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

Percentage Accurate: 96.6% → 99.2%

Time: 6.1s

Alternatives: 13

Speedup: 1.3×

• 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

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: 96.6% 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: 99.2% accurate, 1.2× 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 96.6%

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

2. Taylor expanded in z around 0

   \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{-1 \cdot z} - b\right)} \]
3. 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 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 96.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 96.6

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

4. Applied rewrites 96.6%

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

Alternative 3: 89.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+138}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-a, b, \log z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= t -3.4e+25)
     t_1
     (if (<= t 2.85e+138) (* x (exp (fma (- a) b (* (log z) y)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-t * y)) * x;
	double tmp;
	if (t <= -3.4e+25) {
		tmp = t_1;
	} else if (t <= 2.85e+138) {
		tmp = x * exp(fma(-a, b, (log(z) * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-t) * y)) * x)
	tmp = 0.0
	if (t <= -3.4e+25)
		tmp = t_1;
	elseif (t <= 2.85e+138)
		tmp = Float64(x * exp(fma(Float64(-a), b, Float64(log(z) * y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -3.4e+25], t$95$1, If[LessEqual[t, 2.85e+138], N[(x * N[Exp[N[((-a) * b + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.39999999999999984e25 or 2.84999999999999993e138 < t

   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. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 56.4

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

   4. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.4

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

   6. Applied rewrites 56.4%

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

   if -3.39999999999999984e25 < t < 2.84999999999999993e138

   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. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 79.4

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

   7. Applied rewrites 79.4%

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

Alternative 4: 86.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2600000000000:\\ \;\;\;\;e^{\mathsf{fma}\left(-a, b, \left(-a\right) \cdot z\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (log z) t) y)) x)))
   (if (<= y -3.1e-74)
     t_1
     (if (<= y 2600000000000.0) (* (exp (fma (- a) b (* (- a) z))) 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 <= -3.1e-74) {
		tmp = t_1;
	} else if (y <= 2600000000000.0) {
		tmp = exp(fma(-a, b, (-a * z))) * 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 <= -3.1e-74)
		tmp = t_1;
	elseif (y <= 2600000000000.0)
		tmp = Float64(exp(fma(Float64(-a), b, Float64(Float64(-a) * z))) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3.1e-74], t$95$1, If[LessEqual[y, 2600000000000.0], N[(N[Exp[N[((-a) * b + N[((-a) * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1000000000000002e-74 or 2.6e12 < y

   1. Initial program 96.6%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 72.0

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

   4. Applied rewrites 72.0%

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

   if -3.1000000000000002e-74 < y < 2.6e12

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 58.7

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. sub-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-log1p.f64 62.6

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

   6. Applied rewrites 62.6%

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

   7. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto e^{\mathsf{fma}\left(\mathsf{neg}\left(a\right), b, -1 \cdot \left(a \cdot z\right)\right)} \cdot x \]

      4. lift-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-neg.f64 62.6

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

   9. Applied rewrites 62.6%

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

Alternative 5: 77.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {z}^{y} \cdot x\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 24000000000000:\\ \;\;\;\;e^{\mathsf{fma}\left(-a, b, \left(-a\right) \cdot z\right)} \cdot x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+210}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (pow z y) x)))
   (if (<= y -4.4e-6)
     t_1
     (if (<= y 24000000000000.0)
       (* (exp (fma (- a) b (* (- a) z))) x)
       (if (<= y 3e+210) t_1 (* (exp (* (- t) y)) x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(z, y) * x;
	double tmp;
	if (y <= -4.4e-6) {
		tmp = t_1;
	} else if (y <= 24000000000000.0) {
		tmp = exp(fma(-a, b, (-a * z))) * x;
	} else if (y <= 3e+210) {
		tmp = t_1;
	} else {
		tmp = exp((-t * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64((z ^ y) * x)
	tmp = 0.0
	if (y <= -4.4e-6)
		tmp = t_1;
	elseif (y <= 24000000000000.0)
		tmp = Float64(exp(fma(Float64(-a), b, Float64(Float64(-a) * z))) * x);
	elseif (y <= 3e+210)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(Float64(-t) * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -4.4e-6], t$95$1, If[LessEqual[y, 24000000000000.0], N[(N[Exp[N[((-a) * b + N[((-a) * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 3e+210], t$95$1, N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4000000000000002e-6 or 2.4e13 < y < 3.00000000000000022e210

