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

Percentage Accurate: 96.5% → 99.3%

Time: 5.1s

Alternatives: 10

Speedup: 1.2×

• 41.0% 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: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 1.2× 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 96.5%

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

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 99.3

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

4. Applied rewrites 99.3%

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

Alternative 2: 88.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+236}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\left(-z\right) - b, a, \left(-t\right) \cdot y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2e+236)
   (* (exp (* (- (log z) t) y)) x)
   (* (exp (fma (- (- z) b) a (* (- t) y))) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2e+236) {
		tmp = exp(((log(z) - t) * y)) * x;
	} else {
		tmp = exp(fma((-z - b), a, (-t * y))) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2e+236)
		tmp = Float64(exp(Float64(Float64(log(z) - t) * y)) * x);
	else
		tmp = Float64(exp(fma(Float64(Float64(-z) - b), a, Float64(Float64(-t) * y))) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2e+236], N[(N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a + N[((-t) * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.00000000000000011e236

   1. Initial program 98.0%

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

   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 94.5

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

   4. Applied rewrites 94.5%

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

   if -2.00000000000000011e236 < y

   1. Initial program 96.4%

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

   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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-1 \cdot t\right)} + a \cdot \left(\left(-z\right) - b\right)} \]
   6. 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 87.6

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

   7. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.6

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

   9. Applied rewrites 87.7%

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

Alternative 3: 84.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;e^{\left(-a\right) \cdot b} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (log z) t) y)) x)))
   (if (<= y -1.45e-18) t_1 (if (<= y 1.75e-7) (* (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(((log(z) - t) * y)) * x;
	double tmp;
	if (y <= -1.45e-18) {
		tmp = t_1;
	} else if (y <= 1.75e-7) {
		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(((log(z) - t) * y)) * x
    if (y <= (-1.45d-18)) then
        tmp = t_1
    else if (y <= 1.75d-7) 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(((Math.log(z) - t) * y)) * x;
	double tmp;
	if (y <= -1.45e-18) {
		tmp = t_1;
	} else if (y <= 1.75e-7) {
		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(((math.log(z) - t) * y)) * x
	tmp = 0
	if y <= -1.45e-18:
		tmp = t_1
	elif y <= 1.75e-7:
		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(log(z) - t) * y)) * x)
	tmp = 0.0
	if (y <= -1.45e-18)
		tmp = t_1;
	elseif (y <= 1.75e-7)
		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(((log(z) - t) * y)) * x;
	tmp = 0.0;
	if (y <= -1.45e-18)
		tmp = t_1;
	elseif (y <= 1.75e-7)
		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[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -1.45e-18], t$95$1, If[LessEqual[y, 1.75e-7], N[(N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e-18 or 1.74999999999999992e-7 < y

   1. Initial program 97.7%

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

   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 87.4

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

   4. Applied rewrites 87.4%

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

   if -1.45e-18 < y < 1.74999999999999992e-7

   1. Initial program 95.2%

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

   2. 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-1 \cdot t\right)} + a \cdot \left(\left(-z\right) - b\right)} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in b around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 80.7

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

   12. Applied rewrites 80.7%

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

Alternative 4: 71.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{e^{y \cdot t}}\\ \mathbf{if}\;t \leq -4100:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-44}:\\ \;\;\;\;e^{\left(-a\right) \cdot b} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (exp (* y t)))))
   (if (<= t -4100.0) t_1 (if (<= t 1.1e-44) (* (exp (* (- a) b)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / exp((y * t));
	double tmp;
	if (t <= -4100.0) {
		tmp = t_1;
	} else if (t <= 1.1e-44) {
		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 = x / exp((y * t))
    if (t <= (-4100.0d0)) then
        tmp = t_1
    else if (t <= 1.1d-44) 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 = x / Math.exp((y * t));
	double tmp;
	if (t <= -4100.0) {
		tmp = t_1;
	} else if (t <= 1.1e-44) {
		tmp = Math.exp((-a * b)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / math.exp((y * t))
	tmp = 0
	if t <= -4100.0:
		tmp = t_1
	elif t <= 1.1e-44:
		tmp = math.exp((-a * b)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / exp(Float64(y * t)))
	tmp = 0.0
	if (t <= -4100.0)
		tmp = t_1;
	elseif (t <= 1.1e-44)
		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 = x / exp((y * t));
	tmp = 0.0;
	if (t <= -4100.0)
		tmp = t_1;
	elseif (t <= 1.1e-44)
		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[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4100.0], t$95$1, If[LessEqual[t, 1.1e-44], N[(N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4100 or 1.10000000000000006e-44 < 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. 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

