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

Percentage Accurate: 96.3% → 99.1%

Time: 15.6s

Alternatives: 22

Speedup: 1.5×

• 40.4% 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: 96.3% 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: 99.1% accurate, 1.5× speedup

Math

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

FPCore

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

Python

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) - (a * (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[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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 z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

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

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

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

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

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

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

   9. distribute-lft-out N/A

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

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

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

   11. mul-1-neg N/A

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

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right), \mathsf{*.f64}\left(\left(0 - a\right), \left(b + z\right)\right)\right)\right)\right) \]

   13. --lowering--.f64 N/A

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

   14. +-lowering-+.f64 98.8%

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

5. Simplified 98.8%

   \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right) + \left(0 - a\right) \cdot \left(b + z\right)}} \]

6. Final simplification 98.8%

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

Alternative 2: 83.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00045:\\ \;\;\;\;x \cdot e^{0 - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -2.3e-80)
     t_1
     (if (<= y 0.00045) (* x (exp (- 0.0 (* a 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 <= -2.3e-80) {
		tmp = t_1;
	} else if (y <= 0.00045) {
		tmp = x * exp((0.0 - (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 = x * exp((y * (log(z) - t)))
    if (y <= (-2.3d-80)) then
        tmp = t_1
    else if (y <= 0.00045d0) then
        tmp = x * exp((0.0d0 - (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 = x * Math.exp((y * (Math.log(z) - t)));
	double tmp;
	if (y <= -2.3e-80) {
		tmp = t_1;
	} else if (y <= 0.00045) {
		tmp = x * Math.exp((0.0 - (a * 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 <= -2.3e-80:
		tmp = t_1
	elif y <= 0.00045:
		tmp = x * math.exp((0.0 - (a * 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 <= -2.3e-80)
		tmp = t_1;
	elseif (y <= 0.00045)
		tmp = Float64(x * exp(Float64(0.0 - Float64(a * b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * (log(z) - t)));
	tmp = 0.0;
	if (y <= -2.3e-80)
		tmp = t_1;
	elseif (y <= 0.00045)
		tmp = x * exp((0.0 - (a * b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-80], t$95$1, If[LessEqual[y, 0.00045], N[(x * N[Exp[N[(0.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2999999999999998e-80 or 4.4999999999999999e-4 < y

   1. Initial program 96.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 85.8%

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

   5. Simplified 85.8%

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

   if -2.2999999999999998e-80 < y < 4.4999999999999999e-4

   1. Initial program 95.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 81.8%

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

   5. Simplified 81.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{e^{y \cdot t}}\\ \mathbf{if}\;t \leq -440:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-216}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (exp (* y t)))))
   (if (<= t -440.0)
     t_1
     (if (<= t 8e-216)
       (* x (pow z y))
       (if (<= t 2e+44) (/ 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 / exp((y * t));
	double tmp;
	if (t <= -440.0) {
		tmp = t_1;
	} else if (t <= 8e-216) {
		tmp = x * pow(z, y);
	} else if (t <= 2e+44) {
		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 / exp((y * t))
    if (t <= (-440.0d0)) then
        tmp = t_1
    else if (t <= 8d-216) then
        tmp = x * (z ** y)
    else if (t <= 2d+44) 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.exp((y * t));
	double tmp;
	if (t <= -440.0) {
		tmp = t_1;
	} else if (t <= 8e-216) {
		tmp = x * Math.pow(z, y);
	} else if (t <= 2e+44) {
		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.exp((y * t))
	tmp = 0
	if t <= -440.0:
		tmp = t_1
	elif t <= 8e-216:
		tmp = x * math.pow(z, y)
	elif t <= 2e+44:
		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 / exp(Float64(y * t)))
	tmp = 0.0
	if (t <= -440.0)
		tmp = t_1;
	elseif (t <= 8e-216)
		tmp = Float64(x * (z ^ y));
	elseif (t <= 2e+44)
		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 / exp((y * t));
	tmp = 0.0;
	if (t <= -440.0)
		tmp = t_1;
	elseif (t <= 8e-216)
		tmp = x * (z ^ y);
	elseif (t <= 2e+44)
		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[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -440.0], t$95$1, If[LessEqual[t, 8e-216], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e+44], N[(x / N[Exp[N[(a * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -440 or 2.0000000000000002e44 < t

   1. Initial program 94.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 79.9%

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

   5. Simplified 79.9%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(y \cdot t\right)} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{e^{y \cdot t}}} \]

      3. un-div-inv N/A

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

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

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

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

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

      6. *-lowering-*.f64 79.9%

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

   7. Applied egg-rr 79.9%

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

   if -440 < t < 8.0000000000000003e-216

   1. Initial program 97.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 67.9%

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

   5. Simplified 67.9%

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

   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 67.9%

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

   8. Simplified 67.9%

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

   if 8.0000000000000003e-216 < t < 2.0000000000000002e44

   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 76.3%

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

   5. Simplified 76.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(a \cdot b\right)} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{e^{a \cdot b}}} \]

      3. un-div-inv N/A

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

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

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

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

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

      6. *-lowering-*.f64 76.3%

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

   7. Applied egg-rr 76.3%

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

Alternative 4: 56.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{+211}:\\ \;\;\;\;x \cdot \left(1 + b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right) - a\right)\right)\\ \mathbf{elif}\;a \leq -1.26 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + t\_1\right)\right)}\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+61}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.5 (* a a))))
   (if (<= a -2.2e+211)
     (* x (+ 1.0 (* b (- (* 0.5 (* b (* a a))) a))))
     (if (<= a -1.26e+33)
       (/
        x
        (+
         1.0
         (*
          b
          (+ a (* b (+ (* 0.16666666666666666 (* b (* a (* a a)))) t_1))))))
       (if (<= a 3.6e+61)
         (* x (pow z y))
         (/ x (+ 1.0 (* b (+ a (* b t_1))))))))))

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 (a <= -2.2e+211) {
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)));
	} else if (a <= -1.26e+33) {
		tmp = x / (1.0 + (b * (a + (b * ((0.16666666666666666 * (b * (a * (a * a)))) + t_1)))));
	} else if (a <= 3.6e+61) {
		tmp = x * pow(z, y);
	} else {
		tmp = x / (1.0 + (b * (a + (b * 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 = 0.5d0 * (a * a)
    if (a <= (-2.2d+211)) then
        tmp = x * (1.0d0 + (b * ((0.5d0 * (b * (a * a))) - a)))
    else if (a <= (-1.26d+33)) then
        tmp = x / (1.0d0 + (b * (a + (b * ((0.16666666666666666d0 * (b * (a * (a * a)))) + t_1)))))
    else if (a <= 3.6d+61) then
        tmp = x * (z ** y)
    else
        tmp = x / (1.0d0 + (b * (a + (b * 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 = 0.5 * (a * a);
	double tmp;
	if (a <= -2.2e+211) {
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)));
	} else if (a <= -1.26e+33) {
		tmp = x / (1.0 + (b * (a + (b * ((0.16666666666666666 * (b * (a * (a * a)))) + t_1)))));
	} else if (a <= 3.6e+61) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.5 * (a * a)
	tmp = 0
	if a <= -2.2e+211:
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)))
	elif a <= -1.26e+33:
		tmp = x / (1.0 + (b * (a + (b * ((0.16666666666666666 * (b * (a * (a * a)))) + t_1)))))
	elif a <= 3.6e+61:
		tmp = x * math.pow(z, y)
	else:
		tmp = x / (1.0 + (b * (a + (b * t_1))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.5 * Float64(a * a))
	tmp = 0.0
	if (a <= -2.2e+211)
		tmp = Float64(x * Float64(1.0 + Float64(b * Float64(Float64(0.5 * Float64(b * Float64(a * a))) - a))));
	elseif (a <= -1.26e+33)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(a + Float64(b * Float64(Float64(0.16666666666666666 * Float64(b * Float64(a * Float64(a * a)))) + t_1))))));
	elseif (a <= 3.6e+61)
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(a + Float64(b * t_1)))));
	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 (a <= -2.2e+211)
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)));
	elseif (a <= -1.26e+33)
		tmp = x / (1.0 + (b * (a + (b * ((0.16666666666666666 * (b * (a * (a * a)))) + t_1)))));
	elseif (a <= 3.6e+61)
		tmp = x * (z ^ y);
	else
		tmp = x / (1.0 + (b * (a + (b * t_1))));
	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[a, -2.2e+211], N[(x * N[(1.0 + N[(b * N[(N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.26e+33], N[(x / N[(1.0 + N[(b * N[(a + N[(b * N[(N[(0.16666666666666666 * N[(b * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.6e+61], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(b * N[(a + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.20000000000000004e211

   1. Initial program 90.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 76.6%

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

   5. Simplified 76.6%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 67.2%

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

   8. Simplified 67.2%

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

   if -2.20000000000000004e211 < a < -1.26e33

   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 81.6%

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

   5. Simplified 81.6%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(a \cdot b\right)} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{e^{a \cdot b}}} \]

