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

Percentage Accurate: 96.7% → 99.2%

Time: 5.9s

Alternatives: 14

Speedup: 1.2×

• 42.4% 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: 96.7% 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 96.7%

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

2. Taylor expanded in 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: 97.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+201}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-a, b, \left(\log z - t\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(y, -t, a \cdot \left(\left(-z\right) - b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 5.8e+201)
   (* x (exp (fma (- a) b (* (- (log z) t) y))))
   (* x (exp (fma y (- t) (* a (- (- z) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 5.8e+201) {
		tmp = x * exp(fma(-a, b, ((log(z) - t) * y)));
	} else {
		tmp = x * exp(fma(y, -t, (a * (-z - b))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 5.8e+201)
		tmp = Float64(x * exp(fma(Float64(-a), b, Float64(Float64(log(z) - t) * y))));
	else
		tmp = Float64(x * exp(fma(y, Float64(-t), Float64(a * Float64(Float64(-z) - b)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 5.8e+201], N[(x * N[Exp[N[((-a) * b + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * (-t) + N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.8000000000000003e201

   1. Initial program 97.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + y \cdot \left(\log z - t\right)}} \]
   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 97.3

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

   4. Applied rewrites 97.3%

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

   if 5.8000000000000003e201 < a

   1. Initial program 89.2%

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

   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 96.7

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

   4. Applied rewrites 96.7%

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

   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 92.5

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

   7. Applied rewrites 92.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 95.1

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

   9. Applied rewrites 95.1%

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

Alternative 3: 92.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-29}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(y, -t, a \cdot \left(\left(-z\right) - b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (log z) t) y)) x)))
   (if (<= y -3.3e+109)
     t_1
     (if (<= y 4.3e-29) (* x (exp (fma y (- t) (* a (- (- z) b))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((log(z) - t) * y)) * x;
	double tmp;
	if (y <= -3.3e+109) {
		tmp = t_1;
	} else if (y <= 4.3e-29) {
		tmp = x * exp(fma(y, -t, (a * (-z - b))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(log(z) - t) * y)) * x)
	tmp = 0.0
	if (y <= -3.3e+109)
		tmp = t_1;
	elseif (y <= 4.3e-29)
		tmp = Float64(x * exp(fma(y, Float64(-t), Float64(a * Float64(Float64(-z) - b)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3.3e+109], t$95$1, If[LessEqual[y, 4.3e-29], N[(x * N[Exp[N[(y * (-t) + N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2999999999999999e109 or 4.2999999999999998e-29 < y

   1. Initial program 97.5%

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

   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 88.7

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

   4. Applied rewrites 88.7%

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

   if -3.2999999999999999e109 < y < 4.2999999999999998e-29

   1. Initial program 96.1%

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

   2. 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 95.5

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

   7. Applied rewrites 95.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 95.6

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

   9. Applied rewrites 95.6%

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

Alternative 4: 90.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -7.6 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-29}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-a, b, \left(-t\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (log z) t) y)) x)))
   (if (<= y -7.6e+18)
     t_1
     (if (<= y 4.3e-29) (* x (exp (fma (- a) b (* (- t) y)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((log(z) - t) * y)) * x;
	double tmp;
	if (y <= -7.6e+18) {
		tmp = t_1;
	} else if (y <= 4.3e-29) {
		tmp = x * exp(fma(-a, b, (-t * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(log(z) - t) * y)) * x)
	tmp = 0.0
	if (y <= -7.6e+18)
		tmp = t_1;
	elseif (y <= 4.3e-29)
		tmp = Float64(x * exp(fma(Float64(-a), b, Float64(Float64(-t) * 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[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -7.6e+18], t$95$1, If[LessEqual[y, 4.3e-29], N[(x * N[Exp[N[((-a) * b + N[((-t) * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.6e18 or 4.2999999999999998e-29 < y

   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 88.1

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

   4. Applied rewrites 88.1%

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

   if -7.6e18 < y < 4.2999999999999998e-29

   1. Initial program 95.9%

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

   2. Taylor expanded in 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 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 93.8

