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

Percentage Accurate: 96.8% → 96.8%

Time: 10.3s

Alternatives: 9

Speedup: 1.0×

• 41.2% 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: 96.8% 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: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return exp((((log((1.0 - z)) - b) * a) + ((log(z) - t) * y))) * 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((((log((1.0d0 - z)) - b) * a) + ((log(z) - t) * y))) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 96.9%

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

Alternative 2: 86.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+24}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (log z) t) y)) x)))
   (if (<= y -7.2e-60)
     t_1
     (if (<= y 9e+24) (* (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 <= -7.2e-60) {
		tmp = t_1;
	} else if (y <= 9e+24) {
		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 <= (-7.2d-60)) then
        tmp = t_1
    else if (y <= 9d+24) 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 <= -7.2e-60) {
		tmp = t_1;
	} else if (y <= 9e+24) {
		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 <= -7.2e-60:
		tmp = t_1
	elif y <= 9e+24:
		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 <= -7.2e-60)
		tmp = t_1;
	elseif (y <= 9e+24)
		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 <= -7.2e-60)
		tmp = t_1;
	elseif (y <= 9e+24)
		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, -7.2e-60], t$95$1, If[LessEqual[y, 9e+24], 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 < -7.2e-60 or 9.00000000000000039e24 < y

   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 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%

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

   if -7.2e-60 < y < 9.00000000000000039e24

   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 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 87.5

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

   5. Applied rewrites 87.5%

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

   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 87.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-60}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+24}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-25}:\\ \;\;\;\;e^{\log z \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+69}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\frac{y}{t} \cdot t\right) \cdot \left(-t\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1e-25)
   (* (exp (* (log z) y)) x)
   (if (<= y 6.2e+69)
     (* (exp (* (- (- z) b) a)) x)
     (* (exp (* (* (/ y t) t) (- t))) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1e-25) {
		tmp = exp((log(z) * y)) * x;
	} else if (y <= 6.2e+69) {
		tmp = exp(((-z - b) * a)) * x;
	} else {
		tmp = exp((((y / t) * t) * -t)) * 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 <= (-1d-25)) then
        tmp = exp((log(z) * y)) * x
    else if (y <= 6.2d+69) then
        tmp = exp(((-z - b) * a)) * x
    else
        tmp = exp((((y / t) * t) * -t)) * 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 <= -1e-25) {
		tmp = Math.exp((Math.log(z) * y)) * x;
	} else if (y <= 6.2e+69) {
		tmp = Math.exp(((-z - b) * a)) * x;
	} else {
		tmp = Math.exp((((y / t) * t) * -t)) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1e-25:
		tmp = math.exp((math.log(z) * y)) * x
	elif y <= 6.2e+69:
		tmp = math.exp(((-z - b) * a)) * x
	else:
		tmp = math.exp((((y / t) * t) * -t)) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1e-25)
		tmp = Float64(exp(Float64(log(z) * y)) * x);
	elseif (y <= 6.2e+69)
		tmp = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x);
	else
		tmp = Float64(exp(Float64(Float64(Float64(y / t) * t) * Float64(-t))) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1e-25)
		tmp = exp((log(z) * y)) * x;
	elseif (y <= 6.2e+69)
		tmp = exp(((-z - b) * a)) * x;
	else
		tmp = exp((((y / t) * t) * -t)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1e-25], N[(N[Exp[N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 6.2e+69], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], N[(N[Exp[N[(N[(N[(y / t), $MachinePrecision] * t), $MachinePrecision] * (-t)), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.00000000000000004e-25

   1. Initial program 96.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   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 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 71.5%

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

   if -1.00000000000000004e-25 < y < 6.1999999999999997e69

   1. Initial program 96.1%

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

   3. Taylor expanded in a around 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 82.8

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

   5. Applied rewrites 82.8%

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

   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 82.8%

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

   if 6.1999999999999997e69 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 79.1

