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

Percentage Accurate: 96.2% → 99.7%

Time: 16.7s

Alternatives: 22

Speedup: 1.5×

• 40.2% 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: 96.2% 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: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{\mathsf{fma}\left(y, \log z - t, \left(0 - a\right) \cdot \left(z + b\right)\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (fma y (- (log z) t) (* (- 0.0 a) (+ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(fma(y, (log(z) - t), ((0.0 - a) * (z + b))));
}

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(fma(y, Float64(log(z) - t), Float64(Float64(0.0 - a) * Float64(z + b)))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[(0.0 - a), $MachinePrecision] * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

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

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

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

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. mul-1-neg N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. distribute-lft-out-- N/A

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

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

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

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

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

   13. neg-sub0 N/A

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

   14. --lowering--.f64 99.6

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

   \[\leadsto x \cdot e^{\mathsf{fma}\left(y, \log z - t, \left(0 - a\right) \cdot \left(z + b\right)\right)} \]
7. Add Preprocessing

Alternative 2: 38.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+230}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -10000000:\\ \;\;\;\;0 - x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;t \cdot \left(\frac{x}{t} - x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -4e+230)
     (* y (/ x y))
     (if (<= t_1 -10000000.0)
       (- 0.0 (* x (* y t)))
       (if (<= t_1 2.0)
         (* x (- 1.0 (* y t)))
         (if (<= t_1 2e+288)
           (* t (- (/ x t) (* x y)))
           (* x (- 1.0 (* a b)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -4e+230) {
		tmp = y * (x / y);
	} else if (t_1 <= -10000000.0) {
		tmp = 0.0 - (x * (y * t));
	} else if (t_1 <= 2.0) {
		tmp = x * (1.0 - (y * t));
	} else if (t_1 <= 2e+288) {
		tmp = t * ((x / t) - (x * y));
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_1 <= (-4d+230)) then
        tmp = y * (x / y)
    else if (t_1 <= (-10000000.0d0)) then
        tmp = 0.0d0 - (x * (y * t))
    else if (t_1 <= 2.0d0) then
        tmp = x * (1.0d0 - (y * t))
    else if (t_1 <= 2d+288) then
        tmp = t * ((x / t) - (x * y))
    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 t_1 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -4e+230) {
		tmp = y * (x / y);
	} else if (t_1 <= -10000000.0) {
		tmp = 0.0 - (x * (y * t));
	} else if (t_1 <= 2.0) {
		tmp = x * (1.0 - (y * t));
	} else if (t_1 <= 2e+288) {
		tmp = t * ((x / t) - (x * y));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_1 <= -4e+230:
		tmp = y * (x / y)
	elif t_1 <= -10000000.0:
		tmp = 0.0 - (x * (y * t))
	elif t_1 <= 2.0:
		tmp = x * (1.0 - (y * t))
	elif t_1 <= 2e+288:
		tmp = t * ((x / t) - (x * y))
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -4e+230)
		tmp = Float64(y * Float64(x / y));
	elseif (t_1 <= -10000000.0)
		tmp = Float64(0.0 - Float64(x * Float64(y * t)));
	elseif (t_1 <= 2.0)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	elseif (t_1 <= 2e+288)
		tmp = Float64(t * Float64(Float64(x / t) - Float64(x * y)));
	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)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_1 <= -4e+230)
		tmp = y * (x / y);
	elseif (t_1 <= -10000000.0)
		tmp = 0.0 - (x * (y * t));
	elseif (t_1 <= 2.0)
		tmp = x * (1.0 - (y * t));
	elseif (t_1 <= 2e+288)
		tmp = t * ((x / t) - (x * y));
	else
		tmp = x * (1.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -4e+230], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -10000000.0], N[(0.0 - N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+288], N[(t * N[(N[(x / t), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.0000000000000004e230

   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 t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 55.7

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

   5. Simplified 55.7%

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

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      12. --lowering--.f64 2.6

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(0 - y\right), x\right)} \]

   9. Taylor expanded in y around inf

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

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

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

       2. mul-1-neg N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \frac{x}{y}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \frac{x}{y}\right) \]

       4. mul-1-neg N/A

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

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

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, \frac{x}{y}\right)} \]

       6. mul-1-neg N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{x}{y}\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

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

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

       9. /-lowering-/.f64 6.9

          \[\leadsto y \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 6.9%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(t, 0 - x, \frac{x}{y}\right)} \]

   12. Taylor expanded in t around 0

       \[\leadsto y \cdot \color{blue}{\frac{x}{y}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 27.4

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

   14. Simplified 27.4%

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

   if -4.0000000000000004e230 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 39.2

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

   5. Simplified 39.2%

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

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      12. --lowering--.f64 4.8

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

   8. Simplified 4.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(0 - y\right), x\right)} \]

   9. Taylor expanded in y around inf

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

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

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

       2. mul-1-neg N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \frac{x}{y}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \frac{x}{y}\right) \]

       4. mul-1-neg N/A

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

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

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, \frac{x}{y}\right)} \]

       6. mul-1-neg N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{x}{y}\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

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

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

       9. /-lowering-/.f64 4.7

          \[\leadsto y \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 4.7%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(t, 0 - x, \frac{x}{y}\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. distribute-rgt-neg-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-out N/A

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

       11. distribute-rgt-neg-out N/A

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

       12. *-commutative N/A

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

       13. neg-sub0 N/A

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

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

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

       15. *-lowering-*.f64 20.2

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

   14. Simplified 20.2%

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

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 89.5

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

   5. Simplified 89.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 82.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 82.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 89.5

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

   11. Simplified 89.5%

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

   if 2 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2e288

   1. Initial program 97.2%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 41.6

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

   5. Simplified 41.6%

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

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      12. --lowering--.f64 15.7

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

   8. Simplified 15.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(0 - y\right), x\right)} \]

   9. Taylor expanded in t around inf

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

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

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

          \[\leadsto t \cdot \left(\frac{x}{t} + \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)}\right) \]

       4. unsub-neg N/A

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

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

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

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

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 23.7

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

   11. Simplified 23.7%

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

   if 2e288 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 91.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 73.8

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

   5. Simplified 73.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-lowering-*.f64 55.9

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

   8. Simplified 55.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+230}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -10000000:\\ \;\;\;\;0 - x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+288}:\\ \;\;\;\;t \cdot \left(\frac{x}{t} - x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+125}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(t, 0 - x, \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot \left(y \cdot t\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -10000000.0)
     (* x (* y (* y (* t (* t 0.5)))))
     (if (<= t_1 2.0)
       (* x (- 1.0 (* y t)))
       (if (<= t_1 2e+125)
         (* y (fma t (- 0.0 x) (/ x y)))
         (* 0.5 (* t (* x (* y (* y t))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = x * (y * (y * (t * (t * 0.5))));
	} else if (t_1 <= 2.0) {
		tmp = x * (1.0 - (y * t));
	} else if (t_1 <= 2e+125) {
		tmp = y * fma(t, (0.0 - x), (x / y));
	} else {
		tmp = 0.5 * (t * (x * (y * (y * t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = Float64(x * Float64(y * Float64(y * Float64(t * Float64(t * 0.5)))));
	elseif (t_1 <= 2.0)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	elseif (t_1 <= 2e+125)
		tmp = Float64(y * fma(t, Float64(0.0 - x), Float64(x / y)));
	else
		tmp = Float64(0.5 * Float64(t * Float64(x * Float64(y * Float64(y * t)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -10000000.0], N[(x * N[(y * N[(y * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+125], N[(y * N[(t * N[(0.0 - x), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t * N[(x * N[(y * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7

   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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 46.8

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

   5. Simplified 46.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 2.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 2.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. unpow2 N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(t \cdot \frac{1}{2}\right)\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(\frac{1}{2} \cdot t\right)}\right)\right)\right) \]

