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

Percentage Accurate: 96.5% → 96.4%

Time: 11.3s

Alternatives: 8

Speedup: N/A×

• 41.8% 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: 96.4% 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 98.4%

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

3. Taylor expanded in 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

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-log.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 2: 88.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+137}:\\ \;\;\;\;e^{\mathsf{fma}\left(-a, b, \log z \cdot y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= t -5.4e+54)
     t_1
     (if (<= t 1.4e+137) (* (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 <= -5.4e+54) {
		tmp = t_1;
	} else if (t <= 1.4e+137) {
		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 <= -5.4e+54)
		tmp = t_1;
	elseif (t <= 1.4e+137)
		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, -5.4e+54], t$95$1, If[LessEqual[t, 1.4e+137], 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 < -5.40000000000000022e54 or 1.4e137 < t

   1. Initial program 97.6%

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

   3. Taylor expanded in 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 87.7

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

   5. Applied rewrites 87.7%

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

   if -5.40000000000000022e54 < t < 1.4e137

   1. Initial program 98.8%

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

   3. Taylor expanded in 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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 98.2%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{+54}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+137}:\\ \;\;\;\;e^{\mathsf{fma}\left(-a, b, \log z \cdot y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-90}:\\ \;\;\;\;e^{\left(\left(-b\right) - z\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (log z) t) y)) x)))
   (if (<= y -4.8e+39)
     t_1
     (if (<= y 1.4e-90) (* (exp (* (- (- b) z) 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 <= -4.8e+39) {
		tmp = t_1;
	} else if (y <= 1.4e-90) {
		tmp = exp(((-b - z) * 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 <= (-4.8d+39)) then
        tmp = t_1
    else if (y <= 1.4d-90) then
        tmp = exp(((-b - z) * 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 <= -4.8e+39) {
		tmp = t_1;
	} else if (y <= 1.4e-90) {
		tmp = Math.exp(((-b - z) * 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 <= -4.8e+39:
		tmp = t_1
	elif y <= 1.4e-90:
		tmp = math.exp(((-b - z) * 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 <= -4.8e+39)
		tmp = t_1;
	elseif (y <= 1.4e-90)
		tmp = Float64(exp(Float64(Float64(Float64(-b) - z) * 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 <= -4.8e+39)
		tmp = t_1;
	elseif (y <= 1.4e-90)
		tmp = exp(((-b - z) * 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, -4.8e+39], t$95$1, If[LessEqual[y, 1.4e-90], N[(N[Exp[N[(N[((-b) - z), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000002e39 or 1.3999999999999999e-90 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -4.8000000000000002e39 < y < 1.3999999999999999e-90

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 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 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+39}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-90}:\\ \;\;\;\;e^{\left(\left(-b\right) - z\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\log z \cdot y} \cdot x\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;e^{\left(\left(-b\right) - z\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (log z) y)) x)))
   (if (<= y -6.5e+39)
     t_1
     (if (<= y 1.45e+76) (* (exp (* (- (- b) z) 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 <= -6.5e+39) {
		tmp = t_1;
	} else if (y <= 1.45e+76) {
		tmp = exp(((-b - z) * 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 <= (-6.5d+39)) then
        tmp = t_1
    else if (y <= 1.45d+76) then
        tmp = exp(((-b - z) * 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 <= -6.5e+39) {
		tmp = t_1;
	} else if (y <= 1.45e+76) {
		tmp = Math.exp(((-b - z) * 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 <= -6.5e+39:
		tmp = t_1
	elif y <= 1.45e+76:
		tmp = math.exp(((-b - z) * 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 <= -6.5e+39)
		tmp = t_1;
	elseif (y <= 1.45e+76)
		tmp = Float64(exp(Float64(Float64(Float64(-b) - z) * 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 <= -6.5e+39)
		tmp = t_1;
	elseif (y <= 1.45e+76)
		tmp = exp(((-b - z) * 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, -6.5e+39], t$95$1, If[LessEqual[y, 1.45e+76], N[(N[Exp[N[(N[((-b) - z), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5000000000000001e39 or 1.4500000000000001e76 < 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 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 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.4%

