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

Percentage Accurate: 96.5% → 96.5%

Time: 12.4s

Alternatives: 9

Speedup: N/A×

• 41.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-log.f64 96.2

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

5. Applied rewrites 96.2%

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

6. Final simplification 96.2%

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

Alternative 2: 44.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot x\right) \cdot \left(\left(t \cdot t\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(t \cdot t\right) \cdot y\right) \cdot 0.5\right) \cdot y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y)) -1e-8)
   (* (* (* y y) x) (* (* t t) 0.5))
   (fma (* (* (* (* t t) y) 0.5) y) x x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((log((1.0 - z)) - b) * a) + ((log(z) - t) * y)) <= -1e-8) {
		tmp = ((y * y) * x) * ((t * t) * 0.5);
	} else {
		tmp = fma(((((t * t) * y) * 0.5) * y), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y)) <= -1e-8)
		tmp = Float64(Float64(Float64(y * y) * x) * Float64(Float64(t * t) * 0.5));
	else
		tmp = fma(Float64(Float64(Float64(Float64(t * t) * y) * 0.5) * y), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], -1e-8], N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t * t), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision] * x + x), $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))) < -1e-8

   1. Initial program 97.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left({\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot \mathsf{fma}\left(0.5 \cdot y, {\left(\log z - t\right)}^{2}, \log z - t\right)\right), y, {\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot x\right)} \]

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 4.4%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left({\left(\log z - t\right)}^{2} \cdot y, 0.5, \log z\right) - t\right) \cdot y, \color{blue}{x}, x\right) \]

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 30.0%

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

   if -1e-8 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 y around 0

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

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left({\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot \mathsf{fma}\left(0.5 \cdot y, {\left(\log z - t\right)}^{2}, \log z - t\right)\right), y, {\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot x\right)} \]

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.2%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left({\left(\log z - t\right)}^{2} \cdot y, 0.5, \log z\right) - t\right) \cdot y, \color{blue}{x}, x\right) \]

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 59.6%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(t \cdot t\right) \cdot y\right) \cdot 0.5\right) \cdot y, x, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot x\right) \cdot \left(\left(t \cdot t\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(t \cdot t\right) \cdot y\right) \cdot 0.5\right) \cdot y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;t \leq -8 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+155}:\\ \;\;\;\;e^{\mathsf{fma}\left(-a, b, \log z \cdot y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= t -8e+21)
     t_1
     (if (<= t 1.15e+155) (* (exp (fma (- a) b (* (log z) y))) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-t * y)) * x;
	double tmp;
	if (t <= -8e+21) {
		tmp = t_1;
	} else if (t <= 1.15e+155) {
		tmp = exp(fma(-a, b, (log(z) * y))) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-t) * y)) * x)
	tmp = 0.0
	if (t <= -8e+21)
		tmp = t_1;
	elseif (t <= 1.15e+155)
		tmp = Float64(exp(fma(Float64(-a), b, Float64(log(z) * y))) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -8e+21], t$95$1, If[LessEqual[t, 1.15e+155], N[(N[Exp[N[((-a) * b + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8e21 or 1.14999999999999999e155 < t

   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. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if -8e21 < t < 1.14999999999999999e155

   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 z around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot e^{\mathsf{fma}\left(-a, b, \log z \cdot y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.3%

      \[\leadsto x \cdot e^{\mathsf{fma}\left(-a, b, \log z \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+21}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+155}:\\ \;\;\;\;e^{\mathsf{fma}\left(-a, b, \log z \cdot y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -140:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-47}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (log z) t) y)) x)))
   (if (<= y -140.0)
     t_1
     (if (<= y 4.2e-47) (* (exp (* (- (- z) b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((log(z) - t) * y)) * x;
	double tmp;
	if (y <= -140.0) {
		tmp = t_1;
	} else if (y <= 4.2e-47) {
		tmp = exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp(((log(z) - t) * y)) * x
    if (y <= (-140.0d0)) then
        tmp = t_1
    else if (y <= 4.2d-47) then
        tmp = exp(((-z - b) * a)) * 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 = Math.exp(((Math.log(z) - t) * y)) * x;
	double tmp;
	if (y <= -140.0) {
		tmp = t_1;
	} else if (y <= 4.2e-47) {
		tmp = Math.exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp(((math.log(z) - t) * y)) * x
	tmp = 0
	if y <= -140.0:
		tmp = t_1
	elif y <= 4.2e-47:
		tmp = math.exp(((-z - b) * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(log(z) - t) * y)) * x)
	tmp = 0.0
	if (y <= -140.0)
		tmp = t_1;
	elseif (y <= 4.2e-47)
		tmp = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp(((log(z) - t) * y)) * x;
	tmp = 0.0;
	if (y <= -140.0)
		tmp = t_1;
	elseif (y <= 4.2e-47)
		tmp = exp(((-z - b) * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -140.0], t$95$1, If[LessEqual[y, 4.2e-47], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -140 or 4.2000000000000001e-47 < y

