right)}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(t - \log z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \left(t - \log z\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(t, \log z\right)\right)\right)\right) \]

      5. log-lowering-log.f64 88.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(t, \mathsf{log.f64}\left(z\right)\right)\right)\right)\right) \]

   7. Simplified 88.7%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot \left(t - \log z\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-94}:\\ \;\;\;\;\frac{x}{e^{a \cdot \left(z + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -2.6e-7) t_1 (if (<= y 1.16e-94) (/ x (exp (* a (+ z b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * (log(z) - t)));
	double tmp;
	if (y <= -2.6e-7) {
		tmp = t_1;
	} else if (y <= 1.16e-94) {
		tmp = x / exp((a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * (log(z) - t)))
    if (y <= (-2.6d-7)) then
        tmp = t_1
    else if (y <= 1.16d-94) then
        tmp = x / exp((a * (z + b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * (Math.log(z) - t)));
	double tmp;
	if (y <= -2.6e-7) {
		tmp = t_1;
	} else if (y <= 1.16e-94) {
		tmp = x / Math.exp((a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * (math.log(z) - t)))
	tmp = 0
	if y <= -2.6e-7:
		tmp = t_1
	elif y <= 1.16e-94:
		tmp = x / math.exp((a * (z + b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(log(z) - t))))
	tmp = 0.0
	if (y <= -2.6e-7)
		tmp = t_1;
	elseif (y <= 1.16e-94)
		tmp = Float64(x / exp(Float64(a * Float64(z + b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * (log(z) - t)));
	tmp = 0.0;
	if (y <= -2.6e-7)
		tmp = t_1;
	elseif (y <= 1.16e-94)
		tmp = x / exp((a * (z + b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.6e-7], t$95$1, If[LessEqual[y, 1.16e-94], N[(x / N[Exp[N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.59999999999999999e-7 or 1.16000000000000001e-94 < y

   1. Initial program 97.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(e^{y \cdot \left(\log z - t\right)}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \left(\log z - t\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\log z, t\right)\right)\right)\right) \]

      5. log-lowering-log.f64 88.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right)\right)\right) \]

   5. Simplified 88.1%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   if -2.59999999999999999e-7 < y < 1.16000000000000001e-94

   1. Initial program 95.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right)}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(e^{a \cdot \left(b - \log \left(1 - z\right)\right)}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(a \cdot \left(b - \log \left(1 - z\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \left(b - \log \left(1 - z\right)\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(b, \log \left(1 - z\right)\right)\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(b, \mathsf{log.f64}\left(\left(1 - z\right)\right)\right)\right)\right)\right) \]

      6. --lowering--.f64 87.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(b, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, z\right)\right)\right)\right)\right)\right) \]

   7. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right)}}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b + a \cdot z\right)}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(a \cdot \left(b + z\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \left(b + z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \left(z + b\right)\right)\right)\right) \]

      4. +-lowering-+.f64 92.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(z, b\right)\right)\right)\right) \]

   10. Simplified 92.2%

       \[\leadsto \frac{x}{e^{\color{blue}{a \cdot \left(z + b\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -400:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -400.0) (/ x (exp (* y t))) (* x (exp (- (* y (log z)) (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -400.0) {
		tmp = x / exp((y * t));
	} else {
		tmp = x * exp(((y * log(z)) - (a * b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-400.0d0)) then
        tmp = x / exp((y * t))
    else
        tmp = x * exp(((y * log(z)) - (a * b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -400.0) {
		tmp = x / Math.exp((y * t));
	} else {
		tmp = x * Math.exp(((y * Math.log(z)) - (a * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -400.0:
		tmp = x / math.exp((y * t))
	else:
		tmp = x * math.exp(((y * math.log(z)) - (a * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -400.0)
		tmp = Float64(x / exp(Float64(y * t)));
	else
		tmp = Float64(x * exp(Float64(Float64(y * log(z)) - Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -400.0)
		tmp = x / exp((y * t));
	else
		tmp = x * exp(((y * log(z)) - (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -400.0], N[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - N[(a * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -400

   1. Initial program 96.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 84.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 84.9%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   if -400 < t

   1. Initial program 96.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot e^{-1 \cdot \left(a \cdot b\right) + y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(e^{-1 \cdot \left(a \cdot b\right) + y \cdot \left(\log z - t\right)}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(-1 \cdot \left(a \cdot b\right) + y \cdot \left(\log z - t\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right) + -1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right) + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right) - a \cdot b\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \left(\log z - t\right)\right), \left(a \cdot b\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\log z - t\right)\right), \left(a \cdot b\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\log z, t\right)\right), \left(a \cdot b\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \left(a \cdot b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 95.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 95.8%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot b}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(y \cdot \log z - a \cdot b\right)}\right)\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \log z\right), \left(a \cdot b\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \log z\right), \left(a \cdot b\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{log.f64}\left(z\right)\right), \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 90.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{log.f64}\left(z\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   8. Simplified 90.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \log z - a \cdot b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{x}{e^{a \cdot \left(z + b\right) + y \cdot \left(t - \log z\right)}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ x (exp (+ (* a (+ z b)) (* y (- t (log z)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / exp(((a * (z + b)) + (y * (t - log(z)))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x / exp(((a * (z + b)) + (y * (t - log(z)))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / Math.exp(((a * (z + b)) + (y * (t - Math.log(z)))));
}

Python

def code(x, y, z, t, a, b):
	return x / math.exp(((a * (z + b)) + (y * (t - math.log(z)))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x / exp(Float64(Float64(a * Float64(z + b)) + Float64(y * Float64(t - log(z))))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / exp(((a * (z + b)) + (y * (t - log(z)))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / N[Exp[N[(N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision] + N[(y * N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

3. Simplified 96.4%

   \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b + \left(a \cdot z + y \cdot \left(t - \log z\right)\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(\left(a \cdot b + a \cdot z\right) + y \cdot \left(t - \log z\right)\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(a \cdot b + a \cdot z\right), \left(y \cdot \left(t - \log z\right)\right)\right)\right)\right) \]

   3. distribute-lft-out N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(b + z\right)\right), \left(y \cdot \left(t - \log z\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(b + z\right)\right), \left(y \cdot \left(t - \log z\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, z\right)\right), \left(y \cdot \left(t - \log z\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, z\right)\right), \mathsf{*.f64}\left(y, \left(t - \log z\right)\right)\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, z\right)\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(t, \log z\right)\right)\right)\right)\right) \]

   8. log-lowering-log.f64 98.4%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, z\right)\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(t, \mathsf{log.f64}\left(z\right)\right)\right)\right)\right)\right) \]

7. Simplified 98.4%

   \[\leadsto \frac{x}{e^{\color{blue}{a \cdot \left(b + z\right) + y \cdot \left(t - \log z\right)}}} \]

8. Final simplification 98.4%

   \[\leadsto \frac{x}{e^{a \cdot \left(z + b\right) + y \cdot \left(t - \log z\right)}} \]
9. Add Preprocessing

Alternative 6: 96.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot b} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (- (* y (- (log z) t)) (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) - (a * b)));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) - (a * b)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) - (a * b)));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) - (a * b)))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) - Float64(a * b))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) - (a * b)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] - N[(a * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x \cdot e^{-1 \cdot \left(a \cdot b\right) + y \cdot \left(\log z - t\right)}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(e^{-1 \cdot \left(a \cdot b\right) + y \cdot \left(\log z - t\right)}\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(-1 \cdot \left(a \cdot b\right) + y \cdot \left(\log z - t\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right) + -1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right) + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right) - a \cdot b\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \left(\log z - t\right)\right), \left(a \cdot b\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\log z - t\right)\right), \left(a \cdot b\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\log z, t\right)\right), \left(a \cdot b\right)\right)\right)\right) \]

   9. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \left(a \cdot b\right)\right)\right)\right) \]

   10. *-lowering-*.f64 96.0%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

5. Simplified 96.0%

   \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot b}} \]
6. Add Preprocessing

Alternative 7: 58.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(a \cdot \left(0.5 \cdot \left(b \cdot b\right) + \left(a \cdot -0.16666666666666666\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) - b\right)\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -1.45e-7)
     t_1
     (if (<= y -7e-132)
       (* 2.0 (/ x (* a (* a (* b b)))))
       (if (<= y 1.3e-298)
         (*
          x
          (+
           1.0
           (*
            a
            (-
             (*
              a
              (+ (* 0.5 (* b b)) (* (* a -0.16666666666666666) (* b (* b b)))))
             b))))
         (if (<= y 7.2e-7)
           (/ x (+ 1.0 (* b (+ a (* b (* 0.5 (* a a)))))))
           t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -1.45e-7) {
		tmp = t_1;
	} else if (y <= -7e-132) {
		tmp = 2.0 * (x / (a * (a * (b * b))));
	} else if (y <= 1.3e-298) {
		tmp = x * (1.0 + (a * ((a * ((0.5 * (b * b)) + ((a * -0.16666666666666666) * (b * (b * b))))) - b)));
	} else if (y <= 7.2e-7) {
		tmp = x / (1.0 + (b * (a + (b * (0.5 * (a * a))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z ** y)
    if (y <= (-1.45d-7)) then
        tmp = t_1
    else if (y <= (-7d-132)) then
        tmp = 2.0d0 * (x / (a * (a * (b * b))))
    else if (y <= 1.3d-298) then
        tmp = x * (1.0d0 + (a * ((a * ((0.5d0 * (b * b)) + ((a * (-0.16666666666666666d0)) * (b * (b * b))))) - b)))
    else if (y <= 7.2d-7) then
        tmp = x / (1.0d0 + (b * (a + (b * (0.5d0 * (a * a))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double tmp;
	if (y <= -1.45e-7) {
		tmp = t_1;
	} else if (y <= -7e-132) {
		tmp = 2.0 * (x / (a * (a * (b * b))));
	} else if (y <= 1.3e-298) {
		tmp = x * (1.0 + (a * ((a * ((0.5 * (b * b)) + ((a * -0.16666666666666666) * (b * (b * b))))) - b)));
	} else if (y <= 7.2e-7) {
		tmp = x / (1.0 + (b * (a + (b * (0.5 * (a * a))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if y <= -1.45e-7:
		tmp = t_1
	elif y <= -7e-132:
		tmp = 2.0 * (x / (a * (a * (b * b))))
	elif y <= 1.3e-298:
		tmp = x * (1.0 + (a * ((a * ((0.5 * (b * b)) + ((a * -0.16666666666666666) * (b * (b * b))))) - b)))
	elif y <= 7.2e-7:
		tmp = x / (1.0 + (b * (a + (b * (0.5 * (a * a))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -1.45e-7)
		tmp = t_1;
	elseif (y <= -7e-132)
		tmp = Float64(2.0 * Float64(x / Float64(a * Float64(a * Float64(b * b)))));
	elseif (y <= 1.3e-298)
		tmp = Float64(x * Float64(1.0 + Float64(a * Float64(Float64(a * Float64(Float64(0.5 * Float64(b * b)) + Float64(Float64(a * -0.16666666666666666) * Float64(b * Float64(b * b))))) - b))));
	elseif (y <= 7.2e-7)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(a + Float64(b * Float64(0.5 * Float64(a * a)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -1.45e-7)
		tmp = t_1;
	elseif (y <= -7e-132)
		tmp = 2.0 * (x / (a * (a * (b * b))));
	elseif (y <= 1.3e-298)
		tmp = x * (1.0 + (a * ((a * ((0.5 * (b * b)) + ((a * -0.16666666666666666) * (b * (b * b))))) - b)));
	elseif (y <= 7.2e-7)
		tmp = x / (1.0 + (b * (a + (b * (0.5 * (a * a))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.45e-7], t$95$1, If[LessEqual[y, -7e-132], N[(2.0 * N[(x / N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-298], N[(x * N[(1.0 + N[(a * N[(N[(a * N[(N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-7], N[(x / N[(1.0 + N[(b * N[(a + N[(b * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.4499999999999999e-7 or 7.19999999999999989e-7 < y

   1. Initial program 96.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(e^{y \cdot \left(\log z - t\right)}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \left(\log z - t\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\log z, t\right)\right)\right)\right) \]

      5. log-lowering-log.f64 87.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right)\right)\right) \]

   5. Simplified 87.3%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({z}^{y}\right)}\right) \]

      2. pow-lowering-pow.f64 65.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(z, \color{blue}{y}\right)\right) \]

   8. Simplified 65.4%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -1.4499999999999999e-7 < y < -6.9999999999999999e-132

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 78.6%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 78.6%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 56.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 56.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 67.5%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 67.5%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if -6.9999999999999999e-132 < y < 1.2999999999999999e-298

   1. Initial program 97.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 94.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 94.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

   8. Simplified 72.8%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(a \cdot \left(0.5 \cdot \left(b \cdot b\right) + \left(-0.16666666666666666 \cdot a\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) - b\right)\right)} \]

   if 1.2999999999999999e-298 < y < 7.19999999999999989e-7

   1. Initial program 93.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 75.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 75.5%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + b \cdot \left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{b}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), b\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), b\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 58.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), b\right)\right)\right)\right)\right) \]