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 79.1%

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

   9. Applied rewrites 90.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-25}:\\ \;\;\;\;e^{\log z \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+69}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\frac{y}{t} \cdot t\right) \cdot \left(-t\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\frac{y}{t} \cdot t\right) \cdot \left(-t\right)} \cdot x\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+69}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (* (/ y t) t) (- t))) x)))
   (if (<= y -7.2e-60)
     t_1
     (if (<= y 6.2e+69) (* (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((((y / t) * t) * -t)) * x;
	double tmp;
	if (y <= -7.2e-60) {
		tmp = t_1;
	} else if (y <= 6.2e+69) {
		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((((y / t) * t) * -t)) * x
    if (y <= (-7.2d-60)) then
        tmp = t_1
    else if (y <= 6.2d+69) 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((((y / t) * t) * -t)) * x;
	double tmp;
	if (y <= -7.2e-60) {
		tmp = t_1;
	} else if (y <= 6.2e+69) {
		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((((y / t) * t) * -t)) * x
	tmp = 0
	if y <= -7.2e-60:
		tmp = t_1
	elif y <= 6.2e+69:
		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(Float64(y / t) * t) * Float64(-t))) * x)
	tmp = 0.0
	if (y <= -7.2e-60)
		tmp = t_1;
	elseif (y <= 6.2e+69)
		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((((y / t) * t) * -t)) * x;
	tmp = 0.0;
	if (y <= -7.2e-60)
		tmp = t_1;
	elseif (y <= 6.2e+69)
		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[(y / t), $MachinePrecision] * t), $MachinePrecision] * (-t)), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -7.2e-60], t$95$1, If[LessEqual[y, 6.2e+69], 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 < -7.2e-60 or 6.1999999999999997e69 < y

   1. Initial program 98.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. 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 64.5

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

   5. Applied rewrites 64.5%

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

   7. Applied rewrites 62.0%

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

   9. Applied rewrites 73.6%

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

   if -7.2e-60 < y < 6.1999999999999997e69

   1. Initial program 95.8%

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

   3. Taylor expanded in a around 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 84.4

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

   5. Applied rewrites 84.4%

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

   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 84.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-60}:\\ \;\;\;\;e^{\left(\frac{y}{t} \cdot t\right) \cdot \left(-t\right)} \cdot x\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+69}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\frac{y}{t} \cdot t\right) \cdot \left(-t\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-123}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (- z) b) a)) x)))
   (if (<= a -2.9e-42) t_1 (if (<= a 7.5e-123) (* (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 <= -2.9e-42) {
		tmp = t_1;
	} else if (a <= 7.5e-123) {
		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 <= (-2.9d-42)) then
        tmp = t_1
    else if (a <= 7.5d-123) 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 <= -2.9e-42) {
		tmp = t_1;
	} else if (a <= 7.5e-123) {
		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 <= -2.9e-42:
		tmp = t_1
	elif a <= 7.5e-123:
		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 <= -2.9e-42)
		tmp = t_1;
	elseif (a <= 7.5e-123)
		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 <= -2.9e-42)
		tmp = t_1;
	elseif (a <= 7.5e-123)
		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, -2.9e-42], t$95$1, If[LessEqual[a, 7.5e-123], N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.9000000000000003e-42 or 7.50000000000000011e-123 < a

   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 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 83.4

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

   5. Applied rewrites 83.4%

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

   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 83.4%

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

   if -2.9000000000000003e-42 < a < 7.50000000000000011e-123

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. 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 73.3

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

   5. Applied rewrites 73.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-42}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-123}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-123}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- b) a)) x)))
   (if (<= a -2.9e-42) t_1 (if (<= a 7.5e-123) (* (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 (a <= -2.9e-42) {
		tmp = t_1;
	} else if (a <= 7.5e-123) {
		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 (a <= (-2.9d-42)) then
        tmp = t_1
    else if (a <= 7.5d-123) 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 (a <= -2.9e-42) {
		tmp = t_1;
	} else if (a <= 7.5e-123) {
		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 a <= -2.9e-42:
		tmp = t_1
	elif a <= 7.5e-123:
		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 (a <= -2.9e-42)
		tmp = t_1;
	elseif (a <= 7.5e-123)
		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 (a <= -2.9e-42)
		tmp = t_1;
	elseif (a <= 7.5e-123)
		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[a, -2.9e-42], t$95$1, If[LessEqual[a, 7.5e-123], N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.9000000000000003e-42 or 7.50000000000000011e-123 < a