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

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot t\right)\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(t \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 37.0

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

   11. Simplified 37.0%

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

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 89.5

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

   5. Simplified 89.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 82.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 82.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 89.5

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

   11. Simplified 89.5%

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

   if 2 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.9999999999999998e125

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 31.3

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

   5. Simplified 31.3%

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

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      12. --lowering--.f64 6.5

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

   8. Simplified 6.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(0 - y\right), x\right)} \]

   9. Taylor expanded in y around inf

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

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

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

       2. mul-1-neg N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \frac{x}{y}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \frac{x}{y}\right) \]

       4. mul-1-neg N/A

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

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

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, \frac{x}{y}\right)} \]

       6. mul-1-neg N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{x}{y}\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

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

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

       9. /-lowering-/.f64 31.0

          \[\leadsto y \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 31.0%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(t, 0 - x, \frac{x}{y}\right)} \]

   if 1.9999999999999998e125 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 52.7

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

   5. Simplified 52.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 59.6

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 59.6%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   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. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

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

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

       9. *-commutative N/A

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

       10. unpow2 N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot t\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(y \cdot t\right)\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(t \cdot y\right)}\right)\right)\right) \]

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

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(t \cdot y\right)\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(y \cdot t\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 64.0

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

   11. Simplified 64.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot \left(y \cdot t\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 50.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -10000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot \left(y \cdot t\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* y (* y (* t (* t 0.5))))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -10000000.0)
     t_1
     (if (<= t_2 2.0)
       (* x (- 1.0 (* y t)))
       (if (<= t_2 1e+141) t_1 (* 0.5 (* t (* x (* y (* y t))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (y * (y * (t * (t * 0.5))));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = x * (1.0 - (y * t));
	} else if (t_2 <= 1e+141) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (t * (x * (y * (y * t))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * (y * (y * (t * (t * 0.5d0))))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-10000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = x * (1.0d0 - (y * t))
    else if (t_2 <= 1d+141) then
        tmp = t_1
    else
        tmp = 0.5d0 * (t * (x * (y * (y * t))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (y * (y * (t * (t * 0.5))));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = x * (1.0 - (y * t));
	} else if (t_2 <= 1e+141) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (t * (x * (y * (y * t))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (y * (y * (t * (t * 0.5))))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -10000000.0:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = x * (1.0 - (y * t))
	elif t_2 <= 1e+141:
		tmp = t_1
	else:
		tmp = 0.5 * (t * (x * (y * (y * t))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(y * Float64(y * Float64(t * Float64(t * 0.5)))))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -10000000.0)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	elseif (t_2 <= 1e+141)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(t * Float64(x * Float64(y * Float64(y * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (y * (y * (t * (t * 0.5))));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -10000000.0)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = x * (1.0 - (y * t));
	elseif (t_2 <= 1e+141)
		tmp = t_1;
	else
		tmp = 0.5 * (t * (x * (y * (y * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(y * N[(y * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -10000000.0], t$95$1, If[LessEqual[t$95$2, 2.0], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+141], t$95$1, N[(0.5 * N[(t * N[(x * N[(y * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7 or 2 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.00000000000000002e141

   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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 43.9

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

   5. Simplified 43.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 8.8

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 8.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. unpow2 N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(t \cdot \frac{1}{2}\right)\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(\frac{1}{2} \cdot t\right)}\right)\right)\right) \]

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

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot t\right)\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(t \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 35.0

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

   11. Simplified 35.0%

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

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 89.5

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

   5. Simplified 89.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 82.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 82.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 89.5

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

   11. Simplified 89.5%

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

   if 1.00000000000000002e141 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 51.4

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

   5. Simplified 51.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 59.8

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 59.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   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. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

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

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

       9. *-commutative N/A

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

       10. unpow2 N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot t\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(y \cdot t\right)\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(t \cdot y\right)}\right)\right)\right) \]

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

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(t \cdot y\right)\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(y \cdot t\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 65.6

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

   11. Simplified 65.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot \left(y \cdot t\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 35.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+230}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -10000000:\\ \;\;\;\;0 - x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -4e+230)
     (* y (/ x y))
     (if (<= t_1 -10000000.0)
       (- 0.0 (* x (* y t)))
       (if (<= t_1 5e+250) (* 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 t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -4e+230) {
		tmp = y * (x / y);
	} else if (t_1 <= -10000000.0) {
		tmp = 0.0 - (x * (y * t));
	} else if (t_1 <= 5e+250) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_1 <= (-4d+230)) then
        tmp = y * (x / y)
    else if (t_1 <= (-10000000.0d0)) then
        tmp = 0.0d0 - (x * (y * t))
    else if (t_1 <= 5d+250) 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 t_1 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -4e+230) {
		tmp = y * (x / y);
	} else if (t_1 <= -10000000.0) {
		tmp = 0.0 - (x * (y * t));
	} else if (t_1 <= 5e+250) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_1 <= -4e+230:
		tmp = y * (x / y)
	elif t_1 <= -10000000.0:
		tmp = 0.0 - (x * (y * t))
	elif t_1 <= 5e+250:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -4e+230)
		tmp = Float64(y * Float64(x / y));
	elseif (t_1 <= -10000000.0)
		tmp = Float64(0.0 - Float64(x * Float64(y * t)));
	elseif (t_1 <= 5e+250)
		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)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_1 <= -4e+230)
		tmp = y * (x / y);
	elseif (t_1 <= -10000000.0)
		tmp = 0.0 - (x * (y * t));
	elseif (t_1 <= 5e+250)
		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_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -4e+230], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -10000000.0], N[(0.0 - N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+250], 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 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -4.0000000000000004e230

   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 t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 55.7

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

   5. Simplified 55.7%

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

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      12. --lowering--.f64 2.6

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(0 - y\right), x\right)} \]

   9. Taylor expanded in y around inf

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

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

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

       2. mul-1-neg N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \frac{x}{y}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \frac{x}{y}\right) \]

       4. mul-1-neg N/A

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

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

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, \frac{x}{y}\right)} \]

       6. mul-1-neg N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{x}{y}\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

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

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

       9. /-lowering-/.f64 6.9

          \[\leadsto y \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 6.9%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(t, 0 - x, \frac{x}{y}\right)} \]

   12. Taylor expanded in t around 0

       \[\leadsto y \cdot \color{blue}{\frac{x}{y}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 27.4

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

   14. Simplified 27.4%

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

   if -4.0000000000000004e230 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 39.2

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

   5. Simplified 39.2%

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

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      12. --lowering--.f64 4.8

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

   8. Simplified 4.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(0 - y\right), x\right)} \]

   9. Taylor expanded in y around inf

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

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

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

       2. mul-1-neg N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \frac{x}{y}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \frac{x}{y}\right) \]

       4. mul-1-neg N/A

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

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

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, \frac{x}{y}\right)} \]

       6. mul-1-neg N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{x}{y}\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

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

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

       9. /-lowering-/.f64 4.7

          \[\leadsto y \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 4.7%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(t, 0 - x, \frac{x}{y}\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. distribute-rgt-neg-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-out N/A

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

       11. distribute-rgt-neg-out N/A

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

       12. *-commutative N/A

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

       13. neg-sub0 N/A

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

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

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

       15. *-lowering-*.f64 20.2

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

   14. Simplified 20.2%

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

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.0000000000000002e250

   1. Initial program 94.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 64.8

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

   5. Simplified 64.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 58.9

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 58.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 51.8

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

   11. Simplified 51.8%

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

   if 5.0000000000000002e250 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 69.6