   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. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-pow.f64 51.6

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

   7. Applied rewrites 51.6%

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

   if -4.4000000000000002e-6 < y < 2.4e13

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 58.7

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. sub-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-log1p.f64 62.6

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

   6. Applied rewrites 62.6%

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

   7. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto e^{\mathsf{fma}\left(\mathsf{neg}\left(a\right), b, -1 \cdot \left(a \cdot z\right)\right)} \cdot x \]

      4. lift-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-neg.f64 62.6

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

   9. Applied rewrites 62.6%

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

   if 3.00000000000000022e210 < y

   1. Initial program 96.6%

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

   2. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 56.4

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

   4. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.4

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

   6. Applied rewrites 56.4%

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

Alternative 6: 74.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {z}^{y} \cdot x\\ \mathbf{if}\;y \leq -85000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2600000000000:\\ \;\;\;\;e^{\left(-a\right) \cdot b} \cdot x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+210}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (pow z y) x)))
   (if (<= y -85000000000000.0)
     t_1
     (if (<= y 2600000000000.0)
       (* (exp (* (- a) b)) x)
       (if (<= y 3e+210) t_1 (* (exp (* (- t) y)) x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(z, y) * x;
	double tmp;
	if (y <= -85000000000000.0) {
		tmp = t_1;
	} else if (y <= 2600000000000.0) {
		tmp = exp((-a * b)) * x;
	} else if (y <= 3e+210) {
		tmp = t_1;
	} else {
		tmp = exp((-t * y)) * x;
	}
	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 = (z ** y) * x
    if (y <= (-85000000000000.0d0)) then
        tmp = t_1
    else if (y <= 2600000000000.0d0) then
        tmp = exp((-a * b)) * x
    else if (y <= 3d+210) then
        tmp = t_1
    else
        tmp = exp((-t * y)) * 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 t_1 = Math.pow(z, y) * x;
	double tmp;
	if (y <= -85000000000000.0) {
		tmp = t_1;
	} else if (y <= 2600000000000.0) {
		tmp = Math.exp((-a * b)) * x;
	} else if (y <= 3e+210) {
		tmp = t_1;
	} else {
		tmp = Math.exp((-t * y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(z, y) * x
	tmp = 0
	if y <= -85000000000000.0:
		tmp = t_1
	elif y <= 2600000000000.0:
		tmp = math.exp((-a * b)) * x
	elif y <= 3e+210:
		tmp = t_1
	else:
		tmp = math.exp((-t * y)) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64((z ^ y) * x)
	tmp = 0.0
	if (y <= -85000000000000.0)
		tmp = t_1;
	elseif (y <= 2600000000000.0)
		tmp = Float64(exp(Float64(Float64(-a) * b)) * x);
	elseif (y <= 3e+210)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(Float64(-t) * y)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z ^ y) * x;
	tmp = 0.0;
	if (y <= -85000000000000.0)
		tmp = t_1;
	elseif (y <= 2600000000000.0)
		tmp = exp((-a * b)) * x;
	elseif (y <= 3e+210)
		tmp = t_1;
	else
		tmp = exp((-t * y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -85000000000000.0], t$95$1, If[LessEqual[y, 2600000000000.0], N[(N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 3e+210], t$95$1, N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5e13 or 2.6e12 < y < 3.00000000000000022e210

   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. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-pow.f64 51.6