      3. un-div-inv N/A

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

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

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

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

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

      6. *-lowering-*.f64 81.6%

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

   7. Applied egg-rr 81.6%

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

   8. Taylor expanded in b around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 53.9%

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

   10. Simplified 53.9%

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

   if -1.26e33 < a < 3.6000000000000001e61

   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 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 87.1%

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

   5. Simplified 87.1%

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

   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 65.0%

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

   8. Simplified 65.0%

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

   if 3.6000000000000001e61 < a

   1. Initial program 82.1%

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

   3. Taylor expanded in 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 73.8%

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

   5. Simplified 73.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(a \cdot b\right)} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{e^{a \cdot b}}} \]

      3. un-div-inv N/A

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

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

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

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

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

      6. *-lowering-*.f64 73.8%

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

   7. Applied egg-rr 73.8%

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

   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. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

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

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

   10. Simplified 61.4%

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

Alternative 5: 72.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -0.000225:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -0.000225) t_1 (if (<= y 8.5e+24) (/ 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 <= -0.000225) {
		tmp = t_1;
	} else if (y <= 8.5e+24) {
		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 <= (-0.000225d0)) then
        tmp = t_1
    else if (y <= 8.5d+24) 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 <= -0.000225) {
		tmp = t_1;
	} else if (y <= 8.5e+24) {
		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 <= -0.000225:
		tmp = t_1
	elif y <= 8.5e+24:
		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 <= -0.000225)
		tmp = t_1;
	elseif (y <= 8.5e+24)
		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 <= -0.000225)
		tmp = t_1;
	elseif (y <= 8.5e+24)
		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, -0.000225], t$95$1, If[LessEqual[y, 8.5e+24], N[(x / N[Exp[N[(a * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2499999999999999e-4 or 8.49999999999999959e24 < y

   1. Initial program 96.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 85.9%

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

   5. Simplified 85.9%

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

   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 66.4%

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

   8. Simplified 66.4%

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

   if -2.2499999999999999e-4 < y < 8.49999999999999959e24

   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 78.4%

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

   5. Simplified 78.4%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(a \cdot b\right)} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{e^{a \cdot b}}} \]

      3. un-div-inv N/A

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

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

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

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

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

      6. *-lowering-*.f64 78.4%

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

   7. Applied egg-rr 78.4%

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

Alternative 6: 48.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(a \cdot a\right)\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \left(1 + t \cdot \left(0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y\right)\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-202}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot \left(0.16666666666666666 \cdot \left(b \cdot t\_1\right) + 0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(t\_1 \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (* a a))))
   (if (<= y -3.2e+76)
     (* x (+ 1.0 (* t (- (* 0.5 (* t (* y y))) y))))
     (if (<= y 1.55e-202)
       (/
        x
        (+
         1.0
         (*
          b
          (+ a (* b (+ (* 0.16666666666666666 (* b t_1)) (* 0.5 (* a a))))))))
       (* x (* b (* b (* b (* t_1 -0.16666666666666666)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (a * a);
	double tmp;
	if (y <= -3.2e+76) {
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	} else if (y <= 1.55e-202) {
		tmp = x / (1.0 + (b * (a + (b * ((0.16666666666666666 * (b * t_1)) + (0.5 * (a * a)))))));
	} else {
		tmp = x * (b * (b * (b * (t_1 * -0.16666666666666666))));
	}
	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 = a * (a * a)
    if (y <= (-3.2d+76)) then
        tmp = x * (1.0d0 + (t * ((0.5d0 * (t * (y * y))) - y)))
    else if (y <= 1.55d-202) then
        tmp = x / (1.0d0 + (b * (a + (b * ((0.16666666666666666d0 * (b * t_1)) + (0.5d0 * (a * a)))))))
    else
        tmp = x * (b * (b * (b * (t_1 * (-0.16666666666666666d0)))))
    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 * (a * a);
	double tmp;
	if (y <= -3.2e+76) {
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	} else if (y <= 1.55e-202) {
		tmp = x / (1.0 + (b * (a + (b * ((0.16666666666666666 * (b * t_1)) + (0.5 * (a * a)))))));
	} else {
		tmp = x * (b * (b * (b * (t_1 * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (a * a)
	tmp = 0
	if y <= -3.2e+76:
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)))
	elif y <= 1.55e-202:
		tmp = x / (1.0 + (b * (a + (b * ((0.16666666666666666 * (b * t_1)) + (0.5 * (a * a)))))))
	else:
		tmp = x * (b * (b * (b * (t_1 * -0.16666666666666666))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(a * a))
	tmp = 0.0
	if (y <= -3.2e+76)
		tmp = Float64(x * Float64(1.0 + Float64(t * Float64(Float64(0.5 * Float64(t * Float64(y * y))) - y))));
	elseif (y <= 1.55e-202)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(a + Float64(b * Float64(Float64(0.16666666666666666 * Float64(b * t_1)) + Float64(0.5 * Float64(a * a))))))));
	else
		tmp = Float64(x * Float64(b * Float64(b * Float64(b * Float64(t_1 * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (a * a);
	tmp = 0.0;
	if (y <= -3.2e+76)
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	elseif (y <= 1.55e-202)
		tmp = x / (1.0 + (b * (a + (b * ((0.16666666666666666 * (b * t_1)) + (0.5 * (a * a)))))));
	else
		tmp = x * (b * (b * (b * (t_1 * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+76], N[(x * N[(1.0 + N[(t * N[(N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e-202], N[(x / N[(1.0 + N[(b * N[(a + N[(b * N[(N[(0.16666666666666666 * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(b * N[(b * N[(t$95$1 * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.19999999999999976e76

   1. Initial program 98.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 62.3%

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

   5. Simplified 62.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 60.1%

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

   8. Simplified 60.1%

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

   if -3.19999999999999976e76 < y < 1.55e-202

   1. Initial program 95.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 75.4%

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

   5. Simplified 75.4%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(a \cdot b\right)} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{e^{a \cdot b}}} \]

      3. un-div-inv N/A

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

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

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

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

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

      6. *-lowering-*.f64 75.4%

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

   7. Applied egg-rr 75.4%

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

   8. Taylor expanded in b around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 55.7%

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

   10. Simplified 55.7%

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

   if 1.55e-202 < y

   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 50.0%

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

   5. Simplified 50.0%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

   8. Simplified 26.5%

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

   9. Taylor expanded in b around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 51.9%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   11. Simplified 51.9%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 40.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* (* y y) (* 0.5 (* t t))))))
   (if (<= y -8.2e-30)
     t_1
     (if (<= y 5e-18)
       (* x (- 1.0 (* a b)))
       (if (<= y 9e+157) t_1 (* x (- 0.0 (* a b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * ((y * y) * (0.5 * (t * t)));
	double tmp;
	if (y <= -8.2e-30) {
		tmp = t_1;
	} else if (y <= 5e-18) {
		tmp = x * (1.0 - (a * b));
	} else if (y <= 9e+157) {
		tmp = t_1;
	} else {
		tmp = x * (0.0 - (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) :: t_1
    real(8) :: tmp
    t_1 = x * ((y * y) * (0.5d0 * (t * t)))
    if (y <= (-8.2d-30)) then
        tmp = t_1
    else if (y <= 5d-18) then
        tmp = x * (1.0d0 - (a * b))
    else if (y <= 9d+157) then
        tmp = t_1
    else
        tmp = x * (0.0d0 - (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 t_1 = x * ((y * y) * (0.5 * (t * t)));
	double tmp;
	if (y <= -8.2e-30) {
		tmp = t_1;
	} else if (y <= 5e-18) {
		tmp = x * (1.0 - (a * b));
	} else if (y <= 9e+157) {
		tmp = t_1;
	} else {
		tmp = x * (0.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * ((y * y) * (0.5 * (t * t)))
	tmp = 0
	if y <= -8.2e-30:
		tmp = t_1
	elif y <= 5e-18:
		tmp = x * (1.0 - (a * b))
	elif y <= 9e+157:
		tmp = t_1
	else:
		tmp = x * (0.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(Float64(y * y) * Float64(0.5 * Float64(t * t))))
	tmp = 0.0
	if (y <= -8.2e-30)
		tmp = t_1;
	elseif (y <= 5e-18)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	elseif (y <= 9e+157)
		tmp = t_1;
	else
		tmp = Float64(x * Float64(0.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * ((y * y) * (0.5 * (t * t)));
	tmp = 0.0;
	if (y <= -8.2e-30)
		tmp = t_1;
	elseif (y <= 5e-18)
		tmp = x * (1.0 - (a * b));
	elseif (y <= 9e+157)
		tmp = t_1;
	else
		tmp = x * (0.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(N[(y * y), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.2e-30], t$95$1, If[LessEqual[y, 5e-18], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e+157], t$95$1, N[(x * N[(0.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.2000000000000007e-30 or 5.00000000000000036e-18 < y < 8.9999999999999997e157

   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 t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 55.7%