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

   5. Simplified 69.6%

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

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-lowering-*.f64 45.6

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

   8. Simplified 45.6%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -4 \cdot 10^{+230}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -10000000:\\ \;\;\;\;0 - x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - x \cdot \left(y \cdot t\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -10000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+32}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+250}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 0.0 (* x (* y t))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -10000000.0)
     t_1
     (if (<= t_2 2e+32) x (if (<= t_2 5e+250) t_1 (- 0.0 (* x (* a b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.0 - (x * (y * t));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+32) {
		tmp = x;
	} else if (t_2 <= 5e+250) {
		tmp = t_1;
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.0d0 - (x * (y * t))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-10000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 2d+32) then
        tmp = x
    else if (t_2 <= 5d+250) then
        tmp = t_1
    else
        tmp = 0.0d0 - (x * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.0 - (x * (y * t));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+32) {
		tmp = x;
	} else if (t_2 <= 5e+250) {
		tmp = t_1;
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.0 - (x * (y * t))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -10000000.0:
		tmp = t_1
	elif t_2 <= 2e+32:
		tmp = x
	elif t_2 <= 5e+250:
		tmp = t_1
	else:
		tmp = 0.0 - (x * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.0 - Float64(x * Float64(y * t)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -10000000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+32)
		tmp = x;
	elseif (t_2 <= 5e+250)
		tmp = t_1;
	else
		tmp = Float64(0.0 - Float64(x * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.0 - (x * (y * t));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -10000000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+32)
		tmp = x;
	elseif (t_2 <= 5e+250)
		tmp = t_1;
	else
		tmp = 0.0 - (x * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.0 - N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -10000000.0], t$95$1, If[LessEqual[t$95$2, 2e+32], x, If[LessEqual[t$95$2, 5e+250], t$95$1, N[(0.0 - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7 or 2.00000000000000011e32 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.0000000000000002e250

   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 t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 45.3

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

   5. Simplified 45.3%

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

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      12. --lowering--.f64 8.3

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

   8. Simplified 8.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(0 - y\right), x\right)} \]

   9. Taylor expanded in y around inf

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

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

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

       2. mul-1-neg N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \frac{x}{y}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \frac{x}{y}\right) \]

       4. mul-1-neg N/A

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

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

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, \frac{x}{y}\right)} \]

       6. mul-1-neg N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{x}{y}\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

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

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

       9. /-lowering-/.f64 14.1

          \[\leadsto y \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 14.1%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(t, 0 - x, \frac{x}{y}\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. distribute-rgt-neg-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-out N/A

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

       11. distribute-rgt-neg-out N/A

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

       12. *-commutative N/A

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

       13. neg-sub0 N/A

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

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

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

       15. *-lowering-*.f64 18.2

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

   14. Simplified 18.2%

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

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.00000000000000011e32

   1. Initial program 93.5%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 84.0

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

   5. Simplified 84.0%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 79.5%

      \[\leadsto \color{blue}{x} \]

   if 5.0000000000000002e250 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 69.6

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

   5. Simplified 69.6%

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

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-lowering-*.f64 45.6

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

   8. Simplified 45.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

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

       4. *-lowering-*.f64 45.1

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

   11. Simplified 45.1%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -10000000:\\ \;\;\;\;0 - x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+32}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+250}:\\ \;\;\;\;0 - x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(0 - x \cdot a\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+190}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- 0.0 (* x a))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -500000000.0)
     t_1
     (if (<= t_2 5e+14) x (if (<= t_2 5e+190) t_1 (- 0.0 (* x (* a b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (0.0 - (x * a));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+14) {
		tmp = x;
	} else if (t_2 <= 5e+190) {
		tmp = t_1;
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = b * (0.0d0 - (x * a))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-500000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 5d+14) then
        tmp = x
    else if (t_2 <= 5d+190) then
        tmp = t_1
    else
        tmp = 0.0d0 - (x * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (0.0 - (x * a));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+14) {
		tmp = x;
	} else if (t_2 <= 5e+190) {
		tmp = t_1;
	} else {
		tmp = 0.0 - (x * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (0.0 - (x * a))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -500000000.0:
		tmp = t_1
	elif t_2 <= 5e+14:
		tmp = x
	elif t_2 <= 5e+190:
		tmp = t_1
	else:
		tmp = 0.0 - (x * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(0.0 - Float64(x * a)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -500000000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+14)
		tmp = x;
	elseif (t_2 <= 5e+190)
		tmp = t_1;
	else
		tmp = Float64(0.0 - Float64(x * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (0.0 - (x * a));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -500000000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+14)
		tmp = x;
	elseif (t_2 <= 5e+190)
		tmp = t_1;
	else
		tmp = 0.0 - (x * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(0.0 - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -500000000.0], t$95$1, If[LessEqual[t$95$2, 5e+14], x, If[LessEqual[t$95$2, 5e+190], t$95$1, N[(0.0 - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5e8 or 5e14 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.00000000000000036e190

   1. Initial program 97.0%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 47.5

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

   5. Simplified 47.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-lowering-*.f64 4.5

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

   8. Simplified 4.5%

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

   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. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. mul-1-neg N/A

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

       7. neg-sub0 N/A

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

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

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

       9. *-lowering-*.f64 17.9

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

   11. Simplified 17.9%

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

   if -5e8 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e14

   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

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 86.7

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

   5. Simplified 86.7%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 82.0%

      \[\leadsto \color{blue}{x} \]

   if 5.00000000000000036e190 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.0%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 60.3

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

   5. Simplified 60.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-lowering-*.f64 35.9

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

   8. Simplified 35.9%

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

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

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

       4. *-lowering-*.f64 35.3

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

   11. Simplified 35.3%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -500000000:\\ \;\;\;\;b \cdot \left(0 - x \cdot a\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+190}:\\ \;\;\;\;b \cdot \left(0 - x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \mathsf{fma}\left(a, b \cdot -0.16666666666666666, 0.5\right)\right), 0 - a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \frac{x}{t}, x \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -10000000.0)
     (* x (* y (* y (* t (* t 0.5)))))
     (if (<= t_1 4e+118)
       (*
        x
        (fma
         b
         (fma b (* a (* a (fma a (* b -0.16666666666666666) 0.5))) (- 0.0 a))
         1.0))
       (* t (* t (fma y (- 0.0 (/ x t)) (* x (* 0.5 (* y y))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = x * (y * (y * (t * (t * 0.5))));
	} else if (t_1 <= 4e+118) {
		tmp = x * fma(b, fma(b, (a * (a * fma(a, (b * -0.16666666666666666), 0.5))), (0.0 - a)), 1.0);
	} else {
		tmp = t * (t * fma(y, (0.0 - (x / t)), (x * (0.5 * (y * y)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = Float64(x * Float64(y * Float64(y * Float64(t * Float64(t * 0.5)))));
	elseif (t_1 <= 4e+118)
		tmp = Float64(x * fma(b, fma(b, Float64(a * Float64(a * fma(a, Float64(b * -0.16666666666666666), 0.5))), Float64(0.0 - a)), 1.0));
	else
		tmp = Float64(t * Float64(t * fma(y, Float64(0.0 - Float64(x / t)), Float64(x * Float64(0.5 * Float64(y * y))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -10000000.0], N[(x * N[(y * N[(y * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+118], N[(x * N[(b * N[(b * N[(a * N[(a * N[(a * N[(b * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 - a), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(t * N[(y * N[(0.0 - N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7

   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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 46.8

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

   5. Simplified 46.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 2.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 2.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. unpow2 N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(t \cdot \frac{1}{2}\right)\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(\frac{1}{2} \cdot t\right)}\right)\right)\right) \]

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

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot t\right)\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(t \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 37.0

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

   11. Simplified 37.0%

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

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 3.99999999999999987e118

   1. Initial program 94.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 68.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, a \cdot a, -0.16666666666666666 \cdot \left(b \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), 0 - a\right), 1\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(a \cdot b\right)\right)}, 0 - a\right), 1\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(a \cdot b\right)\right), 0 - a\right), 1\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(a \cdot b\right)\right)\right)}, 0 - a\right), 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(a \cdot b\right)\right)\right)}, 0 - a\right), 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(a \cdot b\right)\right)\right)}, 0 - a\right), 1\right) \]