   1. Initial program 94.8%

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

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

   if -2.9000000000000003e-42 < a < 7.50000000000000011e-123

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. 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 73.3

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

   5. Applied rewrites 73.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-42}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-123}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{if}\;b \leq -9.2 \cdot 10^{-153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-199}:\\ \;\;\;\;e^{\left(-z\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- b) a)) x)))
   (if (<= b -9.2e-153) t_1 (if (<= b 2e-199) (* (exp (* (- z) a)) 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 <= -9.2e-153) {
		tmp = t_1;
	} else if (b <= 2e-199) {
		tmp = exp((-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((-b * a)) * x
    if (b <= (-9.2d-153)) then
        tmp = t_1
    else if (b <= 2d-199) then
        tmp = exp((-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((-b * a)) * x;
	double tmp;
	if (b <= -9.2e-153) {
		tmp = t_1;
	} else if (b <= 2e-199) {
		tmp = Math.exp((-z * a)) * 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 <= -9.2e-153:
		tmp = t_1
	elif b <= 2e-199:
		tmp = math.exp((-z * a)) * 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 <= -9.2e-153)
		tmp = t_1;
	elseif (b <= 2e-199)
		tmp = Float64(exp(Float64(Float64(-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((-b * a)) * x;
	tmp = 0.0;
	if (b <= -9.2e-153)
		tmp = t_1;
	elseif (b <= 2e-199)
		tmp = exp((-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[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -9.2e-153], t$95$1, If[LessEqual[b, 2e-199], N[(N[Exp[N[((-z) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -9.19999999999999988e-153 or 1.99999999999999996e-199 < b

   1. Initial program 99.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. 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 66.7

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

   5. Applied rewrites 66.7%

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

   if -9.19999999999999988e-153 < b < 1.99999999999999996e-199

   1. Initial program 90.1%

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

   3. Taylor expanded in a around 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 52.7

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

   5. Applied rewrites 52.7%

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

   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 52.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 51.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-153}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-199}:\\ \;\;\;\;e^{\left(-z\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-61}:\\ \;\;\;\;e^{a \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-z\right) \cdot a} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.1e-61) (* (exp (* a z)) x) (* (exp (* (- z) a)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.1e-61) {
		tmp = exp((a * z)) * x;
	} else {
		tmp = exp((-z * 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 (b <= (-3.1d-61)) then
        tmp = exp((a * z)) * x
    else
        tmp = exp((-z * 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 (b <= -3.1e-61) {
		tmp = Math.exp((a * z)) * x;
	} else {
		tmp = Math.exp((-z * a)) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -3.1e-61:
		tmp = math.exp((a * z)) * x
	else:
		tmp = math.exp((-z * a)) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.1e-61)
		tmp = Float64(exp(Float64(a * z)) * x);
	else
		tmp = Float64(exp(Float64(Float64(-z) * a)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -3.1e-61)
		tmp = exp((a * z)) * x;
	else
		tmp = exp((-z * a)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.1e-61], N[(N[Exp[N[(a * z), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], N[(N[Exp[N[((-z) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.09999999999999995e-61

   1. Initial program 98.9%

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

   3. Taylor expanded in 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 72.8

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

   5. Applied rewrites 72.8%

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

   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 72.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 10.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 10.2

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

   13. Applied rewrites 29.5%

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

   if -3.09999999999999995e-61 < b

   1. Initial program 95.8%

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

   3. Taylor expanded in a around 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 58.8

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

   5. Applied rewrites 58.8%

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

   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 58.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 44.9%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-61}:\\ \;\;\;\;e^{a \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-z\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 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 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 63.9

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

5. Applied rewrites 63.9%

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

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 63.9%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 32.5%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 32.5

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

13. Applied rewrites 27.1%

    \[\leadsto \color{blue}{e^{a \cdot z} \cdot x} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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))))))