   10. Simplified 58.5%

       \[\leadsto \frac{x}{\color{blue}{1 + b \cdot \left(a + \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot b\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(a \cdot \left(0.5 \cdot \left(b \cdot b\right) + \left(a \cdot -0.16666666666666666\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) - b\right)\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-7}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-94}:\\ \;\;\;\;\frac{x}{e^{a \cdot \left(z + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7.8e-7)
   (* x (pow z y))
   (if (<= y 1.16e-94) (/ x (exp (* a (+ z b)))) (/ x (exp (* y t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.8e-7) {
		tmp = x * pow(z, y);
	} else if (y <= 1.16e-94) {
		tmp = x / exp((a * (z + b)));
	} else {
		tmp = x / exp((y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-7.8d-7)) then
        tmp = x * (z ** y)
    else if (y <= 1.16d-94) then
        tmp = x / exp((a * (z + b)))
    else
        tmp = x / exp((y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.8e-7) {
		tmp = x * Math.pow(z, y);
	} else if (y <= 1.16e-94) {
		tmp = x / Math.exp((a * (z + b)));
	} else {
		tmp = x / Math.exp((y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7.8e-7:
		tmp = x * math.pow(z, y)
	elif y <= 1.16e-94:
		tmp = x / math.exp((a * (z + b)))
	else:
		tmp = x / math.exp((y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7.8e-7)
		tmp = Float64(x * (z ^ y));
	elseif (y <= 1.16e-94)
		tmp = Float64(x / exp(Float64(a * Float64(z + b))));
	else
		tmp = Float64(x / exp(Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -7.8e-7)
		tmp = x * (z ^ y);
	elseif (y <= 1.16e-94)
		tmp = x / exp((a * (z + b)));
	else
		tmp = x / exp((y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7.8e-7], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.16e-94], N[(x / N[Exp[N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.80000000000000049e-7

   1. Initial program 93.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(e^{y \cdot \left(\log z - t\right)}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \left(\log z - t\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\log z, t\right)\right)\right)\right) \]

      5. log-lowering-log.f64 87.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right)\right)\right) \]

   5. Simplified 87.3%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({z}^{y}\right)}\right) \]

      2. pow-lowering-pow.f64 71.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(z, \color{blue}{y}\right)\right) \]

   8. Simplified 71.4%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -7.80000000000000049e-7 < y < 1.16000000000000001e-94

   1. Initial program 95.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right)}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(e^{a \cdot \left(b - \log \left(1 - z\right)\right)}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(a \cdot \left(b - \log \left(1 - z\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \left(b - \log \left(1 - z\right)\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(b, \log \left(1 - z\right)\right)\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(b, \mathsf{log.f64}\left(\left(1 - z\right)\right)\right)\right)\right)\right) \]

      6. --lowering--.f64 87.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(b, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, z\right)\right)\right)\right)\right)\right) \]

   7. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right)}}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b + a \cdot z\right)}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(a \cdot \left(b + z\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \left(b + z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \left(z + b\right)\right)\right)\right) \]

      4. +-lowering-+.f64 92.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(z, b\right)\right)\right)\right) \]

   10. Simplified 92.2%

       \[\leadsto \frac{x}{e^{\color{blue}{a \cdot \left(z + b\right)}}} \]

   if 1.16000000000000001e-94 < y

   1. Initial program 99.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 70.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 70.8%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{e^{a \cdot b}}\\ \mathbf{if}\;b \leq -4.1 \cdot 10^{+167}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4800000:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (exp (* a b)))))
   (if (<= b -4.1e+167) t_1 (if (<= b 4800000.0) (/ x (exp (* y t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / exp((a * b));
	double tmp;
	if (b <= -4.1e+167) {
		tmp = t_1;
	} else if (b <= 4800000.0) {
		tmp = x / exp((y * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / exp((a * b))
    if (b <= (-4.1d+167)) then
        tmp = t_1
    else if (b <= 4800000.0d0) then
        tmp = x / exp((y * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / Math.exp((a * b));
	double tmp;
	if (b <= -4.1e+167) {
		tmp = t_1;
	} else if (b <= 4800000.0) {
		tmp = x / Math.exp((y * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / math.exp((a * b))
	tmp = 0
	if b <= -4.1e+167:
		tmp = t_1
	elif b <= 4800000.0:
		tmp = x / math.exp((y * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / exp(Float64(a * b)))
	tmp = 0.0
	if (b <= -4.1e+167)
		tmp = t_1;
	elseif (b <= 4800000.0)
		tmp = Float64(x / exp(Float64(y * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / exp((a * b));
	tmp = 0.0;
	if (b <= -4.1e+167)
		tmp = t_1;
	elseif (b <= 4800000.0)
		tmp = x / exp((y * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[Exp[N[(a * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.1e+167], t$95$1, If[LessEqual[b, 4800000.0], N[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.1e167 or 4.8e6 < b

   1. Initial program 96.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 90.4%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   if -4.1e167 < b < 4.8e6

   1. Initial program 96.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 72.8%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 73.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+75}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -7.8e-7) t_1 (if (<= y 2.4e+75) (/ x (exp (* a b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -7.8e-7) {
		tmp = t_1;
	} else if (y <= 2.4e+75) {
		tmp = x / exp((a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z ** y)
    if (y <= (-7.8d-7)) then
        tmp = t_1
    else if (y <= 2.4d+75) then
        tmp = x / exp((a * b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double tmp;
	if (y <= -7.8e-7) {
		tmp = t_1;
	} else if (y <= 2.4e+75) {
		tmp = x / Math.exp((a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if y <= -7.8e-7:
		tmp = t_1
	elif y <= 2.4e+75:
		tmp = x / math.exp((a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -7.8e-7)
		tmp = t_1;
	elseif (y <= 2.4e+75)
		tmp = Float64(x / exp(Float64(a * b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -7.8e-7)
		tmp = t_1;
	elseif (y <= 2.4e+75)
		tmp = x / exp((a * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.8e-7], t$95$1, If[LessEqual[y, 2.4e+75], N[(x / N[Exp[N[(a * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.80000000000000049e-7 or 2.4e75 < y

   1. Initial program 96.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(e^{y \cdot \left(\log z - t\right)}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\log z - t\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \left(\log z - t\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\log z, t\right)\right)\right)\right) \]

      5. log-lowering-log.f64 90.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right)\right)\right) \]

   5. Simplified 90.3%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({z}^{y}\right)}\right) \]

      2. pow-lowering-pow.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(z, \color{blue}{y}\right)\right) \]

   8. Simplified 69.2%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -7.80000000000000049e-7 < y < 2.4e75

   1. Initial program 96.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 96.3%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 79.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 79.0%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.5% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+140}:\\ \;\;\;\;x + t \cdot \left(0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - y \cdot x\right)\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(a \cdot \left(0.5 \cdot \left(b \cdot b\right) + \left(a \cdot -0.16666666666666666\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) - b\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot t\_1\right)}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \left(t\_1 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + t \cdot \left(y + t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.5 (* a a))))
   (if (<= y -1.05e+140)
     (+ x (* t (- (* 0.5 (* t (* x (* y y)))) (* y x))))
     (if (<= y -5.6e-132)
       (* 2.0 (/ x (* a (* a (* b b)))))
       (if (<= y 1.3e-298)
         (*
          x
          (+
           1.0
           (*
            a
            (-
             (*
              a
              (+ (* 0.5 (* b b)) (* (* a -0.16666666666666666) (* b (* b b)))))
             b))))
         (if (<= y 6.2e+44)
           (/ x (+ 1.0 (* b (+ a (* b t_1)))))
           (if (<= y 7.5e+153)
             (* x (* t_1 (* b b)))
             (/ x (+ 1.0 (* t (+ y (* t (* 0.5 (* y y))))))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (a * a);
	double tmp;
	if (y <= -1.05e+140) {
		tmp = x + (t * ((0.5 * (t * (x * (y * y)))) - (y * x)));
	} else if (y <= -5.6e-132) {
		tmp = 2.0 * (x / (a * (a * (b * b))));
	} else if (y <= 1.3e-298) {
		tmp = x * (1.0 + (a * ((a * ((0.5 * (b * b)) + ((a * -0.16666666666666666) * (b * (b * b))))) - b)));
	} else if (y <= 6.2e+44) {
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	} else if (y <= 7.5e+153) {
		tmp = x * (t_1 * (b * b));
	} else {
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 * (a * a)
    if (y <= (-1.05d+140)) then
        tmp = x + (t * ((0.5d0 * (t * (x * (y * y)))) - (y * x)))
    else if (y <= (-5.6d-132)) then
        tmp = 2.0d0 * (x / (a * (a * (b * b))))
    else if (y <= 1.3d-298) then
        tmp = x * (1.0d0 + (a * ((a * ((0.5d0 * (b * b)) + ((a * (-0.16666666666666666d0)) * (b * (b * b))))) - b)))
    else if (y <= 6.2d+44) then
        tmp = x / (1.0d0 + (b * (a + (b * t_1))))
    else if (y <= 7.5d+153) then
        tmp = x * (t_1 * (b * b))
    else
        tmp = x / (1.0d0 + (t * (y + (t * (0.5d0 * (y * y))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (a * a);
	double tmp;
	if (y <= -1.05e+140) {
		tmp = x + (t * ((0.5 * (t * (x * (y * y)))) - (y * x)));
	} else if (y <= -5.6e-132) {
		tmp = 2.0 * (x / (a * (a * (b * b))));
	} else if (y <= 1.3e-298) {
		tmp = x * (1.0 + (a * ((a * ((0.5 * (b * b)) + ((a * -0.16666666666666666) * (b * (b * b))))) - b)));
	} else if (y <= 6.2e+44) {
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	} else if (y <= 7.5e+153) {
		tmp = x * (t_1 * (b * b));
	} else {
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.5 * (a * a)
	tmp = 0
	if y <= -1.05e+140:
		tmp = x + (t * ((0.5 * (t * (x * (y * y)))) - (y * x)))
	elif y <= -5.6e-132:
		tmp = 2.0 * (x / (a * (a * (b * b))))
	elif y <= 1.3e-298:
		tmp = x * (1.0 + (a * ((a * ((0.5 * (b * b)) + ((a * -0.16666666666666666) * (b * (b * b))))) - b)))
	elif y <= 6.2e+44:
		tmp = x / (1.0 + (b * (a + (b * t_1))))
	elif y <= 7.5e+153:
		tmp = x * (t_1 * (b * b))
	else:
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.5 * Float64(a * a))
	tmp = 0.0
	if (y <= -1.05e+140)
		tmp = Float64(x + Float64(t * Float64(Float64(0.5 * Float64(t * Float64(x * Float64(y * y)))) - Float64(y * x))));
	elseif (y <= -5.6e-132)
		tmp = Float64(2.0 * Float64(x / Float64(a * Float64(a * Float64(b * b)))));
	elseif (y <= 1.3e-298)
		tmp = Float64(x * Float64(1.0 + Float64(a * Float64(Float64(a * Float64(Float64(0.5 * Float64(b * b)) + Float64(Float64(a * -0.16666666666666666) * Float64(b * Float64(b * b))))) - b))));
	elseif (y <= 6.2e+44)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(a + Float64(b * t_1)))));
	elseif (y <= 7.5e+153)
		tmp = Float64(x * Float64(t_1 * Float64(b * b)));
	else
		tmp = Float64(x / Float64(1.0 + Float64(t * Float64(y + Float64(t * Float64(0.5 * Float64(y * y)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.5 * (a * a);
	tmp = 0.0;
	if (y <= -1.05e+140)
		tmp = x + (t * ((0.5 * (t * (x * (y * y)))) - (y * x)));
	elseif (y <= -5.6e-132)
		tmp = 2.0 * (x / (a * (a * (b * b))));
	elseif (y <= 1.3e-298)
		tmp = x * (1.0 + (a * ((a * ((0.5 * (b * b)) + ((a * -0.16666666666666666) * (b * (b * b))))) - b)));
	elseif (y <= 6.2e+44)
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	elseif (y <= 7.5e+153)
		tmp = x * (t_1 * (b * b));
	else
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+140], N[(x + N[(t * N[(N[(0.5 * N[(t * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.6e-132], N[(2.0 * N[(x / N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-298], N[(x * N[(1.0 + N[(a * N[(N[(a * N[(N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+44], N[(x / N[(1.0 + N[(b * N[(a + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+153], N[(x * N[(t$95$1 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(t * N[(y + N[(t * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y < -1.0500000000000001e140

   1. Initial program 91.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 91.9%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 71.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 71.0%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(t \cdot \left(-1 \cdot \left(t \cdot \left(-1 \cdot \left(y \cdot \left(-1 \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) + \left(\frac{-1}{2} \cdot \left(x \cdot {y}^{3}\right) + \frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right)\right) - \left(-1 \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) - x \cdot y\right)} \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{x + t \cdot \left(t \cdot \left(-\left(t \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot y\right) \cdot y\right)\right) + \left(\left(\left(x \cdot y\right) \cdot y\right) \cdot y\right) \cdot -0.3333333333333333\right) + \left(\left(x \cdot y\right) \cdot y\right) \cdot -0.5\right)\right) - x \cdot y\right)} \]

   11. Taylor expanded in t around 0

       \[\leadsto \color{blue}{x + t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{\left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \left(\left(\mathsf{neg}\left(x \cdot y\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)}\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right) - \color{blue}{x \cdot y}\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right), \color{blue}{\left(x \cdot y\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \left(x \cdot {y}^{2}\right)\right)\right), \left(x \cdot y\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \left({y}^{2}\right)\right)\right)\right), \left(x \cdot y\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \left(y \cdot y\right)\right)\right)\right), \left(x \cdot y\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), \left(x \cdot y\right)\right)\right)\right) \]

       12. *-lowering-*.f64 62.5%

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right)\right) \]

   13. Simplified 62.5%

       \[\leadsto \color{blue}{x + t \cdot \left(0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - x \cdot y\right)} \]

   if -1.0500000000000001e140 < y < -5.60000000000000005e-132

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 57.0%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 34.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 60.1%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 60.1%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if -5.60000000000000005e-132 < y < 1.2999999999999999e-298