       5. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(a \cdot b\right) + \frac{1}{2}\right)}\right), 0 - a\right), 1\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \left(\color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{6}} + \frac{1}{2}\right)\right), 0 - a\right), 1\right) \]

       7. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \left(\color{blue}{a \cdot \left(b \cdot \frac{-1}{6}\right)} + \frac{1}{2}\right)\right), 0 - a\right), 1\right) \]

       8. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(\frac{-1}{6} \cdot b\right)} + \frac{1}{2}\right)\right), 0 - a\right), 1\right) \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{-1}{6} \cdot b, \frac{1}{2}\right)}\right), 0 - a\right), 1\right) \]

       10. *-lowering-*.f64 73.2

           \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \mathsf{fma}\left(a, \color{blue}{-0.16666666666666666 \cdot b}, 0.5\right)\right), 0 - a\right), 1\right) \]

   11. Simplified 73.2%

       \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(a, -0.16666666666666666 \cdot b, 0.5\right)\right)}, 0 - a\right), 1\right) \]

   if 3.99999999999999987e118 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 52.6

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

   5. Simplified 52.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 59.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 59.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in t around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. mul-1-neg N/A

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

       6. *-commutative N/A

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

       7. associate-/l* N/A

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

       8. distribute-rgt-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. neg-sub0 N/A

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(y, \color{blue}{0 - \frac{x}{t}}, \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) \]

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

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(y, \color{blue}{0 - \frac{x}{t}}, \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) \]

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

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \color{blue}{\frac{x}{t}}, \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \frac{x}{t}, \color{blue}{\left(x \cdot {y}^{2}\right) \cdot \frac{1}{2}}\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \frac{x}{t}, \color{blue}{x \cdot \left({y}^{2} \cdot \frac{1}{2}\right)}\right)\right) \]

       17. *-commutative N/A

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \frac{x}{t}, x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right) \]

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

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \frac{x}{t}, \color{blue}{x \cdot \left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right) \]

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

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \frac{x}{t}, x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right) \]

       20. unpow2 N/A

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \frac{x}{t}, x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right) \]

       21. *-lowering-*.f64 68.6

           \[\leadsto t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \frac{x}{t}, x \cdot \left(0.5 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right) \]

   11. Simplified 68.6%

       \[\leadsto \color{blue}{t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \frac{x}{t}, x \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -10000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 4 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \mathsf{fma}\left(a, b \cdot -0.16666666666666666, 0.5\right)\right), 0 - a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t \cdot \mathsf{fma}\left(y, 0 - \frac{x}{t}, x \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \mathsf{fma}\left(a, b \cdot -0.16666666666666666, 0.5\right)\right), 0 - a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -10000000.0)
     (* x (* y (* y (* t (* t 0.5)))))
     (if (<= t_1 4e+118)
       (*
        x
        (fma
         b
         (fma b (* a (* a (fma a (* b -0.16666666666666666) 0.5))) (- 0.0 a))
         1.0))
       (* x (fma t (- (* t (* 0.5 (* y y))) y) 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = x * (y * (y * (t * (t * 0.5))));
	} else if (t_1 <= 4e+118) {
		tmp = x * fma(b, fma(b, (a * (a * fma(a, (b * -0.16666666666666666), 0.5))), (0.0 - a)), 1.0);
	} else {
		tmp = x * fma(t, ((t * (0.5 * (y * y))) - y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = Float64(x * Float64(y * Float64(y * Float64(t * Float64(t * 0.5)))));
	elseif (t_1 <= 4e+118)
		tmp = Float64(x * fma(b, fma(b, Float64(a * Float64(a * fma(a, Float64(b * -0.16666666666666666), 0.5))), Float64(0.0 - a)), 1.0));
	else
		tmp = Float64(x * fma(t, Float64(Float64(t * Float64(0.5 * Float64(y * y))) - y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -10000000.0], N[(x * N[(y * N[(y * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+118], N[(x * N[(b * N[(b * N[(a * N[(a * N[(a * N[(b * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 - a), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * N[(N[(t * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7

   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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 46.8

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

   5. Simplified 46.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 2.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 2.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. unpow2 N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(t \cdot \frac{1}{2}\right)\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(\frac{1}{2} \cdot t\right)}\right)\right)\right) \]

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

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot t\right)\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(t \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 37.0

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

   11. Simplified 37.0%

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

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 3.99999999999999987e118

   1. Initial program 94.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 68.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, a \cdot a, -0.16666666666666666 \cdot \left(b \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), 0 - a\right), 1\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(a \cdot b\right)\right)}, 0 - a\right), 1\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(a \cdot b\right)\right), 0 - a\right), 1\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(a \cdot b\right)\right)\right)}, 0 - a\right), 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(a \cdot b\right)\right)\right)}, 0 - a\right), 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \color{blue}{\left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(a \cdot b\right)\right)\right)}, 0 - a\right), 1\right) \]

       5. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(a \cdot b\right) + \frac{1}{2}\right)}\right), 0 - a\right), 1\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \left(\color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{6}} + \frac{1}{2}\right)\right), 0 - a\right), 1\right) \]

       7. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \left(\color{blue}{a \cdot \left(b \cdot \frac{-1}{6}\right)} + \frac{1}{2}\right)\right), 0 - a\right), 1\right) \]

       8. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(\frac{-1}{6} \cdot b\right)} + \frac{1}{2}\right)\right), 0 - a\right), 1\right) \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{-1}{6} \cdot b, \frac{1}{2}\right)}\right), 0 - a\right), 1\right) \]

       10. *-lowering-*.f64 73.2

           \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \mathsf{fma}\left(a, \color{blue}{-0.16666666666666666 \cdot b}, 0.5\right)\right), 0 - a\right), 1\right) \]

   11. Simplified 73.2%

       \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(a, -0.16666666666666666 \cdot b, 0.5\right)\right)}, 0 - a\right), 1\right) \]

   if 3.99999999999999987e118 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 52.6

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

   5. Simplified 52.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 59.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 59.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-in N/A

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

       3. mul-1-neg N/A

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

       4. associate-*r* N/A

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

       5. associate-+r+ N/A

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

       6. associate-*l* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{{y}^{2}}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \frac{1}{2} \cdot \color{blue}{\left({y}^{2} \cdot {t}^{2}\right)}\right) \]

       10. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot {t}^{2}}\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \left(\frac{1}{2} \cdot {y}^{2}\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \]

       12. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot t\right) \cdot t}\right) \]

       13. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\left(\frac{1}{2} \cdot \left({y}^{2} \cdot t\right)\right)} \cdot t\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \left(\frac{1}{2} \cdot \color{blue}{\left(t \cdot {y}^{2}\right)}\right) \cdot t\right) \]

       15. *-commutative N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)}\right) \]