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

   5. Simplified 55.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. unsub-neg N/A

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

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

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

      7. associate-*r* N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 33.0%

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

   8. Simplified 33.0%

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

   9. Taylor expanded in y around -inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

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

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

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

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

       9. unpow2 N/A

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

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

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

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

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

       12. sub-neg N/A

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

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

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

       14. distribute-neg-frac N/A

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

       15. metadata-eval N/A

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

       16. /-lowering-/.f64 38.2%

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

   11. Simplified 38.2%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right) - \frac{t + \frac{-1}{y}}{y}\right)\right)} \]

   12. Taylor expanded in t around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 44.2%

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

   14. Simplified 44.2%

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

   if -8.2000000000000007e-30 < y < 5.00000000000000036e-18

   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 80.5%

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

   5. Simplified 80.5%

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

   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 43.9%

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

   8. Simplified 43.9%

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

   if 8.9999999999999997e157 < y

   1. Initial program 93.3%

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

   3. Taylor expanded in 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 25.9%

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

   5. Simplified 25.9%

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

   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 13.0%

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

   8. Simplified 13.0%

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

   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. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{a \cdot \left(b \cdot x\right)} \]

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

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 43.2%

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

   11. Simplified 43.2%

       \[\leadsto \color{blue}{0 - x \cdot \left(a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \left(1 + t \cdot \left(0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y\right)\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-202}:\\ \;\;\;\;\frac{x}{1 + b \cdot \left(a + b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -8e+77)
   (* x (+ 1.0 (* t (- (* 0.5 (* t (* y y))) y))))
   (if (<= y 1.55e-202)
     (/ x (+ 1.0 (* b (+ a (* b (* 0.5 (* a a)))))))
     (* x (* b (* b (* b (* (* a (* a a)) -0.16666666666666666))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8e+77) {
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	} else if (y <= 1.55e-202) {
		tmp = x / (1.0 + (b * (a + (b * (0.5 * (a * a))))));
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	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 <= (-8d+77)) then
        tmp = x * (1.0d0 + (t * ((0.5d0 * (t * (y * y))) - y)))
    else if (y <= 1.55d-202) then
        tmp = x / (1.0d0 + (b * (a + (b * (0.5d0 * (a * a))))))
    else
        tmp = x * (b * (b * (b * ((a * (a * a)) * (-0.16666666666666666d0)))))
    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 <= -8e+77) {
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	} else if (y <= 1.55e-202) {
		tmp = x / (1.0 + (b * (a + (b * (0.5 * (a * a))))));
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -8e+77:
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)))
	elif y <= 1.55e-202:
		tmp = x / (1.0 + (b * (a + (b * (0.5 * (a * a))))))
	else:
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -8e+77)
		tmp = Float64(x * Float64(1.0 + Float64(t * Float64(Float64(0.5 * Float64(t * Float64(y * y))) - y))));
	elseif (y <= 1.55e-202)
		tmp = Float64(x / Float64(1.0 + Float64(b * Float64(a + Float64(b * Float64(0.5 * Float64(a * a)))))));
	else
		tmp = Float64(x * Float64(b * Float64(b * Float64(b * Float64(Float64(a * Float64(a * a)) * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -8e+77)
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	elseif (y <= 1.55e-202)
		tmp = x / (1.0 + (b * (a + (b * (0.5 * (a * a))))));
	else
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -8e+77], N[(x * N[(1.0 + N[(t * N[(N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e-202], N[(x / N[(1.0 + N[(b * N[(a + N[(b * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(b * N[(b * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.99999999999999986e77

   1. Initial program 98.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 62.3%

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

   5. Simplified 62.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 60.1%

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

   8. Simplified 60.1%

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

   if -7.99999999999999986e77 < y < 1.55e-202

   1. Initial program 95.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 75.4%

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

   5. Simplified 75.4%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(a \cdot b\right)} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{e^{a \cdot b}}} \]

      3. un-div-inv N/A

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

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

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

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

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

      6. *-lowering-*.f64 75.4%

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

   7. Applied egg-rr 75.4%

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

   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. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

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

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

   10. Simplified 53.0%

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

   if 1.55e-202 < y

   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 50.0%

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

   5. Simplified 50.0%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

   8. Simplified 26.5%

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

   9. Taylor expanded in b around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 51.9%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   11. Simplified 51.9%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 47.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \left(1 + t \cdot \left(0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y\right)\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-202}:\\ \;\;\;\;\frac{x}{1 + a \cdot \left(b + 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5.6e+79)
   (* x (+ 1.0 (* t (- (* 0.5 (* t (* y y))) y))))
   (if (<= y 1.55e-202)
     (/ x (+ 1.0 (* a (+ b (* 0.5 (* a (* b b)))))))
     (* x (* b (* b (* b (* (* a (* a a)) -0.16666666666666666))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5.6e+79) {
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	} else if (y <= 1.55e-202) {
		tmp = x / (1.0 + (a * (b + (0.5 * (a * (b * b))))));
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	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 <= (-5.6d+79)) then
        tmp = x * (1.0d0 + (t * ((0.5d0 * (t * (y * y))) - y)))
    else if (y <= 1.55d-202) then
        tmp = x / (1.0d0 + (a * (b + (0.5d0 * (a * (b * b))))))
    else
        tmp = x * (b * (b * (b * ((a * (a * a)) * (-0.16666666666666666d0)))))
    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 <= -5.6e+79) {
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	} else if (y <= 1.55e-202) {
		tmp = x / (1.0 + (a * (b + (0.5 * (a * (b * b))))));
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5.6e+79:
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)))
	elif y <= 1.55e-202:
		tmp = x / (1.0 + (a * (b + (0.5 * (a * (b * b))))))
	else:
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -5.6e+79)
		tmp = Float64(x * Float64(1.0 + Float64(t * Float64(Float64(0.5 * Float64(t * Float64(y * y))) - y))));
	elseif (y <= 1.55e-202)
		tmp = Float64(x / Float64(1.0 + Float64(a * Float64(b + Float64(0.5 * Float64(a * Float64(b * b)))))));
	else
		tmp = Float64(x * Float64(b * Float64(b * Float64(b * Float64(Float64(a * Float64(a * a)) * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -5.6e+79)
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	elseif (y <= 1.55e-202)
		tmp = x / (1.0 + (a * (b + (0.5 * (a * (b * b))))));
	else
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5.6e+79], N[(x * N[(1.0 + N[(t * N[(N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e-202], N[(x / N[(1.0 + N[(a * N[(b + N[(0.5 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(b * N[(b * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.6000000000000002e79

   1. Initial program 98.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 62.3%

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

   5. Simplified 62.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 60.1%

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

   8. Simplified 60.1%

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

   if -5.6000000000000002e79 < y < 1.55e-202

   1. Initial program 95.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 75.4%

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

   5. Simplified 75.4%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(a \cdot b\right)} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{e^{a \cdot b}}} \]

      3. un-div-inv N/A

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

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

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

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

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

      6. *-lowering-*.f64 75.4%

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

   7. Applied egg-rr 75.4%

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

      1. exp-prod N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

      5. pow-exp N/A

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

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

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

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

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

      8. /-lowering-/.f64 75.4%

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

   9. Applied egg-rr 75.4%

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

   10. Taylor expanded in a around 0

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

       1. distribute-rgt-in N/A

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

       2. *-commutative N/A

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

       3. associate-+r+ N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. associate-*l* N/A

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

       12. associate-+r+ N/A

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

       13. distribute-rgt-in N/A

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

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

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

       15. distribute-rgt-in N/A

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

       16. *-commutative N/A

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

       17. associate-*l* N/A

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

       18. associate-*r* N/A

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

       19. unpow2 N/A

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

       20. *-commutative N/A

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

   12. Simplified 50.9%

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

   if 1.55e-202 < y

   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 50.0%

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

   5. Simplified 50.0%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

   8. Simplified 26.5%

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

   9. Taylor expanded in b around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 51.9%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   11. Simplified 51.9%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 48.3% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \left(1 + t \cdot \left(0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-192}:\\ \;\;\;\;x \cdot \left(1 + b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right) - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -8.2e+71)
   (* x (+ 1.0 (* t (- (* 0.5 (* t (* y y))) y))))
   (if (<= y 8.5e-192)
     (* x (+ 1.0 (* b (- (* 0.5 (* b (* a a))) a))))
     (* x (* b (* b (* b (* (* a (* a a)) -0.16666666666666666))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.2e+71) {
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	} else if (y <= 8.5e-192) {
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)));
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	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 <= (-8.2d+71)) then
        tmp = x * (1.0d0 + (t * ((0.5d0 * (t * (y * y))) - y)))
    else if (y <= 8.5d-192) then
        tmp = x * (1.0d0 + (b * ((0.5d0 * (b * (a * a))) - a)))
    else
        tmp = x * (b * (b * (b * ((a * (a * a)) * (-0.16666666666666666d0)))))
    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 <= -8.2e+71) {
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	} else if (y <= 8.5e-192) {
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)));
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -8.2e+71:
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)))
	elif y <= 8.5e-192:
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)))
	else:
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -8.2e+71)
		tmp = Float64(x * Float64(1.0 + Float64(t * Float64(Float64(0.5 * Float64(t * Float64(y * y))) - y))));
	elseif (y <= 8.5e-192)
		tmp = Float64(x * Float64(1.0 + Float64(b * Float64(Float64(0.5 * Float64(b * Float64(a * a))) - a))));
	else
		tmp = Float64(x * Float64(b * Float64(b * Float64(b * Float64(Float64(a * Float64(a * a)) * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -8.2e+71)
		tmp = x * (1.0 + (t * ((0.5 * (t * (y * y))) - y)));
	elseif (y <= 8.5e-192)
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)));
	else
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -8.2e+71], N[(x * N[(1.0 + N[(t * N[(N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e-192], N[(x * N[(1.0 + N[(b * N[(N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(b * N[(b * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.2000000000000004e71