   1. Initial program 97.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 94.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 94.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

   8. Simplified 72.8%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(a \cdot \left(0.5 \cdot \left(b \cdot b\right) + \left(-0.16666666666666666 \cdot a\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) - b\right)\right)} \]

   if 1.2999999999999999e-298 < y < 6.19999999999999991e44

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 72.4%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + b \cdot \left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{b}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), b\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), b\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), b\right)\right)\right)\right)\right) \]

   10. Simplified 56.9%

       \[\leadsto \frac{x}{\color{blue}{1 + b \cdot \left(a + \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot b\right)}} \]

   if 6.19999999999999991e44 < y < 7.50000000000000065e153

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 44.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 44.0%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 25.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 25.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left({b}^{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left({b}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 51.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 51.1%

       \[\leadsto x \cdot \color{blue}{\left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)} \]

   if 7.50000000000000065e153 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 72.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 72.1%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + t \cdot \left(y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + \color{blue}{y}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(\left(t \cdot {y}^{2}\right) \cdot \frac{1}{2} + y\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(t \cdot \left({y}^{2} \cdot \frac{1}{2}\right) + y\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(t \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + y\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t \cdot \left({y}^{2} \cdot \frac{1}{2}\right) + y\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(\left(t \cdot {y}^{2}\right) \cdot \frac{1}{2} + y\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + y\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + \left(t \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + t \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + t \cdot \left(\frac{1}{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 75.4%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 75.4%

       \[\leadsto \frac{x}{\color{blue}{1 + t \cdot \left(y + t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+140}:\\ \;\;\;\;x + t \cdot \left(0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - y \cdot x\right)\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(a \cdot \left(0.5 \cdot \left(b \cdot b\right) + \left(a \cdot -0.16666666666666666\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) - b\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + t \cdot \left(y + t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(a \cdot a\right)\\ t_2 := a \cdot \left(b \cdot b\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+140}:\\ \;\;\;\;x + t \cdot \left(0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - y \cdot x\right)\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot t\_2}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(0.5 \cdot t\_2 - b\right)\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot t\_1\right)}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \left(t\_1 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + t \cdot \left(y + t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.5 (* a a))) (t_2 (* a (* b b))))
   (if (<= y -1.05e+140)
     (+ x (* t (- (* 0.5 (* t (* x (* y y)))) (* y x))))
     (if (<= y -7.2e-132)
       (* 2.0 (/ x (* a t_2)))
       (if (<= y 4.2e-298)
         (* x (+ 1.0 (* a (- (* 0.5 t_2) b))))
         (if (<= y 5.8e+44)
           (/ x (+ 1.0 (* b (+ a (* b t_1)))))
           (if (<= y 4.4e+153)
             (* x (* t_1 (* b b)))
             (/ x (+ 1.0 (* t (+ y (* t (* 0.5 (* y y))))))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (a * a);
	double t_2 = a * (b * b);
	double tmp;
	if (y <= -1.05e+140) {
		tmp = x + (t * ((0.5 * (t * (x * (y * y)))) - (y * x)));
	} else if (y <= -7.2e-132) {
		tmp = 2.0 * (x / (a * t_2));
	} else if (y <= 4.2e-298) {
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)));
	} else if (y <= 5.8e+44) {
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	} else if (y <= 4.4e+153) {
		tmp = x * (t_1 * (b * b));
	} else {
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.5d0 * (a * a)
    t_2 = a * (b * b)
    if (y <= (-1.05d+140)) then
        tmp = x + (t * ((0.5d0 * (t * (x * (y * y)))) - (y * x)))
    else if (y <= (-7.2d-132)) then
        tmp = 2.0d0 * (x / (a * t_2))
    else if (y <= 4.2d-298) then
        tmp = x * (1.0d0 + (a * ((0.5d0 * t_2) - b)))
    else if (y <= 5.8d+44) then
        tmp = x / (1.0d0 + (b * (a + (b * t_1))))
    else if (y <= 4.4d+153) then
        tmp = x * (t_1 * (b * b))
    else
        tmp = x / (1.0d0 + (t * (y + (t * (0.5d0 * (y * y))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (a * a);
	double t_2 = a * (b * b);
	double tmp;
	if (y <= -1.05e+140) {
		tmp = x + (t * ((0.5 * (t * (x * (y * y)))) - (y * x)));
	} else if (y <= -7.2e-132) {
		tmp = 2.0 * (x / (a * t_2));
	} else if (y <= 4.2e-298) {
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)));
	} else if (y <= 5.8e+44) {
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	} else if (y <= 4.4e+153) {
		tmp = x * (t_1 * (b * b));
	} else {
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.5 * (a * a)
	t_2 = a * (b * b)
	tmp = 0
	if y <= -1.05e+140:
		tmp = x + (t * ((0.5 * (t * (x * (y * y)))) - (y * x)))
	elif y <= -7.2e-132:
		tmp = 2.0 * (x / (a * t_2))
	elif y <= 4.2e-298:
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)))
	elif y <= 5.8e+44:
		tmp = x / (1.0 + (b * (a + (b * t_1))))
	elif y <= 4.4e+153:
		tmp = x * (t_1 * (b * b))
	else:
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.5 * Float64(a * a))
	t_2 = Float64(a * Float64(b * b))
	tmp = 0.0
	if (y <= -1.05e+140)
		tmp = Float64(x + Float64(t * Float64(Float64(0.5 * Float64(t * Float64(x * Float64(y * y)))) - Float64(y * x))));
	elseif (y <= -7.2e-132)
		tmp = Float64(2.0 * Float64(x / Float64(a * t_2)));
	elseif (y <= 4.2e-298)
		tmp = Float64(x * Float64(1.0 + Float64(a * Float64(Float64(0.5 * t_2) - b))));
	elseif (y <= 5.8e+44)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(a + Float64(b * t_1)))));
	elseif (y <= 4.4e+153)
		tmp = Float64(x * Float64(t_1 * Float64(b * b)));
	else
		tmp = Float64(x / Float64(1.0 + Float64(t * Float64(y + Float64(t * Float64(0.5 * Float64(y * y)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.5 * (a * a);
	t_2 = a * (b * b);
	tmp = 0.0;
	if (y <= -1.05e+140)
		tmp = x + (t * ((0.5 * (t * (x * (y * y)))) - (y * x)));
	elseif (y <= -7.2e-132)
		tmp = 2.0 * (x / (a * t_2));
	elseif (y <= 4.2e-298)
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)));
	elseif (y <= 5.8e+44)
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	elseif (y <= 4.4e+153)
		tmp = x * (t_1 * (b * b));
	else
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+140], N[(x + N[(t * N[(N[(0.5 * N[(t * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -7.2e-132], N[(2.0 * N[(x / N[(a * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-298], N[(x * N[(1.0 + N[(a * N[(N[(0.5 * t$95$2), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e+44], N[(x / N[(1.0 + N[(b * N[(a + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e+153], N[(x * N[(t$95$1 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(t * N[(y + N[(t * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y < -1.0500000000000001e140

   1. Initial program 91.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 91.9%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 71.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 71.0%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(t \cdot \left(-1 \cdot \left(t \cdot \left(-1 \cdot \left(y \cdot \left(-1 \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) + \left(\frac{-1}{2} \cdot \left(x \cdot {y}^{3}\right) + \frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right)\right) - \left(-1 \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) - x \cdot y\right)} \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{x + t \cdot \left(t \cdot \left(-\left(t \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot y\right) \cdot y\right)\right) + \left(\left(\left(x \cdot y\right) \cdot y\right) \cdot y\right) \cdot -0.3333333333333333\right) + \left(\left(x \cdot y\right) \cdot y\right) \cdot -0.5\right)\right) - x \cdot y\right)} \]

   11. Taylor expanded in t around 0

       \[\leadsto \color{blue}{x + t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(t \cdot \left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{\left(-1 \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \left(\left(\mathsf{neg}\left(x \cdot y\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)}\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right) - \color{blue}{x \cdot y}\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right), \color{blue}{\left(x \cdot y\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \left(x \cdot {y}^{2}\right)\right)\right), \left(x \cdot y\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \left({y}^{2}\right)\right)\right)\right), \left(x \cdot y\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \left(y \cdot y\right)\right)\right)\right), \left(x \cdot y\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), \left(x \cdot y\right)\right)\right)\right) \]

       12. *-lowering-*.f64 62.5%

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right)\right) \]

   13. Simplified 62.5%

       \[\leadsto \color{blue}{x + t \cdot \left(0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - x \cdot y\right)} \]

   if -1.0500000000000001e140 < y < -7.20000000000000015e-132

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 57.0%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 34.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 60.1%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 60.1%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if -7.20000000000000015e-132 < y < 4.2000000000000001e-298

   1. Initial program 97.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 94.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 94.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 70.2%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   if 4.2000000000000001e-298 < y < 5.8000000000000004e44

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 72.4%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + b \cdot \left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{b}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), b\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), b\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), b\right)\right)\right)\right)\right) \]

   10. Simplified 56.9%

       \[\leadsto \frac{x}{\color{blue}{1 + b \cdot \left(a + \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot b\right)}} \]

   if 5.8000000000000004e44 < y < 4.3999999999999999e153

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 44.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 44.0%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 25.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 25.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left({b}^{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left({b}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 51.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 51.1%

       \[\leadsto x \cdot \color{blue}{\left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)} \]

   if 4.3999999999999999e153 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 72.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 72.1%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + t \cdot \left(y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + \color{blue}{y}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(\left(t \cdot {y}^{2}\right) \cdot \frac{1}{2} + y\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(t \cdot \left({y}^{2} \cdot \frac{1}{2}\right) + y\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(t \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + y\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t \cdot \left({y}^{2} \cdot \frac{1}{2}\right) + y\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(\left(t \cdot {y}^{2}\right) \cdot \frac{1}{2} + y\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + y\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + \left(t \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + t \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + t \cdot \left(\frac{1}{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 75.4%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 75.4%

       \[\leadsto \frac{x}{\color{blue}{1 + t \cdot \left(y + t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+140}:\\ \;\;\;\;x + t \cdot \left(0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - y \cdot x\right)\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + t \cdot \left(y + t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(a \cdot a\right)\\ t_2 := a \cdot \left(b \cdot b\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+140}:\\ \;\;\;\;x + t \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot \left(y \cdot x\right)\right) - x\right)\right)\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot t\_2}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(0.5 \cdot t\_2 - b\right)\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot t\_1\right)}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \left(t\_1 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + t \cdot \left(y + t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.5 (* a a))) (t_2 (* a (* b b))))
   (if (<= y -1.05e+140)
     (+ x (* t (* y (- (* 0.5 (* t (* y x))) x))))
     (if (<= y -6.8e-132)
       (* 2.0 (/ x (* a t_2)))
       (if (<= y 1.65e-298)
         (* x (+ 1.0 (* a (- (* 0.5 t_2) b))))
         (if (<= y 1.1e+45)
           (/ x (+ 1.0 (* b (+ a (* b t_1)))))
           (if (<= y 7.2e+153)
             (* x (* t_1 (* b b)))
             (/ x (+ 1.0 (* t (+ y (* t (* 0.5 (* y y))))))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (a * a);
	double t_2 = a * (b * b);
	double tmp;
	if (y <= -1.05e+140) {
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)));
	} else if (y <= -6.8e-132) {
		tmp = 2.0 * (x / (a * t_2));
	} else if (y <= 1.65e-298) {
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)));
	} else if (y <= 1.1e+45) {
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	} else if (y <= 7.2e+153) {
		tmp = x * (t_1 * (b * b));
	} else {
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.5d0 * (a * a)
    t_2 = a * (b * b)
    if (y <= (-1.05d+140)) then
        tmp = x + (t * (y * ((0.5d0 * (t * (y * x))) - x)))
    else if (y <= (-6.8d-132)) then
        tmp = 2.0d0 * (x / (a * t_2))
    else if (y <= 1.65d-298) then
        tmp = x * (1.0d0 + (a * ((0.5d0 * t_2) - b)))
    else if (y <= 1.1d+45) then
        tmp = x / (1.0d0 + (b * (a + (b * t_1))))
    else if (y <= 7.2d+153) then
        tmp = x * (t_1 * (b * b))
    else
        tmp = x / (1.0d0 + (t * (y + (t * (0.5d0 * (y * y))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (a * a);
	double t_2 = a * (b * b);
	double tmp;
	if (y <= -1.05e+140) {
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)));
	} else if (y <= -6.8e-132) {
		tmp = 2.0 * (x / (a * t_2));
	} else if (y <= 1.65e-298) {
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)));
	} else if (y <= 1.1e+45) {
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	} else if (y <= 7.2e+153) {
		tmp = x * (t_1 * (b * b));
	} else {
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.5 * (a * a)
	t_2 = a * (b * b)
	tmp = 0
	if y <= -1.05e+140:
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)))
	elif y <= -6.8e-132:
		tmp = 2.0 * (x / (a * t_2))
	elif y <= 1.65e-298:
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)))
	elif y <= 1.1e+45:
		tmp = x / (1.0 + (b * (a + (b * t_1))))
	elif y <= 7.2e+153:
		tmp = x * (t_1 * (b * b))
	else:
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.5 * Float64(a * a))
	t_2 = Float64(a * Float64(b * b))
	tmp = 0.0
	if (y <= -1.05e+140)
		tmp = Float64(x + Float64(t * Float64(y * Float64(Float64(0.5 * Float64(t * Float64(y * x))) - x))));
	elseif (y <= -6.8e-132)
		tmp = Float64(2.0 * Float64(x / Float64(a * t_2)));
	elseif (y <= 1.65e-298)
		tmp = Float64(x * Float64(1.0 + Float64(a * Float64(Float64(0.5 * t_2) - b))));
	elseif (y <= 1.1e+45)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(a + Float64(b * t_1)))));
	elseif (y <= 7.2e+153)
		tmp = Float64(x * Float64(t_1 * Float64(b * b)));
	else
		tmp = Float64(x / Float64(1.0 + Float64(t * Float64(y + Float64(t * Float64(0.5 * Float64(y * y)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.5 * (a * a);
	t_2 = a * (b * b);
	tmp = 0.0;
	if (y <= -1.05e+140)
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)));
	elseif (y <= -6.8e-132)
		tmp = 2.0 * (x / (a * t_2));
	elseif (y <= 1.65e-298)
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)));
	elseif (y <= 1.1e+45)
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	elseif (y <= 7.2e+153)
		tmp = x * (t_1 * (b * b));
	else
		tmp = x / (1.0 + (t * (y + (t * (0.5 * (y * y))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+140], N[(x + N[(t * N[(y * N[(N[(0.5 * N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -6.8e-132], N[(2.0 * N[(x / N[(a * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e-298], N[(x * N[(1.0 + N[(a * N[(N[(0.5 * t$95$2), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+45], N[(x / N[(1.0 + N[(b * N[(a + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+153], N[(x * N[(t$95$1 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(t * N[(y + N[(t * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y < -1.0500000000000001e140