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

   7. Applied rewrites 51.6%

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

   if -8.5e13 < y < 2.6e12

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 58.7

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-exp.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 58.1

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

   7. Applied rewrites 58.1%

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

   if 3.00000000000000022e210 < y

   1. Initial program 96.6%

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

   2. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 56.4

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

   4. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.4

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

   6. Applied rewrites 56.4%

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

Alternative 7: 71.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-55}:\\ \;\;\;\;e^{\left(-a\right) \cdot b} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= t -1.25e+108)
     t_1
     (if (<= t 1.25e-55) (* (exp (* (- a) b)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-t * y)) * x;
	double tmp;
	if (t <= -1.25e+108) {
		tmp = t_1;
	} else if (t <= 1.25e-55) {
		tmp = exp((-a * b)) * x;
	} 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 = exp((-t * y)) * x
    if (t <= (-1.25d+108)) then
        tmp = t_1
    else if (t <= 1.25d-55) then
        tmp = exp((-a * b)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-t * y)) * x;
	double tmp;
	if (t <= -1.25e+108) {
		tmp = t_1;
	} else if (t <= 1.25e-55) {
		tmp = Math.exp((-a * b)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-t * y)) * x
	tmp = 0
	if t <= -1.25e+108:
		tmp = t_1
	elif t <= 1.25e-55:
		tmp = math.exp((-a * b)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-t) * y)) * x)
	tmp = 0.0
	if (t <= -1.25e+108)
		tmp = t_1;
	elseif (t <= 1.25e-55)
		tmp = Float64(exp(Float64(Float64(-a) * b)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-t * y)) * x;
	tmp = 0.0;
	if (t <= -1.25e+108)
		tmp = t_1;
	elseif (t <= 1.25e-55)
		tmp = exp((-a * b)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -1.25e+108], t$95$1, If[LessEqual[t, 1.25e-55], N[(N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.24999999999999998e108 or 1.25e-55 < t

   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. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 56.4

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

   4. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.4

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

   6. Applied rewrites 56.4%

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

   if -1.24999999999999998e108 < t < 1.25e-55

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 58.7

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-exp.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 58.1

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

   7. Applied rewrites 58.1%

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

Alternative 8: 58.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift--.f64 58.7

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

4. Applied rewrites 58.7%

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

5. Taylor expanded in z around 0

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

   1. lower-exp.f64 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lift-neg.f64 58.1

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

7. Applied rewrites 58.1%

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

Alternative 9: 32.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, y, 1\right) \cdot x\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{+120}:\\ \;\;\;\;\left(\mathsf{fma}\left(-a, b, \left(-a\right) \cdot z\right) + 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma (- t) y 1.0) x)))
   (if (<= t -3.7e+61)
     t_1
     (if (<= t 1.08e+120) (* (+ (fma (- a) b (* (- a) z)) 1.0) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(-t, y, 1.0) * x;
	double tmp;
	if (t <= -3.7e+61) {
		tmp = t_1;
	} else if (t <= 1.08e+120) {
		tmp = (fma(-a, b, (-a * z)) + 1.0) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(-t), y, 1.0) * x)
	tmp = 0.0
	if (t <= -3.7e+61)
		tmp = t_1;
	elseif (t <= 1.08e+120)
		tmp = Float64(Float64(fma(Float64(-a), b, Float64(Float64(-a) * z)) + 1.0) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[((-t) * y + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -3.7e+61], t$95$1, If[LessEqual[t, 1.08e+120], N[(N[(N[((-a) * b + N[((-a) * z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.70000000000000003e61 or 1.0799999999999999e120 < t

   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. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(1 + y \cdot \left(\log z - t\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\log z - t\right) + 1\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\log z - t\right) \cdot y + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 29.9