   1. Initial program 98.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 61.8%

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

   5. Simplified 61.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 57.8%

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

   8. Simplified 57.8%

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

   if -8.2000000000000004e71 < y < 8.49999999999999985e-192

   1. Initial program 95.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 48.1%

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

   8. Simplified 48.1%

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

   if 8.49999999999999985e-192 < y

   1. Initial program 95.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 48.9%

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

   5. Simplified 48.9%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

   8. Simplified 25.3%

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

   9. Taylor expanded in b around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 52.0%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   11. Simplified 52.0%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 45.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-192}:\\ \;\;\;\;x \cdot \left(1 + b \cdot \left(0.5 \cdot \left(b \cdot \left(a \cdot a\right)\right) - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5.7e+25)
   (* x (* (* y y) (* 0.5 (* t t))))
   (if (<= y 8.5e-192)
     (* x (+ 1.0 (* b (- (* 0.5 (* b (* a a))) a))))
     (* x (* b (* b (* b (* (* a (* a a)) -0.16666666666666666))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5.7e+25) {
		tmp = x * ((y * y) * (0.5 * (t * t)));
	} else if (y <= 8.5e-192) {
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)));
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	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 <= (-5.7d+25)) then
        tmp = x * ((y * y) * (0.5d0 * (t * t)))
    else if (y <= 8.5d-192) then
        tmp = x * (1.0d0 + (b * ((0.5d0 * (b * (a * a))) - a)))
    else
        tmp = x * (b * (b * (b * ((a * (a * a)) * (-0.16666666666666666d0)))))
    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 <= -5.7e+25) {
		tmp = x * ((y * y) * (0.5 * (t * t)));
	} else if (y <= 8.5e-192) {
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)));
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5.7e+25:
		tmp = x * ((y * y) * (0.5 * (t * t)))
	elif y <= 8.5e-192:
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)))
	else:
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -5.7e+25)
		tmp = Float64(x * Float64(Float64(y * y) * Float64(0.5 * Float64(t * t))));
	elseif (y <= 8.5e-192)
		tmp = Float64(x * Float64(1.0 + Float64(b * Float64(Float64(0.5 * Float64(b * Float64(a * a))) - a))));
	else
		tmp = Float64(x * Float64(b * Float64(b * Float64(b * Float64(Float64(a * Float64(a * a)) * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -5.7e+25)
		tmp = x * ((y * y) * (0.5 * (t * t)));
	elseif (y <= 8.5e-192)
		tmp = x * (1.0 + (b * ((0.5 * (b * (a * a))) - a)));
	else
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5.7e+25], N[(x * N[(N[(y * y), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e-192], N[(x * N[(1.0 + N[(b * N[(N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(b * N[(b * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.6999999999999996e25

   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 t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 61.5%

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

   5. Simplified 61.5%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. unsub-neg N/A

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

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

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

      7. associate-*r* N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 35.3%

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

   8. Simplified 35.3%

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

   9. Taylor expanded in y around -inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

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

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

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

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

       9. unpow2 N/A

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

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

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

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

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

       12. sub-neg N/A

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

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

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

       14. distribute-neg-frac N/A

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

       15. metadata-eval N/A

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

       16. /-lowering-/.f64 44.0%

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

   11. Simplified 44.0%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right) - \frac{t + \frac{-1}{y}}{y}\right)\right)} \]

   12. Taylor expanded in t around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 46.9%

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

   14. Simplified 46.9%

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

   if -5.6999999999999996e25 < y < 8.49999999999999985e-192

   1. Initial program 95.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 79.9%

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

   5. Simplified 79.9%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 51.8%

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

   8. Simplified 51.8%

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

   if 8.49999999999999985e-192 < y

   1. Initial program 95.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 48.9%

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

   5. Simplified 48.9%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

   8. Simplified 25.3%

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

   9. Taylor expanded in b around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 52.0%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   11. Simplified 52.0%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 44.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot 0.5\right) - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.6e-33)
   (* x (* (* y y) (* 0.5 (* t t))))
   (if (<= y 1.55e-202)
     (* x (+ 1.0 (* a (- (* (* b b) (* a 0.5)) b))))
     (* x (* b (* b (* b (* (* a (* a a)) -0.16666666666666666))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.6e-33) {
		tmp = x * ((y * y) * (0.5 * (t * t)));
	} else if (y <= 1.55e-202) {
		tmp = x * (1.0 + (a * (((b * b) * (a * 0.5)) - b)));
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	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 <= (-6.6d-33)) then
        tmp = x * ((y * y) * (0.5d0 * (t * t)))
    else if (y <= 1.55d-202) then
        tmp = x * (1.0d0 + (a * (((b * b) * (a * 0.5d0)) - b)))
    else
        tmp = x * (b * (b * (b * ((a * (a * a)) * (-0.16666666666666666d0)))))
    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 <= -6.6e-33) {
		tmp = x * ((y * y) * (0.5 * (t * t)));
	} else if (y <= 1.55e-202) {
		tmp = x * (1.0 + (a * (((b * b) * (a * 0.5)) - b)));
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.6e-33:
		tmp = x * ((y * y) * (0.5 * (t * t)))
	elif y <= 1.55e-202:
		tmp = x * (1.0 + (a * (((b * b) * (a * 0.5)) - b)))
	else:
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.6e-33)
		tmp = Float64(x * Float64(Float64(y * y) * Float64(0.5 * Float64(t * t))));
	elseif (y <= 1.55e-202)
		tmp = Float64(x * Float64(1.0 + Float64(a * Float64(Float64(Float64(b * b) * Float64(a * 0.5)) - b))));
	else
		tmp = Float64(x * Float64(b * Float64(b * Float64(b * Float64(Float64(a * Float64(a * a)) * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.6e-33)
		tmp = x * ((y * y) * (0.5 * (t * t)));
	elseif (y <= 1.55e-202)
		tmp = x * (1.0 + (a * (((b * b) * (a * 0.5)) - b)));
	else
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.6e-33], N[(x * N[(N[(y * y), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e-202], N[(x * N[(1.0 + N[(a * N[(N[(N[(b * b), $MachinePrecision] * N[(a * 0.5), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(b * N[(b * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.6000000000000005e-33

   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 t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 58.8%

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

   5. Simplified 58.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. unsub-neg N/A

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

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

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

      7. associate-*r* N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 33.0%

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

   8. Simplified 33.0%

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

   9. Taylor expanded in y around -inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

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

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

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

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

       9. unpow2 N/A

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

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

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

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

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

       12. sub-neg N/A

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

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

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

       14. distribute-neg-frac N/A

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

       15. metadata-eval N/A

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

       16. /-lowering-/.f64 40.3%

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

   11. Simplified 40.3%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right) - \frac{t + \frac{-1}{y}}{y}\right)\right)} \]

   12. Taylor expanded in t around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 43.7%

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

   14. Simplified 43.7%

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

   if -6.6000000000000005e-33 < y < 1.55e-202

   1. Initial program 94.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 84.1%

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

   5. Simplified 84.1%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 51.6%

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

   8. Simplified 51.6%

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

   if 1.55e-202 < y

   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 50.0%

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

   5. Simplified 50.0%

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

   6. Taylor expanded in b around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right) + \color{blue}{-1 \cdot a}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right) - \color{blue}{a}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{\_.f64}\left(\left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right), \color{blue}{a}\right)\right)\right)\right) \]

   8. Simplified 26.5%

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(-0.16666666666666666 \cdot \left(b \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + 0.5 \cdot \left(a \cdot a\right)\right) - a\right)\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot \left({a}^{3} \cdot {b}^{3}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot \color{blue}{{b}^{3}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({b}^{3} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {a}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\color{blue}{\frac{-1}{6}} \cdot {a}^{3}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(b \cdot {b}^{2}\right) \cdot \left(\frac{-1}{6} \cdot {a}^{3}\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left(\frac{-1}{6} \cdot {a}^{3}\right)\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(b \cdot \left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \color{blue}{\left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot {b}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \left(\left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot b\right) \cdot \color{blue}{b}\right)\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \left(\left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right)\right) \cdot b\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right)\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right)\right)}\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot \color{blue}{b}\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left(\frac{-1}{6} \cdot {a}^{3}\right)}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{-1}{6} \cdot {a}^{3}\right)}\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({a}^{3} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left({a}^{3}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