   1. Initial program 91.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 91.9%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 71.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 71.0%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(t \cdot \left(-1 \cdot \left(t \cdot \left(-1 \cdot \left(y \cdot \left(-1 \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) + \left(\frac{-1}{2} \cdot \left(x \cdot {y}^{3}\right) + \frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right)\right) - \left(-1 \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) - x \cdot y\right)} \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{x + t \cdot \left(t \cdot \left(-\left(t \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot y\right) \cdot y\right)\right) + \left(\left(\left(x \cdot y\right) \cdot y\right) \cdot y\right) \cdot -0.3333333333333333\right) + \left(\left(x \cdot y\right) \cdot y\right) \cdot -0.5\right)\right) - x \cdot y\right)} \]

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right) - x\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right) - x\right)}\right)\right)\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right)\right), \color{blue}{x}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot \left(x \cdot y\right)\right)\right), x\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \left(x \cdot y\right)\right)\right), x\right)\right)\right)\right) \]

       5. *-lowering-*.f64 55.2%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, y\right)\right)\right), x\right)\right)\right)\right) \]

   13. Simplified 55.2%

       \[\leadsto x + t \cdot \color{blue}{\left(y \cdot \left(0.5 \cdot \left(t \cdot \left(x \cdot y\right)\right) - x\right)\right)} \]

   if -1.0500000000000001e140 < y < -6.79999999999999965e-132

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 57.0%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 34.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 60.1%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 60.1%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if -6.79999999999999965e-132 < y < 1.6500000000000001e-298

   1. Initial program 97.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 94.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 94.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 70.2%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   if 1.6500000000000001e-298 < y < 1.1e45

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 72.4%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + b \cdot \left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{b}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), b\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), b\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), b\right)\right)\right)\right)\right) \]

   10. Simplified 56.9%

       \[\leadsto \frac{x}{\color{blue}{1 + b \cdot \left(a + \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot b\right)}} \]

   if 1.1e45 < y < 7.2000000000000001e153

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 44.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 44.0%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 25.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 25.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left({b}^{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left({b}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 51.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 51.1%

       \[\leadsto x \cdot \color{blue}{\left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)} \]

   if 7.2000000000000001e153 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 72.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 72.1%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + t \cdot \left(y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + \color{blue}{y}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(\left(t \cdot {y}^{2}\right) \cdot \frac{1}{2} + y\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(t \cdot \left({y}^{2} \cdot \frac{1}{2}\right) + y\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(t \cdot \left(t \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + y\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t \cdot \left({y}^{2} \cdot \frac{1}{2}\right) + y\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(\left(t \cdot {y}^{2}\right) \cdot \frac{1}{2} + y\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + y\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + \left(t \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + t \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(y + t \cdot \left(\frac{1}{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 75.4%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 75.4%

       \[\leadsto \frac{x}{\color{blue}{1 + t \cdot \left(y + t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+140}:\\ \;\;\;\;x + t \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot \left(y \cdot x\right)\right) - x\right)\right)\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + t \cdot \left(y + t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(a \cdot a\right)\\ t_2 := a \cdot \left(b \cdot b\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+140}:\\ \;\;\;\;x + t \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot \left(y \cdot x\right)\right) - x\right)\right)\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot t\_2}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(0.5 \cdot t\_2 - b\right)\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot t\_1\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(t\_1 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.5 (* a a))) (t_2 (* a (* b b))))
   (if (<= y -1.05e+140)
     (+ x (* t (* y (- (* 0.5 (* t (* y x))) x))))
     (if (<= y -7e-132)
       (* 2.0 (/ x (* a t_2)))
       (if (<= y 3.8e-298)
         (* x (+ 1.0 (* a (- (* 0.5 t_2) b))))
         (if (<= y 2.65e+45)
           (/ x (+ 1.0 (* b (+ a (* b t_1)))))
           (if (<= y 1.2e+134)
             (* x (* t_1 (* b b)))
             (* 2.0 (/ (/ (/ x (* a a)) b) b)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (a * a);
	double t_2 = a * (b * b);
	double tmp;
	if (y <= -1.05e+140) {
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)));
	} else if (y <= -7e-132) {
		tmp = 2.0 * (x / (a * t_2));
	} else if (y <= 3.8e-298) {
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)));
	} else if (y <= 2.65e+45) {
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	} else if (y <= 1.2e+134) {
		tmp = x * (t_1 * (b * b));
	} else {
		tmp = 2.0 * (((x / (a * a)) / b) / b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.5d0 * (a * a)
    t_2 = a * (b * b)
    if (y <= (-1.05d+140)) then
        tmp = x + (t * (y * ((0.5d0 * (t * (y * x))) - x)))
    else if (y <= (-7d-132)) then
        tmp = 2.0d0 * (x / (a * t_2))
    else if (y <= 3.8d-298) then
        tmp = x * (1.0d0 + (a * ((0.5d0 * t_2) - b)))
    else if (y <= 2.65d+45) then
        tmp = x / (1.0d0 + (b * (a + (b * t_1))))
    else if (y <= 1.2d+134) then
        tmp = x * (t_1 * (b * b))
    else
        tmp = 2.0d0 * (((x / (a * a)) / b) / b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (a * a);
	double t_2 = a * (b * b);
	double tmp;
	if (y <= -1.05e+140) {
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)));
	} else if (y <= -7e-132) {
		tmp = 2.0 * (x / (a * t_2));
	} else if (y <= 3.8e-298) {
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)));
	} else if (y <= 2.65e+45) {
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	} else if (y <= 1.2e+134) {
		tmp = x * (t_1 * (b * b));
	} else {
		tmp = 2.0 * (((x / (a * a)) / b) / b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.5 * (a * a)
	t_2 = a * (b * b)
	tmp = 0
	if y <= -1.05e+140:
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)))
	elif y <= -7e-132:
		tmp = 2.0 * (x / (a * t_2))
	elif y <= 3.8e-298:
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)))
	elif y <= 2.65e+45:
		tmp = x / (1.0 + (b * (a + (b * t_1))))
	elif y <= 1.2e+134:
		tmp = x * (t_1 * (b * b))
	else:
		tmp = 2.0 * (((x / (a * a)) / b) / b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.5 * Float64(a * a))
	t_2 = Float64(a * Float64(b * b))
	tmp = 0.0
	if (y <= -1.05e+140)
		tmp = Float64(x + Float64(t * Float64(y * Float64(Float64(0.5 * Float64(t * Float64(y * x))) - x))));
	elseif (y <= -7e-132)
		tmp = Float64(2.0 * Float64(x / Float64(a * t_2)));
	elseif (y <= 3.8e-298)
		tmp = Float64(x * Float64(1.0 + Float64(a * Float64(Float64(0.5 * t_2) - b))));
	elseif (y <= 2.65e+45)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(a + Float64(b * t_1)))));
	elseif (y <= 1.2e+134)
		tmp = Float64(x * Float64(t_1 * Float64(b * b)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(x / Float64(a * a)) / b) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.5 * (a * a);
	t_2 = a * (b * b);
	tmp = 0.0;
	if (y <= -1.05e+140)
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)));
	elseif (y <= -7e-132)
		tmp = 2.0 * (x / (a * t_2));
	elseif (y <= 3.8e-298)
		tmp = x * (1.0 + (a * ((0.5 * t_2) - b)));
	elseif (y <= 2.65e+45)
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	elseif (y <= 1.2e+134)
		tmp = x * (t_1 * (b * b));
	else
		tmp = 2.0 * (((x / (a * a)) / b) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+140], N[(x + N[(t * N[(y * N[(N[(0.5 * N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -7e-132], N[(2.0 * N[(x / N[(a * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-298], N[(x * N[(1.0 + N[(a * N[(N[(0.5 * t$95$2), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.65e+45], N[(x / N[(1.0 + N[(b * N[(a + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+134], N[(x * N[(t$95$1 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y < -1.0500000000000001e140

   1. Initial program 91.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 91.9%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 71.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 71.0%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(t \cdot \left(-1 \cdot \left(t \cdot \left(-1 \cdot \left(y \cdot \left(-1 \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) + \left(\frac{-1}{2} \cdot \left(x \cdot {y}^{3}\right) + \frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right)\right) - \left(-1 \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) - x \cdot y\right)} \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{x + t \cdot \left(t \cdot \left(-\left(t \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot y\right) \cdot y\right)\right) + \left(\left(\left(x \cdot y\right) \cdot y\right) \cdot y\right) \cdot -0.3333333333333333\right) + \left(\left(x \cdot y\right) \cdot y\right) \cdot -0.5\right)\right) - x \cdot y\right)} \]

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right) - x\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right) - x\right)}\right)\right)\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right)\right), \color{blue}{x}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot \left(x \cdot y\right)\right)\right), x\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \left(x \cdot y\right)\right)\right), x\right)\right)\right)\right) \]

       5. *-lowering-*.f64 55.2%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, y\right)\right)\right), x\right)\right)\right)\right) \]

   13. Simplified 55.2%

       \[\leadsto x + t \cdot \color{blue}{\left(y \cdot \left(0.5 \cdot \left(t \cdot \left(x \cdot y\right)\right) - x\right)\right)} \]

   if -1.0500000000000001e140 < y < -6.9999999999999999e-132

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 57.0%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 34.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 60.1%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 60.1%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if -6.9999999999999999e-132 < y < 3.8e-298

   1. Initial program 97.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 94.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 94.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 70.2%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   if 3.8e-298 < y < 2.64999999999999996e45

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 72.4%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + b \cdot \left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{b}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), b\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), b\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), b\right)\right)\right)\right)\right) \]

   10. Simplified 56.9%

       \[\leadsto \frac{x}{\color{blue}{1 + b \cdot \left(a + \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot b\right)}} \]

   if 2.64999999999999996e45 < y < 1.20000000000000003e134

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 48.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 48.6%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 31.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 31.1%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left({b}^{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left({b}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 58.2%

       \[\leadsto x \cdot \color{blue}{\left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)} \]

   if 1.20000000000000003e134 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 32.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 32.7%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 30.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 30.2%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in b around -inf

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} - \color{blue}{\frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left(\frac{\color{blue}{-1 \cdot a - \frac{1}{b}}}{b}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a - \frac{1}{b}\right), \color{blue}{b}\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right), b\right)\right)\right)\right) \]

       13. distribute-neg-frac N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{\mathsf{neg}\left(1\right)}{b}\right), b\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{-1}{b}\right), b\right)\right)\right)\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-1 \cdot a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       16. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(a\right)\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       17. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(0 - a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       18. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       19. /-lowering-/.f64 26.0%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \mathsf{/.f64}\left(-1, b\right)\right), b\right)\right)\right)\right) \]

   13. Simplified 26.0%

       \[\leadsto \frac{x}{\color{blue}{\left(b \cdot b\right) \cdot \left(0.5 \cdot \left(a \cdot a\right) - \frac{\left(0 - a\right) + \frac{-1}{b}}{b}\right)}} \]

   14. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   15. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{\color{blue}{{b}^{2}}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{b \cdot \color{blue}{b}}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{\frac{x}{{a}^{2}}}{b}}{\color{blue}{b}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\frac{x}{{a}^{2}}}{b}\right), \color{blue}{b}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{{a}^{2}}\right), b\right), b\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left({a}^{2}\right)\right), b\right), b\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(a \cdot a\right)\right), b\right), b\right)\right) \]

       9. *-lowering-*.f64 49.5%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(a, a\right)\right), b\right), b\right)\right) \]