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

   3. Applied rewrites 79.6%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x}{e^{-1 \cdot \left(y \cdot \left(\log z - t\right)\right)}}} \]
   5. 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. lower-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in t around inf

      \[\leadsto \frac{x}{e^{t \cdot y}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{e^{y \cdot t}} \]

      2. lower-*.f64 76.4

         \[\leadsto \frac{x}{e^{y \cdot t}} \]

   9. Applied rewrites 76.4%

      \[\leadsto \frac{x}{e^{y \cdot t}} \]

   if -4100 < t < 1.10000000000000006e-44

   1. Initial program 96.4%

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

   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-1 \cdot t\right)} + a \cdot \left(\left(-z\right) - b\right)} \]
   6. 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 74.0

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

   7. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.0

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

   9. Applied rewrites 74.0%

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

   10. Taylor expanded in b around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 64.9

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

   12. Applied rewrites 64.9%

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

Alternative 5: 56.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{e^{y \cdot t}} \end{array} \]

FPCore

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

C

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

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 * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

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

3. Applied rewrites 81.6%

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

4. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{x}{e^{-1 \cdot \left(y \cdot \left(\log z - t\right)\right)}}} \]
5. 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. lower-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 71.5

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

6. Applied rewrites 71.5%

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

7. Taylor expanded in t around inf

   \[\leadsto \frac{x}{e^{t \cdot y}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{x}{e^{y \cdot t}} \]

   2. lower-*.f64 56.5

      \[\leadsto \frac{x}{e^{y \cdot t}} \]

9. Applied rewrites 56.5%

   \[\leadsto \frac{x}{e^{y \cdot t}} \]
10. Add Preprocessing

Alternative 6: 35.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right)\\ \mathbf{if}\;x \cdot e^{t\_1 + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;\left(\left(-t\right) \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* y (- (log z) t))))
   (if (<= (* x (exp (+ t_1 (* a (- (log (- 1.0 z)) b))))) 0.0)
     (* (* (- t) y) x)
     (fma x t_1 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (log(z) - t);
	double tmp;
	if ((x * exp((t_1 + (a * (log((1.0 - z)) - b))))) <= 0.0) {
		tmp = (-t * y) * x;
	} else {
		tmp = fma(x, t_1, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y * Float64(log(z) - t))
	tmp = 0.0
	if (Float64(x * exp(Float64(t_1 + Float64(a * Float64(log(Float64(1.0 - z)) - b))))) <= 0.0)
		tmp = Float64(Float64(Float64(-t) * y) * x);
	else
		tmp = fma(x, t_1, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * N[Exp[N[(t$95$1 + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[((-t) * y), $MachinePrecision] * x), $MachinePrecision], N[(x * t$95$1 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))) < 0.0

   1. Initial program 97.6%

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

   2. 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 71.4

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

   4. Applied rewrites 71.4%

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

   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. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 22.7

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

   7. Applied rewrites 22.7%

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

   8. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

      4. lower-*.f64 17.1

         \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

   10. Applied rewrites 17.1%

       \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

   if 0.0 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))))