       18. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(a \cdot {a}^{2}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       22. *-lowering-*.f64 51.9%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   11. Simplified 51.9%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot 0.5\right) - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.8% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-202}:\\ \;\;\;\;\frac{x}{a \cdot b + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.9e-30)
   (* x (* (* y y) (* 0.5 (* t t))))
   (if (<= y 1.55e-202)
     (/ x (+ (* a b) 1.0))
     (* x (* b (* b (* b (* (* a (* a a)) -0.16666666666666666))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.9e-30) {
		tmp = x * ((y * y) * (0.5 * (t * t)));
	} else if (y <= 1.55e-202) {
		tmp = x / ((a * b) + 1.0);
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	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 <= (-2.9d-30)) then
        tmp = x * ((y * y) * (0.5d0 * (t * t)))
    else if (y <= 1.55d-202) then
        tmp = x / ((a * b) + 1.0d0)
    else
        tmp = x * (b * (b * (b * ((a * (a * a)) * (-0.16666666666666666d0)))))
    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 <= -2.9e-30) {
		tmp = x * ((y * y) * (0.5 * (t * t)));
	} else if (y <= 1.55e-202) {
		tmp = x / ((a * b) + 1.0);
	} else {
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.9e-30:
		tmp = x * ((y * y) * (0.5 * (t * t)))
	elif y <= 1.55e-202:
		tmp = x / ((a * b) + 1.0)
	else:
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.9e-30)
		tmp = Float64(x * Float64(Float64(y * y) * Float64(0.5 * Float64(t * t))));
	elseif (y <= 1.55e-202)
		tmp = Float64(x / Float64(Float64(a * b) + 1.0));
	else
		tmp = Float64(x * Float64(b * Float64(b * Float64(b * Float64(Float64(a * Float64(a * a)) * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.9e-30)
		tmp = x * ((y * y) * (0.5 * (t * t)));
	elseif (y <= 1.55e-202)
		tmp = x / ((a * b) + 1.0);
	else
		tmp = x * (b * (b * (b * ((a * (a * a)) * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.9e-30], N[(x * N[(N[(y * y), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e-202], N[(x / N[(N[(a * b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(b * N[(b * N[(b * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.89999999999999989e-30

   1. Initial program 97.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(t \cdot y\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t \cdot y\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - t \cdot y\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot t\right)\right)\right)\right) \]

      5. *-lowering-*.f64 58.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, t\right)\right)\right)\right) \]

   5. Simplified 58.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(t\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left({t}^{2} \cdot y\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) - \color{blue}{t}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right), \color{blue}{t}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot y\right), t\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {t}^{2}\right), y\right), t\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({t}^{2}\right)\right), y\right), t\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right), y\right), t\right)\right)\right)\right) \]

      11. *-lowering-*.f64 32.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), y\right), t\right)\right)\right)\right) \]

   8. Simplified 32.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot y - t\right)\right)} \]

   9. Taylor expanded in y around -inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2} \cdot \left(-1 \cdot \frac{t - \frac{1}{y}}{y} + \frac{1}{2} \cdot {t}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(-1 \cdot \frac{t - \frac{1}{y}}{y} + \frac{1}{2} \cdot {t}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{-1 \cdot \frac{t - \frac{1}{y}}{y}} + \frac{1}{2} \cdot {t}^{2}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{-1 \cdot \frac{t - \frac{1}{y}}{y}} + \frac{1}{2} \cdot {t}^{2}\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{-1 \cdot \frac{t - \frac{1}{y}}{y}}\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{2} \cdot {t}^{2} + \left(\mathsf{neg}\left(\frac{t - \frac{1}{y}}{y}\right)\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{2} \cdot {t}^{2} - \color{blue}{\frac{t - \frac{1}{y}}{y}}\right)\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot {t}^{2}\right), \color{blue}{\left(\frac{t - \frac{1}{y}}{y}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({t}^{2}\right)\right), \left(\frac{\color{blue}{t - \frac{1}{y}}}{y}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right), \left(\frac{t - \color{blue}{\frac{1}{y}}}{y}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{t - \color{blue}{\frac{1}{y}}}{y}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(t - \frac{1}{y}\right), \color{blue}{y}\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(t + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right), y\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right), y\right)\right)\right)\right) \]

       14. distribute-neg-frac N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{\mathsf{neg}\left(1\right)}{y}\right)\right), y\right)\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{-1}{y}\right)\right), y\right)\right)\right)\right) \]

       16. /-lowering-/.f64 39.5%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(-1, y\right)\right), y\right)\right)\right)\right) \]

   11. Simplified 39.5%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right) - \frac{t + \frac{-1}{y}}{y}\right)\right)} \]

   12. Taylor expanded in t around inf

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \color{blue}{\left(\frac{1}{2} \cdot {t}^{2}\right)}\right)\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 44.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   14. Simplified 44.2%

       \[\leadsto x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right)}\right) \]

   if -2.89999999999999989e-30 < y < 1.55e-202

   1. Initial program 94.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 84.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 84.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(a \cdot b\right)} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{e^{a \cdot b}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{e^{a \cdot b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(e^{a \cdot b}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(a \cdot b\right)\right)\right) \]

      6. *-lowering-*.f64 84.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Applied egg-rr 84.2%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot b}}} \]

   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 50.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, b\right), 1\right)\right) \]

   10. Simplified 50.0%

       \[\leadsto \frac{x}{\color{blue}{a \cdot b + 1}} \]

   if 1.55e-202 < y

   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 50.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 50.0%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right) + \color{blue}{-1 \cdot a}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right) - \color{blue}{a}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{\_.f64}\left(\left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right), \color{blue}{a}\right)\right)\right)\right) \]

   8. Simplified 26.5%

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(-0.16666666666666666 \cdot \left(b \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + 0.5 \cdot \left(a \cdot a\right)\right) - a\right)\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot \left({a}^{3} \cdot {b}^{3}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot \color{blue}{{b}^{3}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({b}^{3} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {a}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\color{blue}{\frac{-1}{6}} \cdot {a}^{3}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(b \cdot {b}^{2}\right) \cdot \left(\frac{-1}{6} \cdot {a}^{3}\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left(\frac{-1}{6} \cdot {a}^{3}\right)\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(b \cdot \left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \color{blue}{\left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot {b}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \left(\left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot b\right) \cdot \color{blue}{b}\right)\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \left(\left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right)\right) \cdot b\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right)\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right)\right)}\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(\left(\frac{-1}{6} \cdot {a}^{3}\right) \cdot \color{blue}{b}\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left(\frac{-1}{6} \cdot {a}^{3}\right)}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{-1}{6} \cdot {a}^{3}\right)}\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({a}^{3} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left({a}^{3}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

       18. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(a \cdot {a}^{2}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       22. *-lowering-*.f64 51.9%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   11. Simplified 51.9%

       \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.16666666666666666\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 42.5% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-215}:\\ \;\;\;\;\frac{x}{a \cdot b + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(-0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4e-29)
   (* x (* (* y y) (* 0.5 (* t t))))
   (if (<= y 2.4e-215)
     (/ x (+ (* a b) 1.0))
     (* x (* a (* a (* a (* -0.16666666666666666 (* b (* b b))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4e-29) {
		tmp = x * ((y * y) * (0.5 * (t * t)));
	} else if (y <= 2.4e-215) {
		tmp = x / ((a * b) + 1.0);
	} else {
		tmp = x * (a * (a * (a * (-0.16666666666666666 * (b * (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) :: tmp
    if (y <= (-4d-29)) then
        tmp = x * ((y * y) * (0.5d0 * (t * t)))
    else if (y <= 2.4d-215) then
        tmp = x / ((a * b) + 1.0d0)
    else
        tmp = x * (a * (a * (a * ((-0.16666666666666666d0) * (b * (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 tmp;
	if (y <= -4e-29) {
		tmp = x * ((y * y) * (0.5 * (t * t)));
	} else if (y <= 2.4e-215) {
		tmp = x / ((a * b) + 1.0);
	} else {
		tmp = x * (a * (a * (a * (-0.16666666666666666 * (b * (b * b))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4e-29:
		tmp = x * ((y * y) * (0.5 * (t * t)))
	elif y <= 2.4e-215:
		tmp = x / ((a * b) + 1.0)
	else:
		tmp = x * (a * (a * (a * (-0.16666666666666666 * (b * (b * b))))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4e-29)
		tmp = Float64(x * Float64(Float64(y * y) * Float64(0.5 * Float64(t * t))));
	elseif (y <= 2.4e-215)
		tmp = Float64(x / Float64(Float64(a * b) + 1.0));
	else
		tmp = Float64(x * Float64(a * Float64(a * Float64(a * Float64(-0.16666666666666666 * Float64(b * Float64(b * b)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4e-29)
		tmp = x * ((y * y) * (0.5 * (t * t)));
	elseif (y <= 2.4e-215)
		tmp = x / ((a * b) + 1.0);
	else
		tmp = x * (a * (a * (a * (-0.16666666666666666 * (b * (b * b))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4e-29], N[(x * N[(N[(y * y), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e-215], N[(x / N[(N[(a * b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * N[(a * N[(a * N[(-0.16666666666666666 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.99999999999999977e-29