   16. Simplified 49.5%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+140}:\\ \;\;\;\;x + t \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot \left(y \cdot x\right)\right) - x\right)\right)\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot b\right)\\ t_2 := 0.5 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+139}:\\ \;\;\;\;x + t \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot \left(y \cdot x\right)\right) - x\right)\right)\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot t\_1}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(0.5 \cdot t\_1 - b\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(b \cdot t\_2\right)}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(t\_2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (* b b))) (t_2 (* 0.5 (* a a))))
   (if (<= y -5.2e+139)
     (+ x (* t (* y (- (* 0.5 (* t (* y x))) x))))
     (if (<= y -5.6e-132)
       (* 2.0 (/ x (* a t_1)))
       (if (<= y 5.4e-298)
         (* x (+ 1.0 (* a (- (* 0.5 t_1) b))))
         (if (<= y 6.2e+44)
           (/ x (+ 1.0 (* b (* b t_2))))
           (if (<= y 1.15e+134)
             (* x (* t_2 (* b b)))
             (* 2.0 (/ (/ (/ x (* a a)) b) b)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (b * b);
	double t_2 = 0.5 * (a * a);
	double tmp;
	if (y <= -5.2e+139) {
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)));
	} else if (y <= -5.6e-132) {
		tmp = 2.0 * (x / (a * t_1));
	} else if (y <= 5.4e-298) {
		tmp = x * (1.0 + (a * ((0.5 * t_1) - b)));
	} else if (y <= 6.2e+44) {
		tmp = x / (1.0 + (b * (b * t_2)));
	} else if (y <= 1.15e+134) {
		tmp = x * (t_2 * (b * b));
	} else {
		tmp = 2.0 * (((x / (a * a)) / b) / b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a * (b * b)
    t_2 = 0.5d0 * (a * a)
    if (y <= (-5.2d+139)) then
        tmp = x + (t * (y * ((0.5d0 * (t * (y * x))) - x)))
    else if (y <= (-5.6d-132)) then
        tmp = 2.0d0 * (x / (a * t_1))
    else if (y <= 5.4d-298) then
        tmp = x * (1.0d0 + (a * ((0.5d0 * t_1) - b)))
    else if (y <= 6.2d+44) then
        tmp = x / (1.0d0 + (b * (b * t_2)))
    else if (y <= 1.15d+134) then
        tmp = x * (t_2 * (b * b))
    else
        tmp = 2.0d0 * (((x / (a * a)) / b) / b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (b * b);
	double t_2 = 0.5 * (a * a);
	double tmp;
	if (y <= -5.2e+139) {
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)));
	} else if (y <= -5.6e-132) {
		tmp = 2.0 * (x / (a * t_1));
	} else if (y <= 5.4e-298) {
		tmp = x * (1.0 + (a * ((0.5 * t_1) - b)));
	} else if (y <= 6.2e+44) {
		tmp = x / (1.0 + (b * (b * t_2)));
	} else if (y <= 1.15e+134) {
		tmp = x * (t_2 * (b * b));
	} else {
		tmp = 2.0 * (((x / (a * a)) / b) / b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (b * b)
	t_2 = 0.5 * (a * a)
	tmp = 0
	if y <= -5.2e+139:
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)))
	elif y <= -5.6e-132:
		tmp = 2.0 * (x / (a * t_1))
	elif y <= 5.4e-298:
		tmp = x * (1.0 + (a * ((0.5 * t_1) - b)))
	elif y <= 6.2e+44:
		tmp = x / (1.0 + (b * (b * t_2)))
	elif y <= 1.15e+134:
		tmp = x * (t_2 * (b * b))
	else:
		tmp = 2.0 * (((x / (a * a)) / b) / b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(b * b))
	t_2 = Float64(0.5 * Float64(a * a))
	tmp = 0.0
	if (y <= -5.2e+139)
		tmp = Float64(x + Float64(t * Float64(y * Float64(Float64(0.5 * Float64(t * Float64(y * x))) - x))));
	elseif (y <= -5.6e-132)
		tmp = Float64(2.0 * Float64(x / Float64(a * t_1)));
	elseif (y <= 5.4e-298)
		tmp = Float64(x * Float64(1.0 + Float64(a * Float64(Float64(0.5 * t_1) - b))));
	elseif (y <= 6.2e+44)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(b * t_2))));
	elseif (y <= 1.15e+134)
		tmp = Float64(x * Float64(t_2 * Float64(b * b)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(x / Float64(a * a)) / b) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (b * b);
	t_2 = 0.5 * (a * a);
	tmp = 0.0;
	if (y <= -5.2e+139)
		tmp = x + (t * (y * ((0.5 * (t * (y * x))) - x)));
	elseif (y <= -5.6e-132)
		tmp = 2.0 * (x / (a * t_1));
	elseif (y <= 5.4e-298)
		tmp = x * (1.0 + (a * ((0.5 * t_1) - b)));
	elseif (y <= 6.2e+44)
		tmp = x / (1.0 + (b * (b * t_2)));
	elseif (y <= 1.15e+134)
		tmp = x * (t_2 * (b * b));
	else
		tmp = 2.0 * (((x / (a * a)) / b) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.2e+139], N[(x + N[(t * N[(y * N[(N[(0.5 * N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.6e-132], N[(2.0 * N[(x / N[(a * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e-298], N[(x * N[(1.0 + N[(a * N[(N[(0.5 * t$95$1), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+44], N[(x / N[(1.0 + N[(b * N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+134], N[(x * N[(t$95$2 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y < -5.20000000000000044e139

   1. Initial program 91.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 91.9%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 71.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 71.0%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + t \cdot \left(t \cdot \left(-1 \cdot \left(t \cdot \left(-1 \cdot \left(y \cdot \left(-1 \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) + \left(\frac{-1}{2} \cdot \left(x \cdot {y}^{3}\right) + \frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right)\right) - \left(-1 \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) - x \cdot y\right)} \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{x + t \cdot \left(t \cdot \left(-\left(t \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot y\right) \cdot y\right)\right) + \left(\left(\left(x \cdot y\right) \cdot y\right) \cdot y\right) \cdot -0.3333333333333333\right) + \left(\left(x \cdot y\right) \cdot y\right) \cdot -0.5\right)\right) - x \cdot y\right)} \]

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right) - x\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right) - x\right)}\right)\right)\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right)\right), \color{blue}{x}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot \left(x \cdot y\right)\right)\right), x\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \left(x \cdot y\right)\right)\right), x\right)\right)\right)\right) \]

       5. *-lowering-*.f64 55.2%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, y\right)\right)\right), x\right)\right)\right)\right) \]

   13. Simplified 55.2%

       \[\leadsto x + t \cdot \color{blue}{\left(y \cdot \left(0.5 \cdot \left(t \cdot \left(x \cdot y\right)\right) - x\right)\right)} \]

   if -5.20000000000000044e139 < y < -5.60000000000000005e-132

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 57.0%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 34.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 60.1%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 60.1%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if -5.60000000000000005e-132 < y < 5.4000000000000002e-298

   1. Initial program 97.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 94.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 94.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 70.2%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   if 5.4000000000000002e-298 < y < 6.19999999999999991e44

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 72.4%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 51.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 51.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot b\right) \cdot \color{blue}{b}\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) \cdot b\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}\right) \cdot b\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\left(b \cdot {a}^{2}\right) \cdot \frac{1}{2}\right) \cdot b\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)\right) \cdot b\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(b \cdot \left(\frac{1}{2} \cdot {a}^{2}\right)\right) \cdot b\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot b\right)}\right)\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(\left({a}^{2} \cdot b\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(\left(b \cdot {a}^{2}\right) \cdot \frac{1}{2}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left({a}^{2} \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({a}^{2}\right)}\right)\right)\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right)\right) \]

       19. *-lowering-*.f64 56.5%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 56.5%

       \[\leadsto \frac{x}{1 + \color{blue}{b \cdot \left(b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}} \]

   if 6.19999999999999991e44 < y < 1.1499999999999999e134

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 48.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 48.6%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 31.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 31.1%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left({b}^{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left({b}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 58.2%

       \[\leadsto x \cdot \color{blue}{\left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)} \]

   if 1.1499999999999999e134 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 32.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 32.7%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 30.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 30.2%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in b around -inf

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} - \color{blue}{\frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left(\frac{\color{blue}{-1 \cdot a - \frac{1}{b}}}{b}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a - \frac{1}{b}\right), \color{blue}{b}\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right), b\right)\right)\right)\right) \]

       13. distribute-neg-frac N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{\mathsf{neg}\left(1\right)}{b}\right), b\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{-1}{b}\right), b\right)\right)\right)\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-1 \cdot a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       16. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(a\right)\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       17. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(0 - a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       18. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       19. /-lowering-/.f64 26.0%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \mathsf{/.f64}\left(-1, b\right)\right), b\right)\right)\right)\right) \]

   13. Simplified 26.0%

       \[\leadsto \frac{x}{\color{blue}{\left(b \cdot b\right) \cdot \left(0.5 \cdot \left(a \cdot a\right) - \frac{\left(0 - a\right) + \frac{-1}{b}}{b}\right)}} \]

   14. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   15. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{\color{blue}{{b}^{2}}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{b \cdot \color{blue}{b}}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{\frac{x}{{a}^{2}}}{b}}{\color{blue}{b}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\frac{x}{{a}^{2}}}{b}\right), \color{blue}{b}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{{a}^{2}}\right), b\right), b\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left({a}^{2}\right)\right), b\right), b\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(a \cdot a\right)\right), b\right), b\right)\right) \]

       9. *-lowering-*.f64 49.5%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(a, a\right)\right), b\right), b\right)\right) \]

   16. Simplified 49.5%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+139}:\\ \;\;\;\;x + t \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot \left(y \cdot x\right)\right) - x\right)\right)\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot b\right)\\ t_2 := 2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ t_3 := 0.5 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{+139}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot t\_1}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(0.5 \cdot t\_1 - b\right)\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(b \cdot t\_3\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(t\_3 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (* b b)))
        (t_2 (* 2.0 (/ (/ (/ x (* a a)) b) b)))
        (t_3 (* 0.5 (* a a))))
   (if (<= y -7e+139)
     t_2
     (if (<= y -4.8e-132)
       (* 2.0 (/ x (* a t_1)))
       (if (<= y 2.6e-298)
         (* x (+ 1.0 (* a (- (* 0.5 t_1) b))))
         (if (<= y 1.05e+45)
           (/ x (+ 1.0 (* b (* b t_3))))
           (if (<= y 1.2e+134) (* x (* t_3 (* b b))) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (b * b);
	double t_2 = 2.0 * (((x / (a * a)) / b) / b);
	double t_3 = 0.5 * (a * a);
	double tmp;
	if (y <= -7e+139) {
		tmp = t_2;
	} else if (y <= -4.8e-132) {
		tmp = 2.0 * (x / (a * t_1));
	} else if (y <= 2.6e-298) {
		tmp = x * (1.0 + (a * ((0.5 * t_1) - b)));
	} else if (y <= 1.05e+45) {
		tmp = x / (1.0 + (b * (b * t_3)));
	} else if (y <= 1.2e+134) {
		tmp = x * (t_3 * (b * b));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = a * (b * b)
    t_2 = 2.0d0 * (((x / (a * a)) / b) / b)
    t_3 = 0.5d0 * (a * a)
    if (y <= (-7d+139)) then
        tmp = t_2
    else if (y <= (-4.8d-132)) then
        tmp = 2.0d0 * (x / (a * t_1))
    else if (y <= 2.6d-298) then
        tmp = x * (1.0d0 + (a * ((0.5d0 * t_1) - b)))
    else if (y <= 1.05d+45) then
        tmp = x / (1.0d0 + (b * (b * t_3)))
    else if (y <= 1.2d+134) then
        tmp = x * (t_3 * (b * b))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (b * b);
	double t_2 = 2.0 * (((x / (a * a)) / b) / b);
	double t_3 = 0.5 * (a * a);
	double tmp;
	if (y <= -7e+139) {
		tmp = t_2;
	} else if (y <= -4.8e-132) {
		tmp = 2.0 * (x / (a * t_1));
	} else if (y <= 2.6e-298) {
		tmp = x * (1.0 + (a * ((0.5 * t_1) - b)));
	} else if (y <= 1.05e+45) {
		tmp = x / (1.0 + (b * (b * t_3)));
	} else if (y <= 1.2e+134) {
		tmp = x * (t_3 * (b * b));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (b * b)
	t_2 = 2.0 * (((x / (a * a)) / b) / b)
	t_3 = 0.5 * (a * a)
	tmp = 0
	if y <= -7e+139:
		tmp = t_2
	elif y <= -4.8e-132:
		tmp = 2.0 * (x / (a * t_1))
	elif y <= 2.6e-298:
		tmp = x * (1.0 + (a * ((0.5 * t_1) - b)))
	elif y <= 1.05e+45:
		tmp = x / (1.0 + (b * (b * t_3)))
	elif y <= 1.2e+134:
		tmp = x * (t_3 * (b * b))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(b * b))
	t_2 = Float64(2.0 * Float64(Float64(Float64(x / Float64(a * a)) / b) / b))
	t_3 = Float64(0.5 * Float64(a * a))
	tmp = 0.0
	if (y <= -7e+139)
		tmp = t_2;
	elseif (y <= -4.8e-132)
		tmp = Float64(2.0 * Float64(x / Float64(a * t_1)));
	elseif (y <= 2.6e-298)
		tmp = Float64(x * Float64(1.0 + Float64(a * Float64(Float64(0.5 * t_1) - b))));
	elseif (y <= 1.05e+45)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(b * t_3))));
	elseif (y <= 1.2e+134)
		tmp = Float64(x * Float64(t_3 * Float64(b * b)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (b * b);
	t_2 = 2.0 * (((x / (a * a)) / b) / b);
	t_3 = 0.5 * (a * a);
	tmp = 0.0;
	if (y <= -7e+139)
		tmp = t_2;
	elseif (y <= -4.8e-132)
		tmp = 2.0 * (x / (a * t_1));
	elseif (y <= 2.6e-298)
		tmp = x * (1.0 + (a * ((0.5 * t_1) - b)));
	elseif (y <= 1.05e+45)
		tmp = x / (1.0 + (b * (b * t_3)));
	elseif (y <= 1.2e+134)
		tmp = x * (t_3 * (b * b));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e+139], t$95$2, If[LessEqual[y, -4.8e-132], N[(2.0 * N[(x / N[(a * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e-298], N[(x * N[(1.0 + N[(a * N[(N[(0.5 * t$95$1), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+45], N[(x / N[(1.0 + N[(b * N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+134], N[(x * N[(t$95$3 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -6.99999999999999957e139 or 1.20000000000000003e134 < y