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

   7. Applied rewrites 29.9%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.4

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

   10. Applied rewrites 27.4%

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

   if -3.70000000000000003e61 < t < 1.0799999999999999e120

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 58.7

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto \left(1 + a \cdot \left(\log \left(1 - z\right) - b\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(a \cdot \left(\log \left(1 - z\right) - b\right) + 1\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\log \left(1 - z\right) - b\right) \cdot a + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log \left(1 - z\right) - b, a, 1\right) \cdot x \]

      4. lift-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log \left(1 - z\right) - b, a, 1\right) \cdot x \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log \left(1 - z\right) - b, a, 1\right) \cdot x \]

      6. lift--.f64 27.7

         \[\leadsto \mathsf{fma}\left(\log \left(1 - z\right) - b, a, 1\right) \cdot x \]

   7. Applied rewrites 27.7%

      \[\leadsto \mathsf{fma}\left(\log \left(1 - z\right) - b, a, 1\right) \cdot x \]

   8. Taylor expanded in z around 0

      \[\leadsto \left(1 + \left(-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)\right)\right) \cdot x \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\left(-1 \cdot a\right) \cdot b + -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-1 \cdot a, b, -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{neg}\left(a\right), b, -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      6. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-a, b, -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      7. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(-a, b, \left(-1 \cdot a\right) \cdot z\right) + 1\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-a, b, \left(-1 \cdot a\right) \cdot z\right) + 1\right) \cdot x \]

      9. mul-1-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(-a, b, \left(\mathsf{neg}\left(a\right)\right) \cdot z\right) + 1\right) \cdot x \]

      10. lift-neg.f64 28.1

          \[\leadsto \left(\mathsf{fma}\left(-a, b, \left(-a\right) \cdot z\right) + 1\right) \cdot x \]

   10. Applied rewrites 28.1%

       \[\leadsto \left(\mathsf{fma}\left(-a, b, \left(-a\right) \cdot z\right) + 1\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 32.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+132}:\\ \;\;\;\;\left(\mathsf{fma}\left(-a, b, \left(-a\right) \cdot z\right) + 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log z - t, y, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -5.2e+132)
   (* (+ (fma (- a) b (* (- a) z)) 1.0) x)
   (* (fma (- (log z) t) y 1.0) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -5.2e+132) {
		tmp = (fma(-a, b, (-a * z)) + 1.0) * x;
	} else {
		tmp = fma((log(z) - t), y, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -5.2e+132)
		tmp = Float64(Float64(fma(Float64(-a), b, Float64(Float64(-a) * z)) + 1.0) * x);
	else
		tmp = Float64(fma(Float64(log(z) - t), y, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -5.2e+132], N[(N[(N[((-a) * b + N[((-a) * z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.2e132

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 58.7

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto \left(1 + a \cdot \left(\log \left(1 - z\right) - b\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(a \cdot \left(\log \left(1 - z\right) - b\right) + 1\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\log \left(1 - z\right) - b\right) \cdot a + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log \left(1 - z\right) - b, a, 1\right) \cdot x \]

      4. lift-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log \left(1 - z\right) - b, a, 1\right) \cdot x \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log \left(1 - z\right) - b, a, 1\right) \cdot x \]

      6. lift--.f64 27.7

         \[\leadsto \mathsf{fma}\left(\log \left(1 - z\right) - b, a, 1\right) \cdot x \]

   7. Applied rewrites 27.7%

      \[\leadsto \mathsf{fma}\left(\log \left(1 - z\right) - b, a, 1\right) \cdot x \]

   8. Taylor expanded in z around 0

      \[\leadsto \left(1 + \left(-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)\right)\right) \cdot x \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\left(-1 \cdot a\right) \cdot b + -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-1 \cdot a, b, -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{neg}\left(a\right), b, -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      6. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-a, b, -1 \cdot \left(a \cdot z\right)\right) + 1\right) \cdot x \]

      7. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(-a, b, \left(-1 \cdot a\right) \cdot z\right) + 1\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-a, b, \left(-1 \cdot a\right) \cdot z\right) + 1\right) \cdot x \]