   11. Simplified 67.9%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right) - y, 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -10000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 4 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, a \cdot \left(a \cdot \mathsf{fma}\left(a, b \cdot -0.16666666666666666, 0.5\right)\right), 0 - a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, a \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(a, b \cdot -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -10000000.0)
     (* x (* y (* y (* t (* t 0.5)))))
     (if (<= t_1 4e+118)
       (*
        x
        (fma
         b
         (* a (fma a (* b (fma a (* b -0.16666666666666666) 0.5)) -1.0))
         1.0))
       (* x (fma t (- (* t (* 0.5 (* y y))) y) 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = x * (y * (y * (t * (t * 0.5))));
	} else if (t_1 <= 4e+118) {
		tmp = x * fma(b, (a * fma(a, (b * fma(a, (b * -0.16666666666666666), 0.5)), -1.0)), 1.0);
	} else {
		tmp = x * fma(t, ((t * (0.5 * (y * y))) - y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = Float64(x * Float64(y * Float64(y * Float64(t * Float64(t * 0.5)))));
	elseif (t_1 <= 4e+118)
		tmp = Float64(x * fma(b, Float64(a * fma(a, Float64(b * fma(a, Float64(b * -0.16666666666666666), 0.5)), -1.0)), 1.0));
	else
		tmp = Float64(x * fma(t, Float64(Float64(t * Float64(0.5 * Float64(y * y))) - y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -10000000.0], N[(x * N[(y * N[(y * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+118], N[(x * N[(b * N[(a * N[(a * N[(b * N[(a * N[(b * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * N[(N[(t * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7

   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

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 46.8

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 46.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 2.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 2.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot y\right) \cdot y\right)}\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)} \]

       4. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]

       6. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot y\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot {t}^{2}\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot {t}^{2}\right)\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left({t}^{2} \cdot \frac{1}{2}\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(t \cdot \frac{1}{2}\right)\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(\frac{1}{2} \cdot t\right)}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot t\right)\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(t \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 37.0

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(t \cdot 0.5\right)}\right)\right)\right) \]

   11. Simplified 37.0%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)} \]

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 3.99999999999999987e118

   1. Initial program 94.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 75.5

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 75.5%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      2. neg-lowering-neg.f64 75.5

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(-a\right)}} \]

   7. Applied egg-rr 75.5%

      \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(-a\right)}} \]

   8. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]

   10. Simplified 69.9%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, a \cdot \mathsf{fma}\left(a, b \cdot \mathsf{fma}\left(a, b \cdot -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   if 3.99999999999999987e118 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 52.6

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 52.6%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 59.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 59.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(1 + y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) \cdot y + \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)}\right) \]

       3. mul-1-neg N/A

          \[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(-1 \cdot t\right)} \cdot y + \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto x \cdot \left(1 + \left(\color{blue}{-1 \cdot \left(t \cdot y\right)} + \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)} \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot y\right) \cdot y\right)}\right) \]

       7. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \frac{1}{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(y \cdot y\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{{y}^{2}}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \frac{1}{2} \cdot \color{blue}{\left({y}^{2} \cdot {t}^{2}\right)}\right) \]

       10. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot {t}^{2}}\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \left(\frac{1}{2} \cdot {y}^{2}\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \]

       12. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot t\right) \cdot t}\right) \]

       13. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\left(\frac{1}{2} \cdot \left({y}^{2} \cdot t\right)\right)} \cdot t\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \left(\frac{1}{2} \cdot \color{blue}{\left(t \cdot {y}^{2}\right)}\right) \cdot t\right) \]

       15. *-commutative N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)}\right) \]

   11. Simplified 67.9%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right) - y, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+125}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -10000000.0)
     (* x (* y (* y (* t (* t 0.5)))))
     (if (<= t_1 2e+125)
       (* x (fma a (- (* 0.5 (* a (* b b))) b) 1.0))
       (* x (fma t (- (* t (* 0.5 (* y y))) y) 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = x * (y * (y * (t * (t * 0.5))));
	} else if (t_1 <= 2e+125) {
		tmp = x * fma(a, ((0.5 * (a * (b * b))) - b), 1.0);
	} else {
		tmp = x * fma(t, ((t * (0.5 * (y * y))) - y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = Float64(x * Float64(y * Float64(y * Float64(t * Float64(t * 0.5)))));
	elseif (t_1 <= 2e+125)
		tmp = Float64(x * fma(a, Float64(Float64(0.5 * Float64(a * Float64(b * b))) - b), 1.0));
	else
		tmp = Float64(x * fma(t, Float64(Float64(t * Float64(0.5 * Float64(y * y))) - y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -10000000.0], N[(x * N[(y * N[(y * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+125], N[(x * N[(a * N[(N[(0.5 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * N[(N[(t * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7

   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

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 46.8

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 46.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 2.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 2.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot y\right) \cdot y\right)}\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)} \]

       4. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]

       6. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot y\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot {t}^{2}\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot {t}^{2}\right)\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left({t}^{2} \cdot \frac{1}{2}\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(t \cdot \frac{1}{2}\right)\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(\frac{1}{2} \cdot t\right)}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot t\right)\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(t \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 37.0

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(t \cdot 0.5\right)}\right)\right)\right) \]

   11. Simplified 37.0%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)} \]

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.9999999999999998e125

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 74.9

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 74.9%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right), 1\right)} \]

   8. Simplified 68.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, a \cdot a, -0.16666666666666666 \cdot \left(b \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), 0 - a\right), 1\right)} \]

   9. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

       2. distribute-rgt-in N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\left(-1 \cdot a\right) \cdot b + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) \cdot b\right)} + 1\right) \]

       3. associate-+l+ N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(-1 \cdot a\right) \cdot b + \left(\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) \cdot b + 1\right)\right)} \]

       4. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\color{blue}{\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot b\right)} \cdot b + 1\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left(b \cdot b\right)} + 1\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\color{blue}{\left({a}^{2} \cdot \frac{1}{2}\right)} \cdot \left(b \cdot b\right) + 1\right)\right) \]

       7. unpow2 N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + 1\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\color{blue}{\left(a \cdot \left(a \cdot \frac{1}{2}\right)\right)} \cdot \left(b \cdot b\right) + 1\right)\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right) \cdot \left(b \cdot b\right) + 1\right)\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\left(a \cdot \left(\frac{1}{2} \cdot a\right)\right) \cdot \color{blue}{{b}^{2}} + 1\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}\right)} + 1\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)} + 1\right)\right) \]

       13. associate-+l+ N/A

           \[\leadsto x \cdot \color{blue}{\left(\left(\left(-1 \cdot a\right) \cdot b + a \cdot \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right) + 1\right)} \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\left(\color{blue}{\left(a \cdot -1\right)} \cdot b + a \cdot \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right) + 1\right) \]

       15. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(\color{blue}{a \cdot \left(-1 \cdot b\right)} + a \cdot \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right) + 1\right) \]

       16. distribute-lft-in N/A

           \[\leadsto x \cdot \left(\color{blue}{a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)} + 1\right) \]

   11. Simplified 67.3%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b, 1\right)} \]

   if 1.9999999999999998e125 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 52.7

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 52.7%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 59.6

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 59.6%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(1 + y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) \cdot y + \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)}\right) \]

       3. mul-1-neg N/A

          \[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(-1 \cdot t\right)} \cdot y + \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto x \cdot \left(1 + \left(\color{blue}{-1 \cdot \left(t \cdot y\right)} + \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)} \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot y\right) \cdot y\right)}\right) \]

       7. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \frac{1}{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(y \cdot y\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{{y}^{2}}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \frac{1}{2} \cdot \color{blue}{\left({y}^{2} \cdot {t}^{2}\right)}\right) \]

       10. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot {t}^{2}}\right) \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \left(\frac{1}{2} \cdot {y}^{2}\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \]

       12. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot t\right) \cdot t}\right) \]

       13. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{\left(\frac{1}{2} \cdot \left({y}^{2} \cdot t\right)\right)} \cdot t\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \left(\frac{1}{2} \cdot \color{blue}{\left(t \cdot {y}^{2}\right)}\right) \cdot t\right) \]

       15. *-commutative N/A

           \[\leadsto x \cdot \left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)}\right) \]