   1. Initial program 95.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 33.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 33.7%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 21.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 21.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in b around -inf

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} - \color{blue}{\frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left(\frac{\color{blue}{-1 \cdot a - \frac{1}{b}}}{b}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a - \frac{1}{b}\right), \color{blue}{b}\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right), b\right)\right)\right)\right) \]

       13. distribute-neg-frac N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{\mathsf{neg}\left(1\right)}{b}\right), b\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{-1}{b}\right), b\right)\right)\right)\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-1 \cdot a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       16. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(a\right)\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       17. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(0 - a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       18. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       19. /-lowering-/.f64 19.2%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \mathsf{/.f64}\left(-1, b\right)\right), b\right)\right)\right)\right) \]

   13. Simplified 19.2%

       \[\leadsto \frac{x}{\color{blue}{\left(b \cdot b\right) \cdot \left(0.5 \cdot \left(a \cdot a\right) - \frac{\left(0 - a\right) + \frac{-1}{b}}{b}\right)}} \]

   14. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   15. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{\color{blue}{{b}^{2}}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{b \cdot \color{blue}{b}}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{\frac{x}{{a}^{2}}}{b}}{\color{blue}{b}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\frac{x}{{a}^{2}}}{b}\right), \color{blue}{b}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{{a}^{2}}\right), b\right), b\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left({a}^{2}\right)\right), b\right), b\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(a \cdot a\right)\right), b\right), b\right)\right) \]

       9. *-lowering-*.f64 50.9%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(a, a\right)\right), b\right), b\right)\right) \]

   16. Simplified 50.9%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}} \]

   if -6.99999999999999957e139 < y < -4.80000000000000031e-132

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 57.0%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 34.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 60.1%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 60.1%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if -4.80000000000000031e-132 < y < 2.5999999999999999e-298

   1. Initial program 97.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 94.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 94.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 70.2%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   if 2.5999999999999999e-298 < y < 1.04999999999999997e45

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 72.4%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 51.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 51.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot b\right) \cdot \color{blue}{b}\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) \cdot b\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}\right) \cdot b\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\left(b \cdot {a}^{2}\right) \cdot \frac{1}{2}\right) \cdot b\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)\right) \cdot b\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(b \cdot \left(\frac{1}{2} \cdot {a}^{2}\right)\right) \cdot b\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot b\right)}\right)\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(\left({a}^{2} \cdot b\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(\left(b \cdot {a}^{2}\right) \cdot \frac{1}{2}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left({a}^{2} \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({a}^{2}\right)}\right)\right)\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right)\right) \]

       19. *-lowering-*.f64 56.5%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 56.5%

       \[\leadsto \frac{x}{1 + \color{blue}{b \cdot \left(b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}} \]

   if 1.04999999999999997e45 < y < 1.20000000000000003e134

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 48.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 48.6%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 31.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 31.1%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left({b}^{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left({b}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 58.2%

       \[\leadsto x \cdot \color{blue}{\left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.9% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(a \cdot a\right)\\ t_2 := 2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(b \cdot t\_1\right)}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(t\_1 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.5 (* a a))) (t_2 (* 2.0 (/ (/ (/ x (* a a)) b) b))))
   (if (<= y -4.6e+139)
     t_2
     (if (<= y -1.08e-94)
       (* 2.0 (/ x (* a (* a (* b b)))))
       (if (<= y 5.5e+44)
         (/ x (+ 1.0 (* b (* b t_1))))
         (if (<= y 1.15e+134) (* x (* t_1 (* b b))) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (a * a);
	double t_2 = 2.0 * (((x / (a * a)) / b) / b);
	double tmp;
	if (y <= -4.6e+139) {
		tmp = t_2;
	} else if (y <= -1.08e-94) {
		tmp = 2.0 * (x / (a * (a * (b * b))));
	} else if (y <= 5.5e+44) {
		tmp = x / (1.0 + (b * (b * t_1)));
	} else if (y <= 1.15e+134) {
		tmp = x * (t_1 * (b * b));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.5d0 * (a * a)
    t_2 = 2.0d0 * (((x / (a * a)) / b) / b)
    if (y <= (-4.6d+139)) then
        tmp = t_2
    else if (y <= (-1.08d-94)) then
        tmp = 2.0d0 * (x / (a * (a * (b * b))))
    else if (y <= 5.5d+44) then
        tmp = x / (1.0d0 + (b * (b * t_1)))
    else if (y <= 1.15d+134) then
        tmp = x * (t_1 * (b * b))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (a * a);
	double t_2 = 2.0 * (((x / (a * a)) / b) / b);
	double tmp;
	if (y <= -4.6e+139) {
		tmp = t_2;
	} else if (y <= -1.08e-94) {
		tmp = 2.0 * (x / (a * (a * (b * b))));
	} else if (y <= 5.5e+44) {
		tmp = x / (1.0 + (b * (b * t_1)));
	} else if (y <= 1.15e+134) {
		tmp = x * (t_1 * (b * b));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.5 * (a * a)
	t_2 = 2.0 * (((x / (a * a)) / b) / b)
	tmp = 0
	if y <= -4.6e+139:
		tmp = t_2
	elif y <= -1.08e-94:
		tmp = 2.0 * (x / (a * (a * (b * b))))
	elif y <= 5.5e+44:
		tmp = x / (1.0 + (b * (b * t_1)))
	elif y <= 1.15e+134:
		tmp = x * (t_1 * (b * b))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.5 * Float64(a * a))
	t_2 = Float64(2.0 * Float64(Float64(Float64(x / Float64(a * a)) / b) / b))
	tmp = 0.0
	if (y <= -4.6e+139)
		tmp = t_2;
	elseif (y <= -1.08e-94)
		tmp = Float64(2.0 * Float64(x / Float64(a * Float64(a * Float64(b * b)))));
	elseif (y <= 5.5e+44)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(b * t_1))));
	elseif (y <= 1.15e+134)
		tmp = Float64(x * Float64(t_1 * Float64(b * b)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.5 * (a * a);
	t_2 = 2.0 * (((x / (a * a)) / b) / b);
	tmp = 0.0;
	if (y <= -4.6e+139)
		tmp = t_2;
	elseif (y <= -1.08e-94)
		tmp = 2.0 * (x / (a * (a * (b * b))));
	elseif (y <= 5.5e+44)
		tmp = x / (1.0 + (b * (b * t_1)));
	elseif (y <= 1.15e+134)
		tmp = x * (t_1 * (b * b));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e+139], t$95$2, If[LessEqual[y, -1.08e-94], N[(2.0 * N[(x / N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e+44], N[(x / N[(1.0 + N[(b * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+134], N[(x * N[(t$95$1 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.6e139 or 1.1499999999999999e134 < y

   1. Initial program 95.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 33.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 33.7%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 21.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 21.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in b around -inf

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} - \color{blue}{\frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left(\frac{\color{blue}{-1 \cdot a - \frac{1}{b}}}{b}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a - \frac{1}{b}\right), \color{blue}{b}\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right), b\right)\right)\right)\right) \]

       13. distribute-neg-frac N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{\mathsf{neg}\left(1\right)}{b}\right), b\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{-1}{b}\right), b\right)\right)\right)\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-1 \cdot a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       16. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(a\right)\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       17. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(0 - a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       18. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       19. /-lowering-/.f64 19.2%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \mathsf{/.f64}\left(-1, b\right)\right), b\right)\right)\right)\right) \]

   13. Simplified 19.2%

       \[\leadsto \frac{x}{\color{blue}{\left(b \cdot b\right) \cdot \left(0.5 \cdot \left(a \cdot a\right) - \frac{\left(0 - a\right) + \frac{-1}{b}}{b}\right)}} \]

   14. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   15. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{\color{blue}{{b}^{2}}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{b \cdot \color{blue}{b}}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{\frac{x}{{a}^{2}}}{b}}{\color{blue}{b}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\frac{x}{{a}^{2}}}{b}\right), \color{blue}{b}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{{a}^{2}}\right), b\right), b\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left({a}^{2}\right)\right), b\right), b\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(a \cdot a\right)\right), b\right), b\right)\right) \]

       9. *-lowering-*.f64 50.9%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(a, a\right)\right), b\right), b\right)\right) \]

   16. Simplified 50.9%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}} \]

   if -4.6e139 < y < -1.08e-94

   1. Initial program 97.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 97.5%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 53.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 53.8%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 32.3%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 32.3%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 59.6%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 59.6%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if -1.08e-94 < y < 5.5000000000000001e44

   1. Initial program 95.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 80.2%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 52.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 52.8%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot b\right) \cdot \color{blue}{b}\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) \cdot b\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\left({a}^{2} \cdot b\right) \cdot \frac{1}{2}\right) \cdot b\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\left(b \cdot {a}^{2}\right) \cdot \frac{1}{2}\right) \cdot b\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(b \cdot \left({a}^{2} \cdot \frac{1}{2}\right)\right) \cdot b\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(b \cdot \left(\frac{1}{2} \cdot {a}^{2}\right)\right) \cdot b\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot b\right)}\right)\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({a}^{2} \cdot b\right)}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)}\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(\left({a}^{2} \cdot b\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(\left(b \cdot {a}^{2}\right) \cdot \frac{1}{2}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left({a}^{2} \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({a}^{2}\right)}\right)\right)\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right)\right) \]

       19. *-lowering-*.f64 55.6%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 55.6%

       \[\leadsto \frac{x}{1 + \color{blue}{b \cdot \left(b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}} \]

   if 5.5000000000000001e44 < y < 1.1499999999999999e134

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 48.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 48.6%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 31.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 31.1%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left({b}^{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left({b}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 58.2%

       \[\leadsto x \cdot \color{blue}{\left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;a \leq -1.9 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -5.6 \cdot 10^{-95}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-173}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;a \leq 80:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* (* 0.5 (* a a)) (* b b)))))
   (if (<= a -1.9e-20)
     t_1
     (if (<= a -5.6e-95)
       (* x (- 1.0 (* a b)))
       (if (<= a 2.8e-173)
         (* 2.0 (/ x (* a (* a (* b b)))))
         (if (<= a 80.0) t_1 (* 2.0 (/ (/ (/ x (* a a)) b) b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * ((0.5 * (a * a)) * (b * b));
	double tmp;
	if (a <= -1.9e-20) {
		tmp = t_1;
	} else if (a <= -5.6e-95) {
		tmp = x * (1.0 - (a * b));
	} else if (a <= 2.8e-173) {
		tmp = 2.0 * (x / (a * (a * (b * b))));
	} else if (a <= 80.0) {
		tmp = t_1;
	} else {
		tmp = 2.0 * (((x / (a * a)) / b) / b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((0.5d0 * (a * a)) * (b * b))
    if (a <= (-1.9d-20)) then
        tmp = t_1
    else if (a <= (-5.6d-95)) then
        tmp = x * (1.0d0 - (a * b))
    else if (a <= 2.8d-173) then
        tmp = 2.0d0 * (x / (a * (a * (b * b))))
    else if (a <= 80.0d0) then
        tmp = t_1
    else
        tmp = 2.0d0 * (((x / (a * a)) / b) / b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * ((0.5 * (a * a)) * (b * b));
	double tmp;
	if (a <= -1.9e-20) {
		tmp = t_1;
	} else if (a <= -5.6e-95) {
		tmp = x * (1.0 - (a * b));
	} else if (a <= 2.8e-173) {
		tmp = 2.0 * (x / (a * (a * (b * b))));
	} else if (a <= 80.0) {
		tmp = t_1;
	} else {
		tmp = 2.0 * (((x / (a * a)) / b) / b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * ((0.5 * (a * a)) * (b * b))
	tmp = 0
	if a <= -1.9e-20:
		tmp = t_1
	elif a <= -5.6e-95:
		tmp = x * (1.0 - (a * b))
	elif a <= 2.8e-173:
		tmp = 2.0 * (x / (a * (a * (b * b))))
	elif a <= 80.0:
		tmp = t_1
	else:
		tmp = 2.0 * (((x / (a * a)) / b) / b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(Float64(0.5 * Float64(a * a)) * Float64(b * b)))
	tmp = 0.0
	if (a <= -1.9e-20)
		tmp = t_1;
	elseif (a <= -5.6e-95)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	elseif (a <= 2.8e-173)
		tmp = Float64(2.0 * Float64(x / Float64(a * Float64(a * Float64(b * b)))));
	elseif (a <= 80.0)
		tmp = t_1;
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(x / Float64(a * a)) / b) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * ((0.5 * (a * a)) * (b * b));
	tmp = 0.0;
	if (a <= -1.9e-20)
		tmp = t_1;
	elseif (a <= -5.6e-95)
		tmp = x * (1.0 - (a * b));
	elseif (a <= 2.8e-173)
		tmp = 2.0 * (x / (a * (a * (b * b))));
	elseif (a <= 80.0)
		tmp = t_1;
	else
		tmp = 2.0 * (((x / (a * a)) / b) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.9e-20], t$95$1, If[LessEqual[a, -5.6e-95], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.8e-173], N[(2.0 * N[(x / N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 80.0], t$95$1, N[(2.0 * N[(N[(N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.8999999999999999e-20 or 2.7999999999999999e-173 < a < 80