   1. Initial program 93.9%

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

   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 71.9

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

   4. Applied rewrites 71.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

Alternative 7: 32.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z - t\\ t_2 := y \cdot t\_1 + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -10000:\\ \;\;\;\;\left(\left(-t\right) \cdot y\right) \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (log z) t)) (t_2 (+ (* y t_1) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -10000.0)
     (* (* (- t) y) x)
     (if (<= t_2 2e+69) (* (fma (log z) y 1.0) x) (* (* t_1 y) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(z) - t;
	double t_2 = (y * t_1) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000.0) {
		tmp = (-t * y) * x;
	} else if (t_2 <= 2e+69) {
		tmp = fma(log(z), y, 1.0) * x;
	} else {
		tmp = (t_1 * y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(z) - t)
	t_2 = Float64(Float64(y * t_1) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -10000.0)
		tmp = Float64(Float64(Float64(-t) * y) * x);
	elseif (t_2 <= 2e+69)
		tmp = Float64(fma(log(z), y, 1.0) * x);
	else
		tmp = Float64(Float64(t_1 * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t$95$1), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -10000.0], N[(N[((-t) * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2e+69], N[(N[(N[Log[z], $MachinePrecision] * y + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(t$95$1 * y), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e4

   1. Initial program 98.3%

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

   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 68.3

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

   4. Applied rewrites 68.3%

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

   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. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

      4. lower-*.f64 15.5

         \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

   10. Applied rewrites 15.5%

       \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

   if -1e4 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000001e69

   1. Initial program 92.2%

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

   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 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-pow.f64 76.0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-log.f64 69.4

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

   10. Applied rewrites 69.4%

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

   if 2.0000000000000001e69 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.3%

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

   2. 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 67.5

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

   4. Applied rewrites 67.5%

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

   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. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 34.8

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 34.4

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

   10. Applied rewrites 34.4%

       \[\leadsto \left(\left(\log z - t\right) \cdot y\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 32.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-t\right) \cdot y\right) \cdot x\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -10000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (- t) y) x))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -10000.0)
     t_1
     (if (<= t_2 5e+270) (* (fma (log z) y 1.0) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (-t * y) * x;
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+270) {
		tmp = fma(log(z), y, 1.0) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(-t) * y) * x)
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -10000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+270)
		tmp = Float64(fma(log(z), y, 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), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -10000.0], t$95$1, If[LessEqual[t$95$2, 5e+270], N[(N[(N[Log[z], $MachinePrecision] * y + 1.0), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e4 or 4.99999999999999976e270 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

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

   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 67.6

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

   4. Applied rewrites 67.6%

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

   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. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 16.2

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

   7. Applied rewrites 16.2%

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

   8. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

      4. lower-*.f64 23.6

         \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

   10. Applied rewrites 23.6%

       \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

   if -1e4 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999976e270

   1. Initial program 94.9%

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

   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 76.8

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-pow.f64 62.4

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

   7. Applied rewrites 62.4%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-log.f64 44.7

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

   10. Applied rewrites 44.7%

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

Alternative 9: 26.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-t\right) \cdot y\right) \cdot x\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -10000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (- t) y) x))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -10000.0) t_1 (if (<= t_2 5e+72) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (-t * y) * x;
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+72) {
		tmp = 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) :: t_2
    real(8) :: tmp
    t_1 = (-t * y) * x
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-10000.0d0)) then
        tmp = t_1
    else if (t_2 <= 5d+72) then
        tmp = 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 = (-t * y) * x;
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+72) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (-t * y) * x
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -10000.0:
		tmp = t_1
	elif t_2 <= 5e+72:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(-t) * y) * x)
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -10000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+72)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (-t * y) * x;
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -10000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+72)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[((-t) * y), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -10000.0], t$95$1, If[LessEqual[t$95$2, 5e+72], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e4 or 4.99999999999999992e72 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.9%

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

   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 68.0

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

   4. Applied rewrites 68.0%

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

   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. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 17.7

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

   7. Applied rewrites 17.7%

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

   8. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

      4. lower-*.f64 21.0

         \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

   10. Applied rewrites 21.0%

       \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]

   if -1e4 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999992e72

   1. Initial program 92.2%

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

   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 82.7

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 67.8%

      \[\leadsto x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 19.2% accurate, 42.1× 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 96.5%

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

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 71.5

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

4. Applied rewrites 71.5%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 19.2%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025114 
(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))))))