   11. Simplified 68.3%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \left(0.5 \cdot \left(y \cdot y\right)\right) - y, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 45.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot \left(y \cdot t\right)\right)\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -10000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.5 (* t (* x (* y (* y t))))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -10000000.0)
     t_1
     (if (<= t_2 5e+19) (* x (- 1.0 (* y t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (t * (x * (y * (y * t))));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+19) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.5d0 * (t * (x * (y * (y * t))))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-10000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 5d+19) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (t * (x * (y * (y * t))));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+19) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.5 * (t * (x * (y * (y * t))))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -10000000.0:
		tmp = t_1
	elif t_2 <= 5e+19:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.5 * Float64(t * Float64(x * Float64(y * Float64(y * t)))))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -10000000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+19)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.5 * (t * (x * (y * (y * t))));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -10000000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+19)
		tmp = x * (1.0 - (y * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.5 * N[(t * N[(x * N[(y * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -10000000.0], t$95$1, If[LessEqual[t$95$2, 5e+19], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7 or 5e19 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 47.2

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 47.2%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 27.1

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 27.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   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. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(x \cdot {y}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot x\right) \cdot {y}^{2}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(\color{blue}{\left(x \cdot t\right)} \cdot {y}^{2}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \frac{1}{2} \cdot \left(t \cdot \color{blue}{\left(x \cdot \left(t \cdot {y}^{2}\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(t \cdot \color{blue}{\left(x \cdot \left(t \cdot {y}^{2}\right)\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \color{blue}{\left({y}^{2} \cdot t\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot t\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(y \cdot t\right)\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(t \cdot y\right)}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(t \cdot y\right)\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \left(t \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(y \cdot t\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 40.0

           \[\leadsto 0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(y \cdot t\right)}\right)\right)\right) \]

   11. Simplified 40.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(t \cdot \left(x \cdot \left(y \cdot \left(y \cdot t\right)\right)\right)\right)} \]

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e19

   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 t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 83.8

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 83.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 79.0

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 79.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)}\right) \]

       2. unsub-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

       4. *-commutative N/A

          \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

       5. *-lowering-*.f64 82.3

          \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

   11. Simplified 82.3%

       \[\leadsto x \cdot \color{blue}{\left(1 - y \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 33.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;0 - x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -10000000.0)
     (- 0.0 (* x (* y t)))
     (if (<= t_1 5e+250) (* 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 t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = 0.0 - (x * (y * t));
	} else if (t_1 <= 5e+250) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_1 <= (-10000000.0d0)) then
        tmp = 0.0d0 - (x * (y * t))
    else if (t_1 <= 5d+250) 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 t_1 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = 0.0 - (x * (y * t));
	} else if (t_1 <= 5e+250) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_1 <= -10000000.0:
		tmp = 0.0 - (x * (y * t))
	elif t_1 <= 5e+250:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = Float64(0.0 - Float64(x * Float64(y * t)));
	elseif (t_1 <= 5e+250)
		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)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_1 <= -10000000.0)
		tmp = 0.0 - (x * (y * t));
	elseif (t_1 <= 5e+250)
		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_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -10000000.0], N[(0.0 - N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+250], 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 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7

   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

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 46.8

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 46.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   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. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      12. --lowering--.f64 3.7

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

   8. Simplified 3.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(0 - y\right), x\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(t \cdot x\right) + \frac{x}{y}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(t \cdot x\right) + \frac{x}{y}\right)} \]

       2. mul-1-neg N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \frac{x}{y}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \frac{x}{y}\right) \]

       4. mul-1-neg N/A

          \[\leadsto y \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \frac{x}{y}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, \frac{x}{y}\right)} \]

       6. mul-1-neg N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{x}{y}\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

       9. /-lowering-/.f64 5.7

          \[\leadsto y \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 5.7%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(t, 0 - x, \frac{x}{y}\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot y\right) \cdot t}\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(y \cdot t\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(t \cdot y\right)}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(t \cdot y\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot t}\right)\right) \]

       7. distribute-rgt-neg-out N/A

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} \]

       8. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(-1 \cdot t\right)}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(-1 \cdot t\right)\right)} \]

       10. mul-1-neg N/A

           \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) \]

       11. distribute-rgt-neg-out N/A

           \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(y \cdot t\right)\right)} \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{t \cdot y}\right)\right) \]

       13. neg-sub0 N/A

           \[\leadsto x \cdot \color{blue}{\left(0 - t \cdot y\right)} \]

       14. --lowering--.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(0 - t \cdot y\right)} \]

       15. *-lowering-*.f64 18.1

           \[\leadsto x \cdot \left(0 - \color{blue}{t \cdot y}\right) \]

   14. Simplified 18.1%

       \[\leadsto \color{blue}{x \cdot \left(0 - t \cdot y\right)} \]

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.0000000000000002e250

   1. Initial program 94.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 64.8

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 64.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 58.9

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 58.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)}\right) \]

       2. unsub-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

       4. *-commutative N/A

          \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

       5. *-lowering-*.f64 51.8

          \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

   11. Simplified 51.8%

       \[\leadsto x \cdot \color{blue}{\left(1 - y \cdot t\right)} \]

   if 5.0000000000000002e250 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 69.6

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 69.6%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      4. *-lowering-*.f64 45.6

         \[\leadsto x \cdot \left(1 - \color{blue}{a \cdot b}\right) \]

   8. Simplified 45.6%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -10000000:\\ \;\;\;\;0 - x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(0 - x \cdot a\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- 0.0 (* x a))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -500000000.0) t_1 (if (<= t_2 5e+14) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (0.0 - (x * a));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+14) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = b * (0.0d0 - (x * a))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-500000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 5d+14) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (0.0 - (x * a));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+14) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (0.0 - (x * a))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -500000000.0:
		tmp = t_1
	elif t_2 <= 5e+14:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(0.0 - Float64(x * a)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -500000000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+14)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (0.0 - (x * a));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -500000000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+14)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(0.0 - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -500000000.0], t$95$1, If[LessEqual[t$95$2, 5e+14], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5e8 or 5e14 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 51.8

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 51.8%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      4. *-lowering-*.f64 15.1

         \[\leadsto x \cdot \left(1 - \color{blue}{a \cdot b}\right) \]

   8. Simplified 15.1%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   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. *-commutative N/A

          \[\leadsto -1 \cdot \left(a \cdot \color{blue}{\left(x \cdot b\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\left(a \cdot x\right) \cdot b\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right)\right) \cdot b} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(a \cdot x\right)\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(a \cdot x\right)\right)} \]

       6. mul-1-neg N/A

          \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} \]

       7. neg-sub0 N/A

          \[\leadsto b \cdot \color{blue}{\left(0 - a \cdot x\right)} \]

       8. --lowering--.f64 N/A

          \[\leadsto b \cdot \color{blue}{\left(0 - a \cdot x\right)} \]

       9. *-lowering-*.f64 20.9

          \[\leadsto b \cdot \left(0 - \color{blue}{a \cdot x}\right) \]

   11. Simplified 20.9%

       \[\leadsto \color{blue}{b \cdot \left(0 - a \cdot x\right)} \]

   if -5e8 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e14

   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

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 86.7

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 86.7%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 82.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -500000000:\\ \;\;\;\;b \cdot \left(0 - x \cdot a\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0 - x \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -10000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -10000000.0)
   (* x (* y (* y (* t (* t 0.5)))))
   (* x (fma a (- (* 0.5 (* a (* b b))) b) 1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -10000000.0) {
		tmp = x * (y * (y * (t * (t * 0.5))));
	} else {
		tmp = x * fma(a, ((0.5 * (a * (b * b))) - b), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -10000000.0)
		tmp = Float64(x * Float64(y * Float64(y * Float64(t * Float64(t * 0.5)))));
	else
		tmp = Float64(x * fma(a, Float64(Float64(0.5 * Float64(a * Float64(b * b))) - b), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], -10000000.0], N[(x * N[(y * N[(y * N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * N[(N[(0.5 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7

   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

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 46.8

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 46.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