   1. Initial program 96.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 62.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 62.6%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \color{blue}{-1 \cdot b}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) - \color{blue}{b}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot {b}^{2}\right)\right), b\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right), b\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right), b\right)\right)\right)\right) \]

      10. *-lowering-*.f64 42.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right), b\right)\right)\right)\right) \]

   8. Simplified 42.1%

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left({b}^{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left({b}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 45.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 45.7%

       \[\leadsto x \cdot \color{blue}{\left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)} \]

   if -1.8999999999999999e-20 < a < -5.5999999999999998e-95

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 82.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 82.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 73.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 73.1%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   if -5.5999999999999998e-95 < a < 2.7999999999999999e-173

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 41.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 41.8%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 37.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 37.2%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 48.9%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 48.9%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if 80 < a

   1. Initial program 92.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 65.6%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 65.6%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 46.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 46.4%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in b around -inf

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} - \color{blue}{\frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left(\frac{\color{blue}{-1 \cdot a - \frac{1}{b}}}{b}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a - \frac{1}{b}\right), \color{blue}{b}\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right), b\right)\right)\right)\right) \]

       13. distribute-neg-frac N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{\mathsf{neg}\left(1\right)}{b}\right), b\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{-1}{b}\right), b\right)\right)\right)\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-1 \cdot a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       16. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(a\right)\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       17. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(0 - a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       18. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       19. /-lowering-/.f64 46.5%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \mathsf{/.f64}\left(-1, b\right)\right), b\right)\right)\right)\right) \]

   13. Simplified 46.5%

       \[\leadsto \frac{x}{\color{blue}{\left(b \cdot b\right) \cdot \left(0.5 \cdot \left(a \cdot a\right) - \frac{\left(0 - a\right) + \frac{-1}{b}}{b}\right)}} \]

   14. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   15. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{\color{blue}{{b}^{2}}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{b \cdot \color{blue}{b}}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{\frac{x}{{a}^{2}}}{b}}{\color{blue}{b}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\frac{x}{{a}^{2}}}{b}\right), \color{blue}{b}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{{a}^{2}}\right), b\right), b\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left({a}^{2}\right)\right), b\right), b\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(a \cdot a\right)\right), b\right), b\right)\right) \]

       9. *-lowering-*.f64 59.2%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(a, a\right)\right), b\right), b\right)\right) \]

   16. Simplified 59.2%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;a \leq -5.6 \cdot 10^{-95}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-173}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;a \leq 80:\\ \;\;\;\;x \cdot \left(\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 39.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-266}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (/ (/ x (* a a)) b) b))))
   (if (<= y -6.5e+139)
     t_1
     (if (<= y -5.6e-132)
       (* 2.0 (/ x (* a (* a (* b b)))))
       (if (<= y 1.1e-266) (* x (- 1.0 (* a b))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * (((x / (a * a)) / b) / b);
	double tmp;
	if (y <= -6.5e+139) {
		tmp = t_1;
	} else if (y <= -5.6e-132) {
		tmp = 2.0 * (x / (a * (a * (b * b))));
	} else if (y <= 1.1e-266) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (((x / (a * a)) / b) / b)
    if (y <= (-6.5d+139)) then
        tmp = t_1
    else if (y <= (-5.6d-132)) then
        tmp = 2.0d0 * (x / (a * (a * (b * b))))
    else if (y <= 1.1d-266) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * (((x / (a * a)) / b) / b);
	double tmp;
	if (y <= -6.5e+139) {
		tmp = t_1;
	} else if (y <= -5.6e-132) {
		tmp = 2.0 * (x / (a * (a * (b * b))));
	} else if (y <= 1.1e-266) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 2.0 * (((x / (a * a)) / b) / b)
	tmp = 0
	if y <= -6.5e+139:
		tmp = t_1
	elif y <= -5.6e-132:
		tmp = 2.0 * (x / (a * (a * (b * b))))
	elif y <= 1.1e-266:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(2.0 * Float64(Float64(Float64(x / Float64(a * a)) / b) / b))
	tmp = 0.0
	if (y <= -6.5e+139)
		tmp = t_1;
	elseif (y <= -5.6e-132)
		tmp = Float64(2.0 * Float64(x / Float64(a * Float64(a * Float64(b * b)))));
	elseif (y <= 1.1e-266)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 2.0 * (((x / (a * a)) / b) / b);
	tmp = 0.0;
	if (y <= -6.5e+139)
		tmp = t_1;
	elseif (y <= -5.6e-132)
		tmp = 2.0 * (x / (a * (a * (b * b))));
	elseif (y <= 1.1e-266)
		tmp = x * (1.0 - (a * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(2.0 * N[(N[(N[(x / N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+139], t$95$1, If[LessEqual[y, -5.6e-132], N[(2.0 * N[(x / N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-266], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5000000000000003e139 or 1.1e-266 < y

   1. Initial program 96.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 50.6%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 50.6%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 31.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 31.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in b around -inf

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({b}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b} + \frac{1}{2} \cdot {a}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}} + \frac{1}{2} \cdot {a}^{2}\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \color{blue}{-1 \cdot \frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot {a}^{2} - \color{blue}{\frac{-1 \cdot a - \frac{1}{b}}{b}}\right)\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot {a}^{2}\right), \color{blue}{\left(\frac{-1 \cdot a - \frac{1}{b}}{b}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({a}^{2}\right)\right), \left(\frac{\color{blue}{-1 \cdot a - \frac{1}{b}}}{b}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(a \cdot a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{-1 \cdot a - \color{blue}{\frac{1}{b}}}{b}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a - \frac{1}{b}\right), \color{blue}{b}\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right), b\right)\right)\right)\right) \]

       13. distribute-neg-frac N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{\mathsf{neg}\left(1\right)}{b}\right), b\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(-1 \cdot a + \frac{-1}{b}\right), b\right)\right)\right)\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-1 \cdot a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       16. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(a\right)\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       17. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(0 - a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       18. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \left(\frac{-1}{b}\right)\right), b\right)\right)\right)\right) \]

       19. /-lowering-/.f64 28.2%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, a\right), \mathsf{/.f64}\left(-1, b\right)\right), b\right)\right)\right)\right) \]

   13. Simplified 28.2%

       \[\leadsto \frac{x}{\color{blue}{\left(b \cdot b\right) \cdot \left(0.5 \cdot \left(a \cdot a\right) - \frac{\left(0 - a\right) + \frac{-1}{b}}{b}\right)}} \]

   14. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   15. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{\color{blue}{{b}^{2}}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{x}{{a}^{2}}}{b \cdot \color{blue}{b}}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\frac{\frac{x}{{a}^{2}}}{b}}{\color{blue}{b}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\frac{x}{{a}^{2}}}{b}\right), \color{blue}{b}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{{a}^{2}}\right), b\right), b\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left({a}^{2}\right)\right), b\right), b\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(a \cdot a\right)\right), b\right), b\right)\right) \]

       9. *-lowering-*.f64 44.8%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(a, a\right)\right), b\right), b\right)\right) \]

   16. Simplified 44.8%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}} \]

   if -6.5000000000000003e139 < y < -5.60000000000000005e-132

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 57.0%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 34.7%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 60.1%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 60.1%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if -5.60000000000000005e-132 < y < 1.1e-266

   1. Initial program 96.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 91.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 91.9%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 48.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 48.7%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-266}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{x}{a \cdot a}}{b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.4% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{-132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{1 + a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ x (* a (* a (* b b)))))))
   (if (<= y -2.8e-132) t_1 (if (<= y 1.28e-49) (/ x (+ 1.0 (* a b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * (x / (a * (a * (b * b))));
	double tmp;
	if (y <= -2.8e-132) {
		tmp = t_1;
	} else if (y <= 1.28e-49) {
		tmp = x / (1.0 + (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (x / (a * (a * (b * b))))
    if (y <= (-2.8d-132)) then
        tmp = t_1
    else if (y <= 1.28d-49) then
        tmp = x / (1.0d0 + (a * b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * (x / (a * (a * (b * b))));
	double tmp;
	if (y <= -2.8e-132) {
		tmp = t_1;
	} else if (y <= 1.28e-49) {
		tmp = x / (1.0 + (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 2.0 * (x / (a * (a * (b * b))))
	tmp = 0
	if y <= -2.8e-132:
		tmp = t_1
	elif y <= 1.28e-49:
		tmp = x / (1.0 + (a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(2.0 * Float64(x / Float64(a * Float64(a * Float64(b * b)))))
	tmp = 0.0
	if (y <= -2.8e-132)
		tmp = t_1;
	elseif (y <= 1.28e-49)
		tmp = Float64(x / Float64(1.0 + Float64(a * b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 2.0 * (x / (a * (a * (b * b))));
	tmp = 0.0;
	if (y <= -2.8e-132)
		tmp = t_1;
	elseif (y <= 1.28e-49)
		tmp = x / (1.0 + (a * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(2.0 * N[(x / N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.8e-132], t$95$1, If[LessEqual[y, 1.28e-49], N[(x / N[(1.0 + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000002e-132 or 1.28e-49 < y

   1. Initial program 97.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 44.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 44.9%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(a \cdot \left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{\left(b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 26.3%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 26.3%

       \[\leadsto \frac{x}{\color{blue}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{{a}^{2} \cdot {b}^{2}}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{x}{{a}^{2} \cdot {b}^{2}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left({b}^{2} \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left({b}^{2} \cdot a\right) \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \left(\left(a \cdot {b}^{2}\right) \cdot a\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(a \cdot {b}^{2}\right), \color{blue}{a}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), a\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), a\right)\right)\right) \]

       10. *-lowering-*.f64 44.3%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right)\right) \]

   13. Simplified 44.3%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a}} \]

   if -2.80000000000000002e-132 < y < 1.28e-49

   1. Initial program 94.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 84.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 84.5%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot b\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(a \cdot b + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(a \cdot b\right), \color{blue}{1}\right)\right) \]

      3. *-lowering-*.f64 45.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, b\right), 1\right)\right) \]

   10. Simplified 45.7%

       \[\leadsto \frac{x}{\color{blue}{a \cdot b + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{1 + a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{x}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 31.1% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -35000000000000:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - b \cdot x\right)\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;x - t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -35000000000000.0)
   (* a (- (/ x a) (* b x)))
   (if (<= b 8.6e+32) (- x (* t (* y x))) (/ x (+ 1.0 (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -35000000000000.0) {
		tmp = a * ((x / a) - (b * x));
	} else if (b <= 8.6e+32) {
		tmp = x - (t * (y * x));
	} else {
		tmp = x / (1.0 + (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-35000000000000.0d0)) then
        tmp = a * ((x / a) - (b * x))
    else if (b <= 8.6d+32) then
        tmp = x - (t * (y * x))
    else
        tmp = x / (1.0d0 + (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -35000000000000.0) {
		tmp = a * ((x / a) - (b * x));
	} else if (b <= 8.6e+32) {
		tmp = x - (t * (y * x));
	} else {
		tmp = x / (1.0 + (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -35000000000000.0:
		tmp = a * ((x / a) - (b * x))
	elif b <= 8.6e+32:
		tmp = x - (t * (y * x))
	else:
		tmp = x / (1.0 + (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -35000000000000.0)
		tmp = Float64(a * Float64(Float64(x / a) - Float64(b * x)));
	elseif (b <= 8.6e+32)
		tmp = Float64(x - Float64(t * Float64(y * x)));
	else
		tmp = Float64(x / Float64(1.0 + Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -35000000000000.0)
		tmp = a * ((x / a) - (b * x));
	elseif (b <= 8.6e+32)
		tmp = x - (t * (y * x));
	else
		tmp = x / (1.0 + (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -35000000000000.0], N[(a * N[(N[(x / a), $MachinePrecision] - N[(b * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e+32], N[(x - N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5e13

   1. Initial program 96.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 74.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 74.1%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 27.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 27.4%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(b \cdot x\right) + \frac{x}{a}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(-1 \cdot \left(b \cdot x\right) + \frac{x}{a}\right)}\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(a, \left(\left(\mathsf{neg}\left(b \cdot x\right)\right) + \frac{\color{blue}{x}}{a}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \left(\frac{x}{a} + \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}\right)\right) \]

       4. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(a, \left(\frac{x}{a} - \color{blue}{b \cdot x}\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(\frac{x}{a}\right), \color{blue}{\left(b \cdot x\right)}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, a\right), \left(\color{blue}{b} \cdot x\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, a\right), \left(x \cdot \color{blue}{b}\right)\right)\right) \]

       8. *-lowering-*.f64 30.7%

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, a\right), \mathsf{*.f64}\left(x, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 30.7%

       \[\leadsto \color{blue}{a \cdot \left(\frac{x}{a} - x \cdot b\right)} \]

   if -3.5e13 < b < 8.5999999999999994e32

   1. Initial program 96.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 71.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 71.9%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto x - \color{blue}{t \cdot \left(x \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(t \cdot \left(x \cdot y\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 32.9%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   10. Simplified 32.9%

       \[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)} \]

   if 8.5999999999999994e32 < b

   1. Initial program 96.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 91.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 91.7%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot b\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(a \cdot b + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(a \cdot b\right), \color{blue}{1}\right)\right) \]

      3. *-lowering-*.f64 44.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, b\right), 1\right)\right) \]