   1. Initial program 97.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(t \cdot y\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t \cdot y\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - t \cdot y\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot t\right)\right)\right)\right) \]

      5. *-lowering-*.f64 58.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, t\right)\right)\right)\right) \]

   5. Simplified 58.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(t\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left({t}^{2} \cdot y\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) - \color{blue}{t}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right), \color{blue}{t}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot y\right), t\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {t}^{2}\right), y\right), t\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({t}^{2}\right)\right), y\right), t\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right), y\right), t\right)\right)\right)\right) \]

      11. *-lowering-*.f64 32.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), y\right), t\right)\right)\right)\right) \]

   8. Simplified 32.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot y - t\right)\right)} \]

   9. Taylor expanded in y around -inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2} \cdot \left(-1 \cdot \frac{t - \frac{1}{y}}{y} + \frac{1}{2} \cdot {t}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(-1 \cdot \frac{t - \frac{1}{y}}{y} + \frac{1}{2} \cdot {t}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{-1 \cdot \frac{t - \frac{1}{y}}{y}} + \frac{1}{2} \cdot {t}^{2}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{-1 \cdot \frac{t - \frac{1}{y}}{y}} + \frac{1}{2} \cdot {t}^{2}\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{-1 \cdot \frac{t - \frac{1}{y}}{y}}\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{2} \cdot {t}^{2} + \left(\mathsf{neg}\left(\frac{t - \frac{1}{y}}{y}\right)\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{2} \cdot {t}^{2} - \color{blue}{\frac{t - \frac{1}{y}}{y}}\right)\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot {t}^{2}\right), \color{blue}{\left(\frac{t - \frac{1}{y}}{y}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({t}^{2}\right)\right), \left(\frac{\color{blue}{t - \frac{1}{y}}}{y}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right), \left(\frac{t - \color{blue}{\frac{1}{y}}}{y}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{t - \color{blue}{\frac{1}{y}}}{y}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(t - \frac{1}{y}\right), \color{blue}{y}\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(t + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right), y\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right), y\right)\right)\right)\right) \]

       14. distribute-neg-frac N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{\mathsf{neg}\left(1\right)}{y}\right)\right), y\right)\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{-1}{y}\right)\right), y\right)\right)\right)\right) \]

       16. /-lowering-/.f64 39.5%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(-1, y\right)\right), y\right)\right)\right)\right) \]

   11. Simplified 39.5%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right) - \frac{t + \frac{-1}{y}}{y}\right)\right)} \]

   12. Taylor expanded in t around inf

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \color{blue}{\left(\frac{1}{2} \cdot {t}^{2}\right)}\right)\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 44.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   14. Simplified 44.2%

       \[\leadsto x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right)}\right) \]

   if -3.99999999999999977e-29 < y < 2.4000000000000001e-215

   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 85.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 85.5%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(a \cdot b\right)} \]

      2. exp-neg N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{e^{a \cdot b}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{e^{a \cdot b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(e^{a \cdot b}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\left(a \cdot b\right)\right)\right) \]

      6. *-lowering-*.f64 85.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. Applied egg-rr 85.5%

      \[\leadsto \color{blue}{\frac{x}{e^{a \cdot b}}} \]

   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 52.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, b\right), 1\right)\right) \]

   10. Simplified 52.4%

       \[\leadsto \frac{x}{\color{blue}{a \cdot b + 1}} \]

   if 2.4000000000000001e-215 < y

   1. Initial program 95.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 51.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 51.1%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right) + \color{blue}{-1 \cdot a}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right) - \color{blue}{a}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{\_.f64}\left(\left(b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right), \color{blue}{a}\right)\right)\right)\right) \]

   8. Simplified 26.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(-0.16666666666666666 \cdot \left(b \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + 0.5 \cdot \left(a \cdot a\right)\right) - a\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({a}^{3} \cdot \left(-1 \cdot \frac{b}{{a}^{2}} + \left(\frac{-1}{6} \cdot {b}^{3} + \left(\frac{1}{2} \cdot \frac{{b}^{2}}{a} + \frac{1}{{a}^{3}}\right)\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{-1 \cdot \frac{b}{{a}^{2}}} + \left(\frac{-1}{6} \cdot {b}^{3} + \left(\frac{1}{2} \cdot \frac{{b}^{2}}{a} + \frac{1}{{a}^{3}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(a \cdot {a}^{2}\right) \cdot \left(-1 \cdot \color{blue}{\frac{b}{{a}^{2}}} + \left(\frac{-1}{6} \cdot {b}^{3} + \left(\frac{1}{2} \cdot \frac{{b}^{2}}{a} + \frac{1}{{a}^{3}}\right)\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(a \cdot \color{blue}{\left({a}^{2} \cdot \left(-1 \cdot \frac{b}{{a}^{2}} + \left(\frac{-1}{6} \cdot {b}^{3} + \left(\frac{1}{2} \cdot \frac{{b}^{2}}{a} + \frac{1}{{a}^{3}}\right)\right)\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2} \cdot \left(-1 \cdot \frac{b}{{a}^{2}} + \left(\frac{-1}{6} \cdot {b}^{3} + \left(\frac{1}{2} \cdot \frac{{b}^{2}}{a} + \frac{1}{{a}^{3}}\right)\right)\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(-1 \cdot \frac{b}{{a}^{2}} + \left(\frac{-1}{6} \cdot {b}^{3} + \left(\frac{1}{2} \cdot \frac{{b}^{2}}{a} + \frac{1}{{a}^{3}}\right)\right)\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{-1 \cdot \frac{b}{{a}^{2}}} + \left(\frac{-1}{6} \cdot {b}^{3} + \left(\frac{1}{2} \cdot \frac{{b}^{2}}{a} + \frac{1}{{a}^{3}}\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{-1 \cdot \frac{b}{{a}^{2}}} + \left(\frac{-1}{6} \cdot {b}^{3} + \left(\frac{1}{2} \cdot \frac{{b}^{2}}{a} + \frac{1}{{a}^{3}}\right)\right)\right)\right)\right)\right) \]

       8. associate-+r+ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(-1 \cdot \frac{b}{{a}^{2}} + \left(\left(\frac{-1}{6} \cdot {b}^{3} + \frac{1}{2} \cdot \frac{{b}^{2}}{a}\right) + \color{blue}{\frac{1}{{a}^{3}}}\right)\right)\right)\right)\right) \]

       9. associate-+r+ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(-1 \cdot \frac{b}{{a}^{2}} + \left(\frac{-1}{6} \cdot {b}^{3} + \frac{1}{2} \cdot \frac{{b}^{2}}{a}\right)\right) + \color{blue}{\frac{1}{{a}^{3}}}\right)\right)\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\left(-1 \cdot \frac{b}{{a}^{2}} + \left(\frac{-1}{6} \cdot {b}^{3} + \frac{1}{2} \cdot \frac{{b}^{2}}{a}\right)\right), \color{blue}{\left(\frac{1}{{a}^{3}}\right)}\right)\right)\right)\right) \]

   11. Simplified 20.6%

       \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\left(a \cdot a\right) \cdot \left(\left(\left(b \cdot \left(\left(b \cdot b\right) \cdot -0.16666666666666666\right) + \frac{\left(b \cdot b\right) \cdot 0.5}{a}\right) - \frac{b}{a \cdot a}\right) + \frac{1}{a \cdot \left(a \cdot a\right)}\right)\right)\right)} \]

   12. Taylor expanded in a around inf

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{6} \cdot \left({a}^{2} \cdot {b}^{3}\right)\right)}\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(\left({a}^{2} \cdot {b}^{3}\right) \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(\left(\left(a \cdot a\right) \cdot {b}^{3}\right) \cdot \frac{-1}{6}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(\left(a \cdot \left(a \cdot {b}^{3}\right)\right) \cdot \frac{-1}{6}\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(\left(a \cdot {b}^{3}\right) \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(a \cdot {b}^{3}\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right)\right)}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(\left(a \cdot {b}^{3}\right) \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left({b}^{3} \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{-1}{6} \cdot \color{blue}{{b}^{3}}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{6} \cdot {b}^{3}\right)}\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({b}^{3} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left({b}^{3}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(b \cdot \left(b \cdot b\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(b \cdot {b}^{2}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left({b}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot b\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 47.2%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   14. Simplified 47.2%