      11. --lowering--.f64 2.4

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{0 - t}\right), 1\right) \]

   8. Simplified 2.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), 0 - t\right), 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot y\right) \cdot y\right)}\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) \cdot y\right)} \]

       4. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]

       6. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot y\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot {t}^{2}\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot {t}^{2}\right)\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left({t}^{2} \cdot \frac{1}{2}\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(t \cdot \frac{1}{2}\right)\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(\frac{1}{2} \cdot t\right)}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot t\right)\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(t \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 37.0

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(t \cdot \color{blue}{\left(t \cdot 0.5\right)}\right)\right)\right) \]

   11. Simplified 37.0%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(t \cdot \left(t \cdot 0.5\right)\right)\right)\right)} \]

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 65.7

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 65.7%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + b \cdot \left(\frac{-1}{6} \cdot \left({a}^{3} \cdot b\right) + \frac{1}{2} \cdot {a}^{2}\right), 1\right)} \]

   8. Simplified 60.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, a \cdot a, -0.16666666666666666 \cdot \left(b \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), 0 - a\right), 1\right)} \]

   9. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

       2. distribute-rgt-in N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\left(-1 \cdot a\right) \cdot b + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) \cdot b\right)} + 1\right) \]

       3. associate-+l+ N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(-1 \cdot a\right) \cdot b + \left(\left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) \cdot b + 1\right)\right)} \]

       4. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\color{blue}{\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot b\right)} \cdot b + 1\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left(b \cdot b\right)} + 1\right)\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\color{blue}{\left({a}^{2} \cdot \frac{1}{2}\right)} \cdot \left(b \cdot b\right) + 1\right)\right) \]

       7. unpow2 N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + 1\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\color{blue}{\left(a \cdot \left(a \cdot \frac{1}{2}\right)\right)} \cdot \left(b \cdot b\right) + 1\right)\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right) \cdot \left(b \cdot b\right) + 1\right)\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\left(a \cdot \left(\frac{1}{2} \cdot a\right)\right) \cdot \color{blue}{{b}^{2}} + 1\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(\color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}\right)} + 1\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(-1 \cdot a\right) \cdot b + \left(a \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)} + 1\right)\right) \]

       13. associate-+l+ N/A

           \[\leadsto x \cdot \color{blue}{\left(\left(\left(-1 \cdot a\right) \cdot b + a \cdot \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right) + 1\right)} \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\left(\color{blue}{\left(a \cdot -1\right)} \cdot b + a \cdot \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right) + 1\right) \]

       15. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(\color{blue}{a \cdot \left(-1 \cdot b\right)} + a \cdot \left(\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right) + 1\right) \]

       16. distribute-lft-in N/A

           \[\leadsto x \cdot \left(\color{blue}{a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)} + 1\right) \]

   11. Simplified 64.8%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, 0.5 \cdot \left(a \cdot \left(b \cdot b\right)\right) - b, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 34.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -10000000:\\ \;\;\;\;0 - x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -10000000.0)
   (- 0.0 (* x (* 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 (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -10000000.0) {
		tmp = 0.0 - (x * (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 (((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))) <= (-10000000.0d0)) then
        tmp = 0.0d0 - (x * (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 (((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))) <= -10000000.0) {
		tmp = 0.0 - (x * (y * t));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))) <= -10000000.0:
		tmp = 0.0 - (x * (y * t))
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -10000000.0)
		tmp = Float64(0.0 - Float64(x * 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 (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -10000000.0)
		tmp = 0.0 - (x * (y * t));
	else
		tmp = x * (1.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], -10000000.0], N[(0.0 - N[(x * 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 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e7

   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

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 46.8

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 46.8%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   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. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)\right)} + x \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)} + x \]

      4. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(x \cdot y\right), x\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x \cdot y\right)}, x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(-1 \cdot y\right)}, x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{x \cdot \left(-1 \cdot y\right)}, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, x\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

      12. --lowering--.f64 3.7

          \[\leadsto \mathsf{fma}\left(t, x \cdot \color{blue}{\left(0 - y\right)}, x\right) \]

   8. Simplified 3.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x \cdot \left(0 - y\right), x\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(t \cdot x\right) + \frac{x}{y}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(t \cdot x\right) + \frac{x}{y}\right)} \]

       2. mul-1-neg N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \frac{x}{y}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \frac{x}{y}\right) \]

       4. mul-1-neg N/A

          \[\leadsto y \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \frac{x}{y}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, \frac{x}{y}\right)} \]

       6. mul-1-neg N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{x}{y}\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, \frac{x}{y}\right) \]

       9. /-lowering-/.f64 5.7

          \[\leadsto y \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 5.7%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(t, 0 - x, \frac{x}{y}\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(t \cdot \left(x \cdot y\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot y\right) \cdot t}\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(y \cdot t\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(t \cdot y\right)}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(t \cdot y\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot t}\right)\right) \]

       7. distribute-rgt-neg-out N/A

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} \]

       8. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(-1 \cdot t\right)}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(-1 \cdot t\right)\right)} \]

       10. mul-1-neg N/A

           \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) \]

       11. distribute-rgt-neg-out N/A

           \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(y \cdot t\right)\right)} \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{t \cdot y}\right)\right) \]

       13. neg-sub0 N/A

           \[\leadsto x \cdot \color{blue}{\left(0 - t \cdot y\right)} \]

       14. --lowering--.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(0 - t \cdot y\right)} \]

       15. *-lowering-*.f64 18.1

           \[\leadsto x \cdot \left(0 - \color{blue}{t \cdot y}\right) \]

   14. Simplified 18.1%

       \[\leadsto \color{blue}{x \cdot \left(0 - t \cdot y\right)} \]

   if -1e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 65.7

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 65.7%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      4. *-lowering-*.f64 46.9

         \[\leadsto x \cdot \left(1 - \color{blue}{a \cdot b}\right) \]

   8. Simplified 46.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -10000000:\\ \;\;\;\;0 - x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 84.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -7 \cdot 10^{-93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-53}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -7e-93)
     t_1
     (if (<= y 2.9e-53) (* x (exp (* a (- (log (- 1.0 z)) b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * (log(z) - t)));
	double tmp;
	if (y <= -7e-93) {
		tmp = t_1;
	} else if (y <= 2.9e-53) {
		tmp = x * exp((a * (log((1.0 - z)) - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * (log(z) - t)))
    if (y <= (-7d-93)) then
        tmp = t_1
    else if (y <= 2.9d-53) then
        tmp = x * exp((a * (log((1.0d0 - z)) - b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * (Math.log(z) - t)));
	double tmp;
	if (y <= -7e-93) {
		tmp = t_1;
	} else if (y <= 2.9e-53) {
		tmp = x * Math.exp((a * (Math.log((1.0 - z)) - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * (math.log(z) - t)))
	tmp = 0
	if y <= -7e-93:
		tmp = t_1
	elif y <= 2.9e-53:
		tmp = x * math.exp((a * (math.log((1.0 - z)) - b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(log(z) - t))))
	tmp = 0.0
	if (y <= -7e-93)
		tmp = t_1;
	elseif (y <= 2.9e-53)
		tmp = Float64(x * exp(Float64(a * Float64(log(Float64(1.0 - z)) - b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * (log(z) - t)));
	tmp = 0.0;
	if (y <= -7e-93)
		tmp = t_1;
	elseif (y <= 2.9e-53)
		tmp = x * exp((a * (log((1.0 - z)) - b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e-93], t$95$1, If[LessEqual[y, 2.9e-53], N[(x * N[Exp[N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7e-93 or 2.8999999999999998e-53 < 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 y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 85.3

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 85.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   if -7e-93 < y < 2.8999999999999998e-53