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

   if -6.5000000000000001e39 < y < 1.4500000000000001e76

   1. Initial program 99.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 84.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+39}:\\ \;\;\;\;e^{\log z \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;e^{\left(\left(-b\right) - z\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\log z \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-90}:\\ \;\;\;\;e^{\left(\left(-b\right) - z\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= y -5e+39)
     t_1
     (if (<= y 1.4e-90) (* (exp (* (- (- b) z) a)) 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 (y <= -5e+39) {
		tmp = t_1;
	} else if (y <= 1.4e-90) {
		tmp = exp(((-b - z) * 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((-t * y)) * x
    if (y <= (-5d+39)) then
        tmp = t_1
    else if (y <= 1.4d-90) then
        tmp = exp(((-b - z) * 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((-t * y)) * x;
	double tmp;
	if (y <= -5e+39) {
		tmp = t_1;
	} else if (y <= 1.4e-90) {
		tmp = Math.exp(((-b - z) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-t * y)) * x
	tmp = 0
	if y <= -5e+39:
		tmp = t_1
	elif y <= 1.4e-90:
		tmp = math.exp(((-b - z) * a)) * 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 (y <= -5e+39)
		tmp = t_1;
	elseif (y <= 1.4e-90)
		tmp = Float64(exp(Float64(Float64(Float64(-b) - z) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-t * y)) * x;
	tmp = 0.0;
	if (y <= -5e+39)
		tmp = t_1;
	elseif (y <= 1.4e-90)
		tmp = exp(((-b - z) * 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[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -5e+39], t$95$1, If[LessEqual[y, 1.4e-90], N[(N[Exp[N[(N[((-b) - z), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000015e39 or 1.3999999999999999e-90 < y

   1. Initial program 97.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. 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 64.8

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

   5. Applied rewrites 64.8%

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

   if -5.00000000000000015e39 < y < 1.3999999999999999e-90

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 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 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-90}:\\ \;\;\;\;e^{\left(\left(-b\right) - z\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-90}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= y -5e+39) t_1 (if (<= y 1.4e-90) (* (exp (* (- b) a)) 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 (y <= -5e+39) {
		tmp = t_1;
	} else if (y <= 1.4e-90) {
		tmp = exp((-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((-t * y)) * x
    if (y <= (-5d+39)) then
        tmp = t_1
    else if (y <= 1.4d-90) then
        tmp = exp((-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((-t * y)) * x;
	double tmp;
	if (y <= -5e+39) {
		tmp = t_1;
	} else if (y <= 1.4e-90) {
		tmp = Math.exp((-b * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-t * y)) * x
	tmp = 0
	if y <= -5e+39:
		tmp = t_1
	elif y <= 1.4e-90:
		tmp = math.exp((-b * a)) * 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 (y <= -5e+39)
		tmp = t_1;
	elseif (y <= 1.4e-90)
		tmp = Float64(exp(Float64(Float64(-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((-t * y)) * x;
	tmp = 0.0;
	if (y <= -5e+39)
		tmp = t_1;
	elseif (y <= 1.4e-90)
		tmp = exp((-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[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -5e+39], t$95$1, If[LessEqual[y, 1.4e-90], N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000015e39 or 1.3999999999999999e-90 < y

   1. Initial program 97.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. 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 64.8

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

   5. Applied rewrites 64.8%

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

   if -5.00000000000000015e39 < y < 1.3999999999999999e-90

   1. Initial program 99.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. 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 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-90}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return exp((-b * a)) * 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 = exp((-b * a)) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in b around inf

   \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
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 60.2

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

5. Applied rewrites 60.2%

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

6. Final simplification 60.2%

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

Alternative 8: 28.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ e^{a \cdot z} \cdot x \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return exp((a * z)) * 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 = exp((a * z)) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in 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 61.6

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

5. Applied rewrites 61.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 61.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 61.6

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

10. Applied rewrites 63.2%

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

11. Taylor expanded in b around 0

    \[\leadsto e^{a \cdot z} \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 30.4%

    \[\leadsto e^{z \cdot a} \cdot x \]

14. Final simplification 30.4%

    \[\leadsto e^{a \cdot z} \cdot x \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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))))))