       \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot -0.16666666666666666\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-215}:\\ \;\;\;\;\frac{x}{a \cdot b + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(-0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.9% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot t\_1\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.5 (* t t))))
   (if (<= y -6.2e-30)
     (* x (* (* y y) t_1))
     (if (<= y 1.5e-17) (* x (- 1.0 (* a b))) (* x (* y (* y t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (t * t);
	double tmp;
	if (y <= -6.2e-30) {
		tmp = x * ((y * y) * t_1);
	} else if (y <= 1.5e-17) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (y * (y * 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 = 0.5d0 * (t * t)
    if (y <= (-6.2d-30)) then
        tmp = x * ((y * y) * t_1)
    else if (y <= 1.5d-17) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = x * (y * (y * 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 = 0.5 * (t * t);
	double tmp;
	if (y <= -6.2e-30) {
		tmp = x * ((y * y) * t_1);
	} else if (y <= 1.5e-17) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (y * (y * t_1));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.5 * (t * t)
	tmp = 0
	if y <= -6.2e-30:
		tmp = x * ((y * y) * t_1)
	elif y <= 1.5e-17:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = x * (y * (y * t_1))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.5 * Float64(t * t))
	tmp = 0.0
	if (y <= -6.2e-30)
		tmp = Float64(x * Float64(Float64(y * y) * t_1));
	elseif (y <= 1.5e-17)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(x * Float64(y * Float64(y * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.5 * (t * t);
	tmp = 0.0;
	if (y <= -6.2e-30)
		tmp = x * ((y * y) * t_1);
	elseif (y <= 1.5e-17)
		tmp = x * (1.0 - (a * b));
	else
		tmp = x * (y * (y * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e-30], N[(x * N[(N[(y * y), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e-17], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.19999999999999982e-30

   1. Initial program 97.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(t \cdot y\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t \cdot y\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - t \cdot y\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot t\right)\right)\right)\right) \]

      5. *-lowering-*.f64 58.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, t\right)\right)\right)\right) \]

   5. Simplified 58.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(t\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left({t}^{2} \cdot y\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) - \color{blue}{t}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right), \color{blue}{t}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot y\right), t\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {t}^{2}\right), y\right), t\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({t}^{2}\right)\right), y\right), t\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right), y\right), t\right)\right)\right)\right) \]

      11. *-lowering-*.f64 32.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), y\right), t\right)\right)\right)\right) \]

   8. Simplified 32.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot y - t\right)\right)} \]

   9. Taylor expanded in y around -inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2} \cdot \left(-1 \cdot \frac{t - \frac{1}{y}}{y} + \frac{1}{2} \cdot {t}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(-1 \cdot \frac{t - \frac{1}{y}}{y} + \frac{1}{2} \cdot {t}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{-1 \cdot \frac{t - \frac{1}{y}}{y}} + \frac{1}{2} \cdot {t}^{2}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{-1 \cdot \frac{t - \frac{1}{y}}{y}} + \frac{1}{2} \cdot {t}^{2}\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{-1 \cdot \frac{t - \frac{1}{y}}{y}}\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{2} \cdot {t}^{2} + \left(\mathsf{neg}\left(\frac{t - \frac{1}{y}}{y}\right)\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{2} \cdot {t}^{2} - \color{blue}{\frac{t - \frac{1}{y}}{y}}\right)\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot {t}^{2}\right), \color{blue}{\left(\frac{t - \frac{1}{y}}{y}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({t}^{2}\right)\right), \left(\frac{\color{blue}{t - \frac{1}{y}}}{y}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right), \left(\frac{t - \color{blue}{\frac{1}{y}}}{y}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{t - \color{blue}{\frac{1}{y}}}{y}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(t - \frac{1}{y}\right), \color{blue}{y}\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(t + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right), y\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right), y\right)\right)\right)\right) \]

       14. distribute-neg-frac N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{\mathsf{neg}\left(1\right)}{y}\right)\right), y\right)\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{-1}{y}\right)\right), y\right)\right)\right)\right) \]

       16. /-lowering-/.f64 39.5%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(-1, y\right)\right), y\right)\right)\right)\right) \]

   11. Simplified 39.5%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right) - \frac{t + \frac{-1}{y}}{y}\right)\right)} \]

   12. Taylor expanded in t around inf

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \color{blue}{\left(\frac{1}{2} \cdot {t}^{2}\right)}\right)\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 44.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   14. Simplified 44.2%

       \[\leadsto x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right)}\right) \]

   if -6.19999999999999982e-30 < y < 1.50000000000000003e-17

   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 80.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 80.5%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   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 43.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 43.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   if 1.50000000000000003e-17 < y

   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 t around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(t \cdot y\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t \cdot y\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - t \cdot y\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot t\right)\right)\right)\right) \]

      5. *-lowering-*.f64 55.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, t\right)\right)\right)\right) \]

   5. Simplified 55.2%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(t\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left({t}^{2} \cdot y\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) - \color{blue}{t}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right), \color{blue}{t}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot y\right), t\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {t}^{2}\right), y\right), t\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({t}^{2}\right)\right), y\right), t\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right), y\right), t\right)\right)\right)\right) \]

      11. *-lowering-*.f64 30.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), y\right), t\right)\right)\right)\right) \]

   8. Simplified 30.5%

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot y - t\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left({t}^{2} \cdot x\right) \cdot \color{blue}{{y}^{2}}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot \color{blue}{{y}^{2}} \]

       3. *-commutative N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right)} \]

       4. associate-*r* N/A

          \[\leadsto {y}^{2} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{2} \cdot {t}^{2}\right)\right) \cdot \color{blue}{x} \]

       6. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{2}\right) \cdot {t}^{2}\right) \cdot x \]

       7. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot {t}^{2}\right) \cdot x \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot {t}^{2}\right), \color{blue}{x}\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left({y}^{2} \cdot {t}^{2}\right)\right), x\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right), x\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(y \cdot y\right)\right)\right), x\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \left(\left({t}^{2} \cdot y\right) \cdot y\right)\right), x\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right), x\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right), x\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right), x\right) \]

       16. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot y\right)\right), x\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} \cdot {t}^{2}\right)\right)\right), x\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot {t}^{2}\right)\right)\right), x\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \left({t}^{2}\right)\right)\right)\right), x\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right)\right)\right), x\right) \]

       21. *-lowering-*.f64 45.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), x\right) \]

   11. Simplified 45.9%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.8% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+22}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.2e+22)
   (- x (* t (* x y)))
   (if (<= y 1.1e-17) (* x (- 1.0 (* a b))) (* x (- 0.0 (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.2e+22) {
		tmp = x - (t * (x * y));
	} else if (y <= 1.1e-17) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (0.0 - (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 (y <= (-6.2d+22)) then
        tmp = x - (t * (x * y))
    else if (y <= 1.1d-17) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = x * (0.0d0 - (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 (y <= -6.2e+22) {
		tmp = x - (t * (x * y));
	} else if (y <= 1.1e-17) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (0.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.2e+22:
		tmp = x - (t * (x * y))
	elif y <= 1.1e-17:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = x * (0.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.2e+22)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	elseif (y <= 1.1e-17)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(x * Float64(0.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.2e+22)
		tmp = x - (t * (x * y));
	elseif (y <= 1.1e-17)
		tmp = x * (1.0 - (a * b));
	else
		tmp = x * (0.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.2e+22], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-17], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2000000000000004e22

   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 t around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(t \cdot y\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t \cdot y\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - t \cdot y\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot t\right)\right)\right)\right) \]

      5. *-lowering-*.f64 60.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, t\right)\right)\right)\right) \]

   5. Simplified 60.5%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. 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 27.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 27.0%

      \[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)} \]

   if -6.2000000000000004e22 < y < 1.1e-17

   1. Initial program 96.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 78.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 78.4%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   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 40.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 40.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   if 1.1e-17 < y

   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 36.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 36.9%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   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 9.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 9.2%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   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. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{a \cdot \left(b \cdot x\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(a \cdot \left(b \cdot x\right)\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\left(b \cdot x\right) \cdot \color{blue}{a}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\left(x \cdot b\right) \cdot a\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(x \cdot \color{blue}{\left(b \cdot a\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(x \cdot \left(a \cdot \color{blue}{b}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       9. *-lowering-*.f64 29.8%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 29.8%

       \[\leadsto \color{blue}{0 - x \cdot \left(a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+22}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 33.1% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-28}:\\ \;\;\;\;0 - y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.45e-28)
   (- 0.0 (* y (* x t)))
   (if (<= y 1.8e-16) (* x (- 1.0 (* a b))) (* x (- 0.0 (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.45e-28) {
		tmp = 0.0 - (y * (x * t));
	} else if (y <= 1.8e-16) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (0.0 - (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 (y <= (-1.45d-28)) then
        tmp = 0.0d0 - (y * (x * t))
    else if (y <= 1.8d-16) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = x * (0.0d0 - (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 (y <= -1.45e-28) {
		tmp = 0.0 - (y * (x * t));
	} else if (y <= 1.8e-16) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (0.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.45e-28:
		tmp = 0.0 - (y * (x * t))
	elif y <= 1.8e-16:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = x * (0.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.45e-28)
		tmp = Float64(0.0 - Float64(y * Float64(x * t)));
	elseif (y <= 1.8e-16)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(x * Float64(0.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.45e-28)
		tmp = 0.0 - (y * (x * t));
	elseif (y <= 1.8e-16)
		tmp = x * (1.0 - (a * b));
	else
		tmp = x * (0.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.45e-28], N[(0.0 - N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e-16], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45000000000000006e-28