   10. Simplified 44.0%

       \[\leadsto \frac{x}{\color{blue}{a \cdot b + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -35000000000000:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - b \cdot x\right)\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;x - t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 33.3% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+49}:\\ \;\;\;\;x - t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{1 + a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -9.5e+49)
   (- x (* t (* y x)))
   (if (<= y 5.5e+44) (/ x (+ 1.0 (* a b))) (- 0.0 (* x (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9.5e+49) {
		tmp = x - (t * (y * x));
	} else if (y <= 5.5e+44) {
		tmp = x / (1.0 + (a * b));
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-9.5d+49)) then
        tmp = x - (t * (y * x))
    else if (y <= 5.5d+44) then
        tmp = x / (1.0d0 + (a * b))
    else
        tmp = 0.0d0 - (x * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9.5e+49) {
		tmp = x - (t * (y * x));
	} else if (y <= 5.5e+44) {
		tmp = x / (1.0 + (a * b));
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -9.5e+49:
		tmp = x - (t * (y * x))
	elif y <= 5.5e+44:
		tmp = x / (1.0 + (a * b))
	else:
		tmp = 0.0 - (x * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -9.5e+49)
		tmp = Float64(x - Float64(t * Float64(y * x)));
	elseif (y <= 5.5e+44)
		tmp = Float64(x / Float64(1.0 + Float64(a * b)));
	else
		tmp = Float64(0.0 - Float64(x * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -9.5e+49)
		tmp = x - (t * (y * x));
	elseif (y <= 5.5e+44)
		tmp = x / (1.0 + (a * b));
	else
		tmp = 0.0 - (x * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -9.5e+49], N[(x - N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e+44], N[(x / N[(1.0 + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.49999999999999969e49

   1. Initial program 94.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 66.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 66.9%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto x - \color{blue}{t \cdot \left(x \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(t \cdot \left(x \cdot y\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 28.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   10. Simplified 28.0%

       \[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)} \]

   if -9.49999999999999969e49 < y < 5.5000000000000001e44

   1. Initial program 95.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.5%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 76.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 76.8%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + a \cdot b\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(a \cdot b + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(a \cdot b\right), \color{blue}{1}\right)\right) \]

      3. *-lowering-*.f64 39.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, b\right), 1\right)\right) \]

   10. Simplified 39.8%

       \[\leadsto \frac{x}{\color{blue}{a \cdot b + 1}} \]

   if 5.5000000000000001e44 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 39.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 39.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 10.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 10.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(a \cdot b\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{a \cdot b}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       4. *-lowering-*.f64 28.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 28.4%

       \[\leadsto x \cdot \color{blue}{\left(0 - a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+49}:\\ \;\;\;\;x - t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{1 + a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 32.2% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+50}:\\ \;\;\;\;x - t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+41}:\\ \;\;\;\;x - a \cdot \left(b \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.8e+50)
   (- x (* t (* y x)))
   (if (<= y 2.3e+41) (- x (* a (* b x))) (- 0.0 (* x (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.8e+50) {
		tmp = x - (t * (y * x));
	} else if (y <= 2.3e+41) {
		tmp = x - (a * (b * x));
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-6.8d+50)) then
        tmp = x - (t * (y * x))
    else if (y <= 2.3d+41) then
        tmp = x - (a * (b * x))
    else
        tmp = 0.0d0 - (x * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.8e+50) {
		tmp = x - (t * (y * x));
	} else if (y <= 2.3e+41) {
		tmp = x - (a * (b * x));
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.8e+50:
		tmp = x - (t * (y * x))
	elif y <= 2.3e+41:
		tmp = x - (a * (b * x))
	else:
		tmp = 0.0 - (x * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.8e+50)
		tmp = Float64(x - Float64(t * Float64(y * x)));
	elseif (y <= 2.3e+41)
		tmp = Float64(x - Float64(a * Float64(b * x)));
	else
		tmp = Float64(0.0 - Float64(x * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.8e+50)
		tmp = x - (t * (y * x));
	elseif (y <= 2.3e+41)
		tmp = x - (a * (b * x));
	else
		tmp = 0.0 - (x * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.8e+50], N[(x - N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e+41], N[(x - N[(a * N[(b * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.7999999999999997e50

   1. Initial program 93.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(t \cdot y\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 68.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, t\right)\right)\right) \]

   7. Simplified 68.2%

      \[\leadsto \frac{x}{e^{\color{blue}{y \cdot t}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto x - \color{blue}{t \cdot \left(x \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(t \cdot \left(x \cdot y\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 28.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   10. Simplified 28.5%

       \[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)} \]

   if -6.7999999999999997e50 < y < 2.2999999999999998e41

   1. Initial program 95.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.5%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 76.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 76.8%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto x - \color{blue}{a \cdot \left(b \cdot x\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(a \cdot \left(b \cdot x\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(a, \color{blue}{\left(b \cdot x\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(a, \left(x \cdot \color{blue}{b}\right)\right)\right) \]

      6. *-lowering-*.f64 32.6%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \color{blue}{b}\right)\right)\right) \]

   10. Simplified 32.6%

       \[\leadsto \color{blue}{x - a \cdot \left(x \cdot b\right)} \]

   if 2.2999999999999998e41 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 40.6%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 10.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 10.8%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(a \cdot b\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{a \cdot b}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       4. *-lowering-*.f64 28.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 28.0%

       \[\leadsto x \cdot \color{blue}{\left(0 - a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+50}:\\ \;\;\;\;x - t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+41}:\\ \;\;\;\;x - a \cdot \left(b \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 32.2% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-24}:\\ \;\;\;\;0 - a \cdot \left(b \cdot x\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.95e-24)
   (- 0.0 (* a (* b x)))
   (if (<= y 2e+41) (* x (- 1.0 (* a b))) (- 0.0 (* x (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.95e-24) {
		tmp = 0.0 - (a * (b * x));
	} else if (y <= 2e+41) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.95d-24)) then
        tmp = 0.0d0 - (a * (b * x))
    else if (y <= 2d+41) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = 0.0d0 - (x * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.95e-24) {
		tmp = 0.0 - (a * (b * x));
	} else if (y <= 2e+41) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.95e-24:
		tmp = 0.0 - (a * (b * x))
	elif y <= 2e+41:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = 0.0 - (x * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.95e-24)
		tmp = Float64(0.0 - Float64(a * Float64(b * x)));
	elseif (y <= 2e+41)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(0.0 - Float64(x * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.95e-24)
		tmp = 0.0 - (a * (b * x));
	elseif (y <= 2e+41)
		tmp = x * (1.0 - (a * b));
	else
		tmp = 0.0 - (x * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.95e-24], N[(0.0 - N[(a * N[(b * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+41], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.95e-24

   1. Initial program 94.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 38.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 38.5%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 10.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 10.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto a \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \left(0 - \color{blue}{b \cdot x}\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(0, \color{blue}{\left(b \cdot x\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(0, \left(x \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 18.4%

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 18.4%

       \[\leadsto \color{blue}{a \cdot \left(0 - x \cdot b\right)} \]

   if -1.95e-24 < y < 2.00000000000000001e41

   1. Initial program 95.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 80.2%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 35.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 35.6%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   if 2.00000000000000001e41 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 40.6%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 10.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 10.8%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(a \cdot b\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{a \cdot b}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       4. *-lowering-*.f64 28.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 28.0%

       \[\leadsto x \cdot \color{blue}{\left(0 - a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-24}:\\ \;\;\;\;0 - a \cdot \left(b \cdot x\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 27.1% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;0 - a \cdot \left(b \cdot x\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.4e-8)
   (- 0.0 (* a (* b x)))
   (if (<= y 5e+44) x (- 0.0 (* x (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.4e-8) {
		tmp = 0.0 - (a * (b * x));
	} else if (y <= 5e+44) {
		tmp = x;
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-4.4d-8)) then
        tmp = 0.0d0 - (a * (b * x))
    else if (y <= 5d+44) then
        tmp = x
    else
        tmp = 0.0d0 - (x * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.4e-8) {
		tmp = 0.0 - (a * (b * x));
	} else if (y <= 5e+44) {
		tmp = x;
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4.4e-8:
		tmp = 0.0 - (a * (b * x))
	elif y <= 5e+44:
		tmp = x
	else:
		tmp = 0.0 - (x * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.4e-8)
		tmp = Float64(0.0 - Float64(a * Float64(b * x)));
	elseif (y <= 5e+44)
		tmp = x;
	else
		tmp = Float64(0.0 - Float64(x * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4.4e-8)
		tmp = 0.0 - (a * (b * x));
	elseif (y <= 5e+44)
		tmp = x;
	else
		tmp = 0.0 - (x * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.4e-8], N[(0.0 - N[(a * N[(b * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+44], x, N[(0.0 - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.3999999999999997e-8

   1. Initial program 93.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 37.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 37.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 11.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 11.4%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto a \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \left(0 - \color{blue}{b \cdot x}\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(0, \color{blue}{\left(b \cdot x\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(0, \left(x \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 19.2%

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 19.2%

       \[\leadsto \color{blue}{a \cdot \left(0 - x \cdot b\right)} \]

   if -4.3999999999999997e-8 < y < 4.9999999999999996e44

   1. Initial program 95.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 79.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 79.5%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x} \]
   9. Step-by-step derivation

   10. Simplified 27.1%

       \[\leadsto \color{blue}{x} \]

   if 4.9999999999999996e44 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 39.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 39.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 10.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 10.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(a \cdot b\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{a \cdot b}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       4. *-lowering-*.f64 28.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 28.4%

       \[\leadsto x \cdot \color{blue}{\left(0 - a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;0 - a \cdot \left(b \cdot x\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 27.1% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - a \cdot \left(b \cdot x\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 0.0 (* a (* b x)))))
   (if (<= y -7e-10) t_1 (if (<= y 9.8e+33) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.0 - (a * (b * x));
	double tmp;
	if (y <= -7e-10) {
		tmp = t_1;
	} else if (y <= 9.8e+33) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.0d0 - (a * (b * x))
    if (y <= (-7d-10)) then
        tmp = t_1
    else if (y <= 9.8d+33) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.0 - (a * (b * x));
	double tmp;
	if (y <= -7e-10) {
		tmp = t_1;
	} else if (y <= 9.8e+33) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.0 - (a * (b * x))
	tmp = 0
	if y <= -7e-10:
		tmp = t_1
	elif y <= 9.8e+33:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.0 - Float64(a * Float64(b * x)))
	tmp = 0.0
	if (y <= -7e-10)
		tmp = t_1;
	elseif (y <= 9.8e+33)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.0 - (a * (b * x));
	tmp = 0.0;
	if (y <= -7e-10)
		tmp = t_1;
	elseif (y <= 9.8e+33)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.0 - N[(a * N[(b * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e-10], t$95$1, If[LessEqual[y, 9.8e+33], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999961e-10 or 9.80000000000000027e33 < y

   1. Initial program 97.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 40.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 10.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 10.8%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto a \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \left(0 - \color{blue}{b \cdot x}\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(0, \color{blue}{\left(b \cdot x\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(0, \left(x \cdot \color{blue}{b}\right)\right)\right) \]

       7. *-lowering-*.f64 23.0%

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 23.0%

       \[\leadsto \color{blue}{a \cdot \left(0 - x \cdot b\right)} \]

   if -6.99999999999999961e-10 < y < 9.80000000000000027e33

   1. Initial program 95.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 79.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 79.4%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x} \]
   9. Step-by-step derivation

   10. Simplified 27.9%

       \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-10}:\\ \;\;\;\;0 - a \cdot \left(b \cdot x\right)\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - a \cdot \left(b \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 30.3% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+41}:\\ \;\;\;\;x - a \cdot \left(b \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 2.15e+41) (- x (* a (* b x))) (- 0.0 (* x (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 2.15e+41) {
		tmp = x - (a * (b * x));
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= 2.15d+41) then
        tmp = x - (a * (b * x))
    else
        tmp = 0.0d0 - (x * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 2.15e+41) {
		tmp = x - (a * (b * x));
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 2.15e+41:
		tmp = x - (a * (b * x))
	else:
		tmp = 0.0 - (x * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 2.15e+41)
		tmp = Float64(x - Float64(a * Float64(b * x)));
	else
		tmp = Float64(0.0 - Float64(x * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 2.15e+41)
		tmp = x - (a * (b * x));
	else
		tmp = 0.0 - (x * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 2.15e+41], N[(x - N[(a * N[(b * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.15000000000000012e41

   1. Initial program 95.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   3. Simplified 95.1%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 65.6%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Simplified 65.6%

      \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto x - \color{blue}{a \cdot \left(b \cdot x\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(a \cdot \left(b \cdot x\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(a, \color{blue}{\left(b \cdot x\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(a, \left(x \cdot \color{blue}{b}\right)\right)\right) \]

      6. *-lowering-*.f64 27.6%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \color{blue}{b}\right)\right)\right) \]

   10. Simplified 27.6%

       \[\leadsto \color{blue}{x - a \cdot \left(x \cdot b\right)} \]

   if 2.15000000000000012e41 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - a \cdot b\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot b\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 40.6%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{a \cdot b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      4. *-lowering-*.f64 10.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 10.8%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)}\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(a \cdot b\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{a \cdot b}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       4. *-lowering-*.f64 28.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 28.0%

       \[\leadsto x \cdot \color{blue}{\left(0 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+41}:\\ \;\;\;\;x - a \cdot \left(b \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 19.5% accurate, 315.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 96.4%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

3. Simplified 96.4%

   \[\leadsto \color{blue}{\frac{x}{e^{a \cdot \left(b - \log \left(1 - z\right)\right) + y \cdot \left(t - \log z\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf

   \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(a \cdot b\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 59.2%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

7. Simplified 59.2%

   \[\leadsto \frac{x}{e^{\color{blue}{a \cdot b}}} \]

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{x} \]
9. Step-by-step derivation

10. Simplified 15.6%

    \[\leadsto \color{blue}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))