   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 y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. rem-exp-log N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(e^{\log \left(1 - z\right)}\right)} - b\right)} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\log \left(e^{\log \left(1 - z\right)}\right)} - b\right)} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 - z\right)} - b\right)} \]

      6. --lowering--.f64 83.9

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 - z\right)} - b\right)} \]

   5. Simplified 83.9%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 82.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{-179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-54}:\\ \;\;\;\;x \cdot e^{0 - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -8.5e-179)
     t_1
     (if (<= y 3.6e-54) (* 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 <= -8.5e-179) {
		tmp = t_1;
	} else if (y <= 3.6e-54) {
		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 <= (-8.5d-179)) then
        tmp = t_1
    else if (y <= 3.6d-54) 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 <= -8.5e-179) {
		tmp = t_1;
	} else if (y <= 3.6e-54) {
		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 <= -8.5e-179:
		tmp = t_1
	elif y <= 3.6e-54:
		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 <= -8.5e-179)
		tmp = t_1;
	elseif (y <= 3.6e-54)
		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 <= -8.5e-179)
		tmp = t_1;
	elseif (y <= 3.6e-54)
		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, -8.5e-179], t$95$1, If[LessEqual[y, 3.6e-54], 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 < -8.49999999999999932e-179 or 3.59999999999999976e-54 < y

   1. Initial program 97.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 83.4

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 83.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   if -8.49999999999999932e-179 < y < 3.59999999999999976e-54

   1. Initial program 92.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 86.0

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 86.0%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      2. neg-lowering-neg.f64 86.0

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(-a\right)}} \]

   7. Applied egg-rr 86.0%

      \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(-a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-179}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-54}:\\ \;\;\;\;x \cdot e^{0 - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 70.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{0 - y \cdot t}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-165}:\\ \;\;\;\;x \cdot e^{0 - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (- 0.0 (* y t))))))
   (if (<= t -4.5e+23)
     t_1
     (if (<= t 7.8e-165) (* 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((0.0 - (y * t)));
	double tmp;
	if (t <= -4.5e+23) {
		tmp = t_1;
	} else if (t <= 7.8e-165) {
		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((0.0d0 - (y * t)))
    if (t <= (-4.5d+23)) then
        tmp = t_1
    else if (t <= 7.8d-165) 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((0.0 - (y * t)));
	double tmp;
	if (t <= -4.5e+23) {
		tmp = t_1;
	} else if (t <= 7.8e-165) {
		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((0.0 - (y * t)))
	tmp = 0
	if t <= -4.5e+23:
		tmp = t_1
	elif t <= 7.8e-165:
		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(0.0 - Float64(y * t))))
	tmp = 0.0
	if (t <= -4.5e+23)
		tmp = t_1;
	elseif (t <= 7.8e-165)
		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((0.0 - (y * t)));
	tmp = 0.0;
	if (t <= -4.5e+23)
		tmp = t_1;
	elseif (t <= 7.8e-165)
		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[(0.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.5e+23], t$95$1, If[LessEqual[t, 7.8e-165], 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 t < -4.49999999999999979e23 or 7.7999999999999997e-165 < t

   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 t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. neg-sub0 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{0 - t \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

      5. *-lowering-*.f64 79.3

         \[\leadsto x \cdot e^{0 - \color{blue}{y \cdot t}} \]

   5. Simplified 79.3%

      \[\leadsto x \cdot e^{\color{blue}{0 - y \cdot t}} \]

   if -4.49999999999999979e23 < t < 7.7999999999999997e-165

   1. Initial program 93.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 66.2

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 66.2%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      2. neg-lowering-neg.f64 66.2

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(-a\right)}} \]

   7. Applied egg-rr 66.2%

      \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(-a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+23}:\\ \;\;\;\;x \cdot e^{0 - y \cdot t}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-165}:\\ \;\;\;\;x \cdot e^{0 - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{0 - y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 72.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.86 \cdot 10^{+109}:\\ \;\;\;\;x \cdot e^{0 - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -2.3e-7)
     t_1
     (if (<= y 1.86e+109) (* 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 * pow(z, y);
	double tmp;
	if (y <= -2.3e-7) {
		tmp = t_1;
	} else if (y <= 1.86e+109) {
		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 * (z ** y)
    if (y <= (-2.3d-7)) then
        tmp = t_1
    else if (y <= 1.86d+109) 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.pow(z, y);
	double tmp;
	if (y <= -2.3e-7) {
		tmp = t_1;
	} else if (y <= 1.86e+109) {
		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.pow(z, y)
	tmp = 0
	if y <= -2.3e-7:
		tmp = t_1
	elif y <= 1.86e+109:
		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 * (z ^ y))
	tmp = 0.0
	if (y <= -2.3e-7)
		tmp = t_1;
	elseif (y <= 1.86e+109)
		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 * (z ^ y);
	tmp = 0.0;
	if (y <= -2.3e-7)
		tmp = t_1;
	elseif (y <= 1.86e+109)
		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[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-7], t$95$1, If[LessEqual[y, 1.86e+109], 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 < -2.29999999999999995e-7 or 1.86000000000000008e109 < y

   1. Initial program 97.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 90.8

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 90.8%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. pow-lowering-pow.f64 67.8

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 67.8%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -2.29999999999999995e-7 < y < 1.86000000000000008e109

   1. Initial program 94.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 76.9

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 76.9%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      2. neg-lowering-neg.f64 76.9

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(-a\right)}} \]

   7. Applied egg-rr 76.9%

      \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(-a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-7}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 1.86 \cdot 10^{+109}:\\ \;\;\;\;x \cdot e^{0 - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 58.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 320000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, b \cdot b, \left(a \cdot -0.16666666666666666\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right), 0 - b\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -2.5e-10)
     t_1
     (if (<= y 320000000000.0)
       (*
        x
        (fma
         a
         (fma
          a
          (fma 0.5 (* b b) (* (* a -0.16666666666666666) (* b (* b b))))
          (- 0.0 b))
         1.0))
       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 <= -2.5e-10) {
		tmp = t_1;
	} else if (y <= 320000000000.0) {
		tmp = x * fma(a, fma(a, fma(0.5, (b * b), ((a * -0.16666666666666666) * (b * (b * b)))), (0.0 - b)), 1.0);
	} 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 <= -2.5e-10)
		tmp = t_1;
	elseif (y <= 320000000000.0)
		tmp = Float64(x * fma(a, fma(a, fma(0.5, Float64(b * b), Float64(Float64(a * -0.16666666666666666) * Float64(b * Float64(b * b)))), Float64(0.0 - b)), 1.0));
	else
		tmp = t_1;
	end
	return 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, -2.5e-10], t$95$1, If[LessEqual[y, 320000000000.0], N[(x * N[(a * N[(a * N[(0.5 * N[(b * b), $MachinePrecision] + N[(N[(a * -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 - b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.50000000000000016e-10 or 3.2e11 < y

   1. Initial program 96.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 86.4

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 86.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. pow-lowering-pow.f64 65.6

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -2.50000000000000016e-10 < y < 3.2e11

   1. Initial program 94.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      6. --lowering--.f64 77.8

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   5. Simplified 77.8%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b + a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + a \cdot \left(\frac{-1}{6} \cdot \left(a \cdot {b}^{3}\right) + \frac{1}{2} \cdot {b}^{2}\right), 1\right)} \]

   8. Simplified 57.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, b \cdot b, \left(-0.16666666666666666 \cdot a\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right), 0 - b\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-10}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 320000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, b \cdot b, \left(a \cdot -0.16666666666666666\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right), 0 - b\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 19.6% accurate, 328.0× 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 95.4%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}} \]

   2. *-commutative N/A

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-1 \cdot a\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}} \]

   5. neg-sub0 N/A

      \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

   6. --lowering--.f64 59.9

      \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

5. Simplified 59.9%

   \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(0 - a\right)}} \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 21.8%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(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))))))