   1. Initial program 97.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(t \cdot y\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t \cdot y\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - t \cdot y\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot t\right)\right)\right)\right) \]

      5. *-lowering-*.f64 58.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, t\right)\right)\right)\right) \]

   5. Simplified 58.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. 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 23.9%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 23.9%

      \[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto -1 \cdot \left(\left(t \cdot x\right) \cdot \color{blue}{y}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t \cdot x\right)\right) \cdot \color{blue}{y} \]

       3. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)}\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(t \cdot x\right)\right)\right) \]

       6. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(0 - \color{blue}{t \cdot x}\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(0, \color{blue}{\left(t \cdot x\right)}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(0, \left(x \cdot \color{blue}{t}\right)\right)\right) \]

       9. *-lowering-*.f64 19.7%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \color{blue}{t}\right)\right)\right) \]

   11. Simplified 19.7%

       \[\leadsto \color{blue}{y \cdot \left(0 - x \cdot t\right)} \]

   if -1.45000000000000006e-28 < y < 1.79999999999999991e-16

   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 80.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 80.5%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   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 43.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 43.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   if 1.79999999999999991e-16 < y

   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 36.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 36.9%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   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 9.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 9.2%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   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. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{a \cdot \left(b \cdot x\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(a \cdot \left(b \cdot x\right)\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\left(b \cdot x\right) \cdot \color{blue}{a}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\left(x \cdot b\right) \cdot a\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(x \cdot \color{blue}{\left(b \cdot a\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(x \cdot \left(a \cdot \color{blue}{b}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       9. *-lowering-*.f64 29.8%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 29.8%

       \[\leadsto \color{blue}{0 - x \cdot \left(a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-28}:\\ \;\;\;\;0 - y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - y \cdot \left(x \cdot t\right)\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{-28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 0.0 (* y (* x t)))))
   (if (<= y -1.25e-28) t_1 (if (<= y 1.75e-16) (* 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 = 0.0 - (y * (x * t));
	double tmp;
	if (y <= -1.25e-28) {
		tmp = t_1;
	} else if (y <= 1.75e-16) {
		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 = 0.0d0 - (y * (x * t))
    if (y <= (-1.25d-28)) then
        tmp = t_1
    else if (y <= 1.75d-16) 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 = 0.0 - (y * (x * t));
	double tmp;
	if (y <= -1.25e-28) {
		tmp = t_1;
	} else if (y <= 1.75e-16) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.0 - (y * (x * t))
	tmp = 0
	if y <= -1.25e-28:
		tmp = t_1
	elif y <= 1.75e-16:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.0 - Float64(y * Float64(x * t)))
	tmp = 0.0
	if (y <= -1.25e-28)
		tmp = t_1;
	elseif (y <= 1.75e-16)
		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 = 0.0 - (y * (x * t));
	tmp = 0.0;
	if (y <= -1.25e-28)
		tmp = t_1;
	elseif (y <= 1.75e-16)
		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[(0.0 - N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e-28], t$95$1, If[LessEqual[y, 1.75e-16], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e-28 or 1.75000000000000009e-16 < y

   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 t around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(t \cdot y\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t \cdot y\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - t \cdot y\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot t\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, t\right)\right)\right)\right) \]

   5. Simplified 56.9%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. 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 17.2%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 17.2%

      \[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto -1 \cdot \left(\left(t \cdot x\right) \cdot \color{blue}{y}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t \cdot x\right)\right) \cdot \color{blue}{y} \]

       3. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)}\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(t \cdot x\right)\right)\right) \]

       6. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(0 - \color{blue}{t \cdot x}\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(0, \color{blue}{\left(t \cdot x\right)}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(0, \left(x \cdot \color{blue}{t}\right)\right)\right) \]

       9. *-lowering-*.f64 19.7%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \color{blue}{t}\right)\right)\right) \]

   11. Simplified 19.7%

       \[\leadsto \color{blue}{y \cdot \left(0 - x \cdot t\right)} \]

   if -1.25e-28 < y < 1.75000000000000009e-16

   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 80.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 80.5%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   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 43.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 43.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-28}:\\ \;\;\;\;0 - y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot \left(x \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 27.6% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-192}:\\ \;\;\;\;t \cdot \left(\frac{x}{t} - x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 8.5e-192) (* t (- (/ x t) (* x y))) (* x (- 0.0 (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 8.5e-192) {
		tmp = t * ((x / t) - (x * y));
	} else {
		tmp = x * (0.0 - (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 (y <= 8.5d-192) then
        tmp = t * ((x / t) - (x * y))
    else
        tmp = x * (0.0d0 - (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 (y <= 8.5e-192) {
		tmp = t * ((x / t) - (x * y));
	} else {
		tmp = x * (0.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 8.5e-192:
		tmp = t * ((x / t) - (x * y))
	else:
		tmp = x * (0.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 8.5e-192)
		tmp = Float64(t * Float64(Float64(x / t) - Float64(x * y)));
	else
		tmp = Float64(x * Float64(0.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 8.5e-192)
		tmp = t * ((x / t) - (x * y));
	else
		tmp = x * (0.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 8.5e-192], N[(t * N[(N[(x / t), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.49999999999999985e-192

   1. Initial program 96.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(t \cdot y\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t \cdot y\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - t \cdot y\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot t\right)\right)\right)\right) \]

      5. *-lowering-*.f64 55.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, t\right)\right)\right)\right) \]

   5. Simplified 55.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. 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 34.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 34.0%

      \[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{x}{t} - x \cdot y\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{x}{t} - x \cdot y\right)}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\left(\frac{x}{t}\right), \color{blue}{\left(x \cdot y\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, t\right), \left(\color{blue}{x} \cdot y\right)\right)\right) \]

       4. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, t\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   11. Simplified 36.7%

       \[\leadsto \color{blue}{t \cdot \left(\frac{x}{t} - x \cdot y\right)} \]

   if 8.49999999999999985e-192 < y

   1. Initial program 95.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 48.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 48.9%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   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 17.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 17.5%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   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. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{a \cdot \left(b \cdot x\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(a \cdot \left(b \cdot x\right)\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\left(b \cdot x\right) \cdot \color{blue}{a}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\left(x \cdot b\right) \cdot a\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(x \cdot \color{blue}{\left(b \cdot a\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(x \cdot \left(a \cdot \color{blue}{b}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       9. *-lowering-*.f64 28.9%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 28.9%

       \[\leadsto \color{blue}{0 - x \cdot \left(a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-192}:\\ \;\;\;\;t \cdot \left(\frac{x}{t} - x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 30.4% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6400000:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 6400000.0) (* x (- 1.0 (* y t))) (* x (- 1.0 (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 6400000.0) {
		tmp = x * (1.0 - (y * t));
	} 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 <= 6400000.0d0) then
        tmp = x * (1.0d0 - (y * t))
    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 <= 6400000.0) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 6400000.0:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 6400000.0)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	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 <= 6400000.0)
		tmp = x * (1.0 - (y * t));
	else
		tmp = x * (1.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 6400000.0], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4e6

   1. Initial program 95.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(t \cdot y\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t \cdot y\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(0 - t \cdot y\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot t\right)\right)\right)\right) \]

      5. *-lowering-*.f64 59.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, t\right)\right)\right)\right) \]

   5. Simplified 59.5%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(t \cdot y\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{t \cdot y}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(t \cdot y\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(y \cdot \color{blue}{t}\right)\right)\right) \]

      5. *-lowering-*.f64 29.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   8. Simplified 29.3%

      \[\leadsto x \cdot \color{blue}{\left(1 - y \cdot t\right)} \]

   if 6.4e6 < b

   1. Initial program 98.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 67.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

   5. Simplified 67.5%

      \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

   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 26.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 26.7%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 28.4% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - a \cdot b\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* x (- 1.0 (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * (1.0 - (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 * (1.0d0 - (a * b))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * (1.0 - (a * b));
}

Python

def code(x, y, z, t, a, b):
	return x * (1.0 - (a * b))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * Float64(1.0 - Float64(a * b)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * (1.0 - (a * b));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 58.8%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, b\right)\right)\right)\right) \]

5. Simplified 58.8%

   \[\leadsto x \cdot e^{\color{blue}{0 - a \cdot b}} \]

6. Taylor expanded in a around 0

   \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}\right) \]
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 25.1%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

8. Simplified 25.1%

   \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
9. Add Preprocessing

Alternative 22: 20.0% 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 96.0%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 69.6%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), t\right)\right)\right)\right) \]

5. Simplified 69.6%

   \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 18.0%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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))))))