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

Percentage Accurate: 96.8% → 99.3%

Time: 6.0s

Alternatives: 7

Speedup: 1.4×

• 41.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.3

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

5. Applied rewrites 99.3%

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

Alternative 2: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{\left(\log z - t\right) \cdot y}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8000000000000:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(y, -t, a \cdot \left(\left(-z\right) - b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* (- (log z) t) y)))))
   (if (<= y -3.2e+97)
     t_1
     (if (<= y 8000000000000.0)
       (* x (exp (fma y (- t) (* a (- (- z) b)))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp(((log(z) - t) * y));
	double tmp;
	if (y <= -3.2e+97) {
		tmp = t_1;
	} else if (y <= 8000000000000.0) {
		tmp = x * exp(fma(y, -t, (a * (-z - b))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(Float64(log(z) - t) * y)))
	tmp = 0.0
	if (y <= -3.2e+97)
		tmp = t_1;
	elseif (y <= 8000000000000.0)
		tmp = Float64(x * exp(fma(y, Float64(-t), Float64(a * Float64(Float64(-z) - b)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+97], t$95$1, If[LessEqual[y, 8000000000000.0], N[(x * N[Exp[N[(y * (-t) + N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.20000000000000016e97 or 8e12 < y

   1. Initial program 98.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if -3.20000000000000016e97 < y < 8e12

   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 z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 95.5

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

   8. Applied rewrites 95.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 95.5

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

   10. Applied rewrites 95.5%

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

Alternative 3: 72.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-237}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \mathbf{elif}\;t \leq 1000000000:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* (- t) y)))))
   (if (<= t -2e+32)
     t_1
     (if (<= t -8.5e-237)
       (* x (exp (* (- a) b)))
       (if (<= t 1000000000.0) (* x (pow z y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((-t * y));
	double tmp;
	if (t <= -2e+32) {
		tmp = t_1;
	} else if (t <= -8.5e-237) {
		tmp = x * exp((-a * b));
	} else if (t <= 1000000000.0) {
		tmp = x * pow(z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((-t * y))
    if (t <= (-2d+32)) then
        tmp = t_1
    else if (t <= (-8.5d-237)) then
        tmp = x * exp((-a * b))
    else if (t <= 1000000000.0d0) then
        tmp = x * (z ** y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((-t * y));
	double tmp;
	if (t <= -2e+32) {
		tmp = t_1;
	} else if (t <= -8.5e-237) {
		tmp = x * Math.exp((-a * b));
	} else if (t <= 1000000000.0) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((-t * y))
	tmp = 0
	if t <= -2e+32:
		tmp = t_1
	elif t <= -8.5e-237:
		tmp = x * math.exp((-a * b))
	elif t <= 1000000000.0:
		tmp = x * math.pow(z, y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(Float64(-t) * y)))
	tmp = 0.0
	if (t <= -2e+32)
		tmp = t_1;
	elseif (t <= -8.5e-237)
		tmp = Float64(x * exp(Float64(Float64(-a) * b)));
	elseif (t <= 1000000000.0)
		tmp = Float64(x * (z ^ y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((-t * y));
	tmp = 0.0;
	if (t <= -2e+32)
		tmp = t_1;
	elseif (t <= -8.5e-237)
		tmp = x * exp((-a * b));
	elseif (t <= 1000000000.0)
		tmp = x * (z ^ y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e+32], t$95$1, If[LessEqual[t, -8.5e-237], N[(x * N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1000000000.0], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.00000000000000011e32 or 1e9 < t

   1. Initial program 96.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if -2.00000000000000011e32 < t < -8.49999999999999951e-237

   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 b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 66.2

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

   5. Applied rewrites 66.2%

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

   if -8.49999999999999951e-237 < t < 1e9

   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 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. *-commutative N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in a around 0

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

      1. lift-pow.f64 65.7

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

   8. Applied rewrites 65.7%

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -55000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -55000.0) t_1 (if (<= y 3.7e+34) (* x (exp (* (- a) b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -55000.0) {
		tmp = t_1;
	} else if (y <= 3.7e+34) {
		tmp = x * exp((-a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z ** y)
    if (y <= (-55000.0d0)) then
        tmp = t_1
    else if (y <= 3.7d+34) then
        tmp = x * exp((-a * b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double tmp;
	if (y <= -55000.0) {
		tmp = t_1;
	} else if (y <= 3.7e+34) {
		tmp = x * Math.exp((-a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if y <= -55000.0:
		tmp = t_1
	elif y <= 3.7e+34:
		tmp = x * math.exp((-a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -55000.0)
		tmp = t_1;
	elseif (y <= 3.7e+34)
		tmp = Float64(x * exp(Float64(Float64(-a) * b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -55000.0)
		tmp = t_1;
	elseif (y <= 3.7e+34)
		tmp = x * exp((-a * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -55000.0], t$95$1, If[LessEqual[y, 3.7e+34], N[(x * N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -55000 or 3.70000000000000009e34 < y

   1. Initial program 98.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. *-commutative N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in a around 0

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

      1. lift-pow.f64 67.7

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

   8. Applied rewrites 67.7%

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

   if -55000 < y < 3.70000000000000009e34

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 78.0

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

   5. Applied rewrites 78.0%

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

Alternative 5: 87.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 z around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 87.3

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

8. Applied rewrites 87.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 87.5

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

10. Applied rewrites 87.5%

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

Alternative 6: 51.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 * (z ** y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. *-commutative N/A

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

   4. pow-to-exp N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift--.f64 62.6

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

5. Applied rewrites 62.6%

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

6. Taylor expanded in a around 0

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

   1. lift-pow.f64 51.2

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

8. Applied rewrites 51.2%

   \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
9. Add Preprocessing

Alternative 7: 18.7% accurate, 328.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 96.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. exp-diff N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 74.9%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

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

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

   4. lift-log.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-exp.f64 71.7

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

7. Applied rewrites 71.7%

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

8. Taylor expanded in y around 0

   \[\leadsto x \]
9. Step-by-step derivation

10. Applied rewrites 18.7%

    \[\leadsto x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025091 
(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))))))