   1. Initial program 96.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -140 < y < 4.2000000000000001e-47

   1. Initial program 96.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 88.5

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 88.5%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 88.5%

      \[\leadsto x \cdot e^{\left(\left(-z\right) - b\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -140:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-47}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\log z \cdot y} \cdot x\\ \mathbf{if}\;y \leq -140:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+68}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (log z) y)) x)))
   (if (<= y -140.0)
     t_1
     (if (<= y 2.7e+68) (* (exp (* (- (- z) b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((log(z) * y)) * x;
	double tmp;
	if (y <= -140.0) {
		tmp = t_1;
	} else if (y <= 2.7e+68) {
		tmp = exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((log(z) * y)) * x
    if (y <= (-140.0d0)) then
        tmp = t_1
    else if (y <= 2.7d+68) then
        tmp = exp(((-z - b) * a)) * 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 = Math.exp((Math.log(z) * y)) * x;
	double tmp;
	if (y <= -140.0) {
		tmp = t_1;
	} else if (y <= 2.7e+68) {
		tmp = Math.exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((math.log(z) * y)) * x
	tmp = 0
	if y <= -140.0:
		tmp = t_1
	elif y <= 2.7e+68:
		tmp = math.exp(((-z - b) * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(log(z) * y)) * x)
	tmp = 0.0
	if (y <= -140.0)
		tmp = t_1;
	elseif (y <= 2.7e+68)
		tmp = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((log(z) * y)) * x;
	tmp = 0.0;
	if (y <= -140.0)
		tmp = t_1;
	elseif (y <= 2.7e+68)
		tmp = exp(((-z - b) * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -140.0], t$95$1, If[LessEqual[y, 2.7e+68], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -140 or 2.69999999999999991e68 < y

   1. Initial program 95.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.6%

      \[\leadsto x \cdot e^{\log z \cdot y} \]

   if -140 < y < 2.69999999999999991e68

   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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 82.2

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 82.2%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.2%

      \[\leadsto x \cdot e^{\left(\left(-z\right) - b\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -140:\\ \;\;\;\;e^{\log z \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+68}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\log z \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{if}\;a \leq -5.9 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+26}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (- z) b) a)) x)))
   (if (<= a -5.9e+82) t_1 (if (<= a 1.05e+26) (* (exp (* (- t) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((-z - b) * a)) * x;
	double tmp;
	if (a <= -5.9e+82) {
		tmp = t_1;
	} else if (a <= 1.05e+26) {
		tmp = exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp(((-z - b) * a)) * x
    if (a <= (-5.9d+82)) then
        tmp = t_1
    else if (a <= 1.05d+26) then
        tmp = exp((-t * y)) * 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 = Math.exp(((-z - b) * a)) * x;
	double tmp;
	if (a <= -5.9e+82) {
		tmp = t_1;
	} else if (a <= 1.05e+26) {
		tmp = Math.exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp(((-z - b) * a)) * x
	tmp = 0
	if a <= -5.9e+82:
		tmp = t_1
	elif a <= 1.05e+26:
		tmp = math.exp((-t * y)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x)
	tmp = 0.0
	if (a <= -5.9e+82)
		tmp = t_1;
	elseif (a <= 1.05e+26)
		tmp = Float64(exp(Float64(Float64(-t) * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp(((-z - b) * a)) * x;
	tmp = 0.0;
	if (a <= -5.9e+82)
		tmp = t_1;
	elseif (a <= 1.05e+26)
		tmp = exp((-t * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[a, -5.9e+82], t$95$1, If[LessEqual[a, 1.05e+26], N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.8999999999999997e82 or 1.05e26 < a

   1. Initial program 91.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 78.6

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 78.6%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.6%

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

   if -5.8999999999999997e82 < a < 1.05e26

   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. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 69.0