      9. mul-1-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(-a, b, \left(\mathsf{neg}\left(a\right)\right) \cdot z\right) + 1\right) \cdot x \]

      10. lift-neg.f64 28.1

          \[\leadsto \left(\mathsf{fma}\left(-a, b, \left(-a\right) \cdot z\right) + 1\right) \cdot x \]

   10. Applied rewrites 28.1%

       \[\leadsto \left(\mathsf{fma}\left(-a, b, \left(-a\right) \cdot z\right) + 1\right) \cdot x \]

   if -5.2e132 < a

   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. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(1 + y \cdot \left(\log z - t\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\log z - t\right) + 1\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\log z - t\right) \cdot y + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 29.9

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

   7. Applied rewrites 29.9%

      \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 31.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, y, 1\right) \cdot x\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(-a, b, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma (- t) y 1.0) x)))
   (if (<= t -3.6e+61) t_1 (if (<= t 1.08e+120) (* (fma (- a) b 1.0) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(-t, y, 1.0) * x;
	double tmp;
	if (t <= -3.6e+61) {
		tmp = t_1;
	} else if (t <= 1.08e+120) {
		tmp = fma(-a, b, 1.0) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(-t), y, 1.0) * x)
	tmp = 0.0
	if (t <= -3.6e+61)
		tmp = t_1;
	elseif (t <= 1.08e+120)
		tmp = Float64(fma(Float64(-a), b, 1.0) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[((-t) * y + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -3.6e+61], t$95$1, If[LessEqual[t, 1.08e+120], N[(N[((-a) * b + 1.0), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.6000000000000001e61 or 1.0799999999999999e120 < t

   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. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(1 + y \cdot \left(\log z - t\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\log z - t\right) + 1\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\log z - t\right) \cdot y + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 29.9

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

   7. Applied rewrites 29.9%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.4

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

   10. Applied rewrites 27.4%

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

   if -3.6000000000000001e61 < t < 1.0799999999999999e120

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 58.7

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-exp.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 58.1

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

   7. Applied rewrites 58.1%

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

   8. Taylor expanded in a around 0

      \[\leadsto \left(1 + -1 \cdot \left(a \cdot b\right)\right) \cdot x \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot b\right) + 1\right) \cdot x \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot a\right) \cdot b + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot a, b, 1\right) \cdot x \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), b, 1\right) \cdot x \]

      5. lift-neg.f64 27.7

         \[\leadsto \mathsf{fma}\left(-a, b, 1\right) \cdot x \]

   10. Applied rewrites 27.7%

       \[\leadsto \mathsf{fma}\left(-a, b, 1\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 27.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-t, y, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (fma (- t) y 1.0) x))

C

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

Julia

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

Wolfram

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

Derivation

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. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 72.0

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

4. Applied rewrites 72.0%

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

5. Taylor expanded in y around 0

   \[\leadsto \left(1 + y \cdot \left(\log z - t\right)\right) \cdot x \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(y \cdot \left(\log z - t\right) + 1\right) \cdot x \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\log z - t\right) \cdot y + 1\right) \cdot x \]

   3. lower-fma.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift--.f64 29.9

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

7. Applied rewrites 29.9%

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

8. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 27.4

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

10. Applied rewrites 27.4%

    \[\leadsto \mathsf{fma}\left(-t, y, 1\right) \cdot x \]
11. Add Preprocessing

Alternative 13: 18.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * 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 = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Python

def code(x, y, z, t, a, b):
	return 1.0 * x

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(1.0 * x), $MachinePrecision]

Derivation

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. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 72.0

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

4. Applied rewrites 72.0%

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

5. Taylor expanded in y around 0

   \[\leadsto 1 \cdot x \]
6. Step-by-step derivation

7. Applied rewrites 18.7%

   \[\leadsto 1 \cdot x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(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))))))