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

   5. Applied rewrites 69.0%

      \[\leadsto x \cdot e^{\color{blue}{\left(-t\right) \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{+82}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+26}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 10^{-40}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- b) a)) x)))
   (if (<= b -1.9e+48) t_1 (if (<= b 1e-40) (* (exp (* (- t) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-b * a)) * x;
	double tmp;
	if (b <= -1.9e+48) {
		tmp = t_1;
	} else if (b <= 1e-40) {
		tmp = exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-b * a)) * x
    if (b <= (-1.9d+48)) then
        tmp = t_1
    else if (b <= 1d-40) then
        tmp = exp((-t * y)) * 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 = Math.exp((-b * a)) * x;
	double tmp;
	if (b <= -1.9e+48) {
		tmp = t_1;
	} else if (b <= 1e-40) {
		tmp = Math.exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-b * a)) * x
	tmp = 0
	if b <= -1.9e+48:
		tmp = t_1
	elif b <= 1e-40:
		tmp = math.exp((-t * y)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-b) * a)) * x)
	tmp = 0.0
	if (b <= -1.9e+48)
		tmp = t_1;
	elseif (b <= 1e-40)
		tmp = Float64(exp(Float64(Float64(-t) * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-b * a)) * x;
	tmp = 0.0;
	if (b <= -1.9e+48)
		tmp = t_1;
	elseif (b <= 1e-40)
		tmp = exp((-t * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -1.9e+48], t$95$1, If[LessEqual[b, 1e-40], N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9e48 or 9.9999999999999993e-41 < b

   1. Initial program 95.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if -1.9e48 < b < 9.9999999999999993e-41

   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. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 67.5

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

   5. Applied rewrites 67.5%

      \[\leadsto x \cdot e^{\color{blue}{\left(-t\right) \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+48}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{elif}\;b \leq 10^{-40}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+123}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot x\right) \cdot \left(\left(t \cdot t\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.5e+123)
   (* (* (* y y) x) (* (* t t) 0.5))
   (* (exp (* (- b) a)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.5e+123) {
		tmp = ((y * y) * x) * ((t * t) * 0.5);
	} else {
		tmp = exp((-b * a)) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.5d+123)) then
        tmp = ((y * y) * x) * ((t * t) * 0.5d0)
    else
        tmp = exp((-b * a)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.5e+123) {
		tmp = ((y * y) * x) * ((t * t) * 0.5);
	} else {
		tmp = Math.exp((-b * a)) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.5e+123:
		tmp = ((y * y) * x) * ((t * t) * 0.5)
	else:
		tmp = math.exp((-b * a)) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.5e+123)
		tmp = Float64(Float64(Float64(y * y) * x) * Float64(Float64(t * t) * 0.5));
	else
		tmp = Float64(exp(Float64(Float64(-b) * a)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.5e+123)
		tmp = ((y * y) * x) * ((t * t) * 0.5);
	else
		tmp = exp((-b * a)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.5e+123], N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.50000000000000004e123

   1. Initial program 89.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left({\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot \mathsf{fma}\left(0.5 \cdot y, {\left(\log z - t\right)}^{2}, \log z - t\right)\right), y, {\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot x\right)} \]

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left({\left(\log z - t\right)}^{2} \cdot y, 0.5, \log z\right) - t\right) \cdot y, \color{blue}{x}, x\right) \]

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 59.8%

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

   if -1.50000000000000004e123 < y

   1. Initial program 97.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 59.7

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

   5. Applied rewrites 59.7%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+123}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot x\right) \cdot \left(\left(t \cdot t\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(y \cdot y\right) \cdot x\right) \cdot \left(\left(t \cdot t\right) \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (* (* y y) x) (* (* t t) 0.5)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((y * y) * x) * ((t * t) * 0.5);
}

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 = ((y * y) * x) * ((t * t) * 0.5d0)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((y * y) * x) * ((t * t) * 0.5);
}

Python

def code(x, y, z, t, a, b):
	return ((y * y) * x) * ((t * t) * 0.5)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(y * y) * x) * Float64(Float64(t * t) * 0.5))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((y * y) * x) * ((t * t) * 0.5);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left({\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot \mathsf{fma}\left(0.5 \cdot y, {\left(\log z - t\right)}^{2}, \log z - t\right)\right), y, {\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot x\right)} \]

6. Taylor expanded in a around 0

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

8. Applied rewrites 41.5%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left({\left(\log z - t\right)}^{2} \cdot y, 0.5, \log z\right) - t\right) \cdot y, \color{blue}{x}, x\right) \]

9. Taylor expanded in t around inf

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

11. Applied rewrites 30.9%

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

12. Final simplification 30.9%

    \[\leadsto \left(\left(y \cdot y\right) \cdot x\right) \cdot \left(\left(t \cdot t\right) \cdot 0.5\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(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))))))