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

   7. Applied rewrites 93.8%

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

Alternative 5: 83.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+191}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-a, b, \left(-t\right) \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.1e+191) (* (pow z y) x) (* x (exp (fma (- a) b (* (- t) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.1e+191) {
		tmp = pow(z, y) * x;
	} else {
		tmp = x * exp(fma(-a, b, (-t * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.1e+191)
		tmp = Float64((z ^ y) * x);
	else
		tmp = Float64(x * exp(fma(Float64(-a), b, Float64(Float64(-t) * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.1e+191], N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision], N[(x * N[Exp[N[((-a) * b + N[((-t) * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e191

   1. Initial program 97.5%

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

   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 92.5

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-pow.f64 72.0

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

   7. Applied rewrites 72.0%

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

   if -1.1e191 < y

   1. Initial program 96.7%

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

   2. Taylor expanded in 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 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 84.9

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

   7. Applied rewrites 84.9%

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

Alternative 6: 73.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {z}^{y} \cdot x\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+60}:\\ \;\;\;\;e^{\left(-a\right) \cdot b} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (pow z y) x)))
   (if (<= y -5.8e+16) t_1 (if (<= y 7.8e+60) (* (exp (* (- a) b)) x) t_1))))

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 <= -5.8e+16) {
		tmp = t_1;
	} else if (y <= 7.8e+60) {
		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 = (z ** y) * x
    if (y <= (-5.8d+16)) then
        tmp = t_1
    else if (y <= 7.8d+60) 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.pow(z, y) * x;
	double tmp;
	if (y <= -5.8e+16) {
		tmp = t_1;
	} else if (y <= 7.8e+60) {
		tmp = Math.exp((-a * b)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(z, y) * x
	tmp = 0
	if y <= -5.8e+16:
		tmp = t_1
	elif y <= 7.8e+60:
		tmp = math.exp((-a * b)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64((z ^ y) * x)
	tmp = 0.0
	if (y <= -5.8e+16)
		tmp = t_1;
	elseif (y <= 7.8e+60)
		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 = (z ^ y) * x;
	tmp = 0.0;
	if (y <= -5.8e+16)
		tmp = t_1;
	elseif (y <= 7.8e+60)
		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[Power[z, y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -5.8e+16], t$95$1, If[LessEqual[y, 7.8e+60], N[(N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.8e16 or 7.8000000000000006e60 < y

   1. Initial program 97.8%

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

   2. 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 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-pow.f64 70.5

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

   7. Applied rewrites 70.5%

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

   if -5.8e16 < y < 7.8000000000000006e60

   1. Initial program 95.9%

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

   2. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 75.3

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

   4. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 75.3

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

   6. Applied rewrites 75.3%

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

Alternative 7: 68.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-a\right) \cdot b} \cdot x\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+66}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- a) b)) x)))
   (if (<= a -6.6e-11) t_1 (if (<= a 9.5e+66) (* (exp (* (- t) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-a * b)) * x;
	double tmp;
	if (a <= -6.6e-11) {
		tmp = t_1;
	} else if (a <= 9.5e+66) {
		tmp = exp((-t * y)) * 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((-a * b)) * x
    if (a <= (-6.6d-11)) then
        tmp = t_1
    else if (a <= 9.5d+66) then
        tmp = exp((-t * y)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-a * b)) * x;
	double tmp;
	if (a <= -6.6e-11) {
		tmp = t_1;
	} else if (a <= 9.5e+66) {
		tmp = Math.exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-a * b)) * x
	tmp = 0
	if a <= -6.6e-11:
		tmp = t_1
	elif a <= 9.5e+66:
		tmp = math.exp((-t * y)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-a) * b)) * x)
	tmp = 0.0
	if (a <= -6.6e-11)
		tmp = t_1;
	elseif (a <= 9.5e+66)
		tmp = Float64(exp(Float64(Float64(-t) * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-a * b)) * x;
	tmp = 0.0;
	if (a <= -6.6e-11)
		tmp = t_1;
	elseif (a <= 9.5e+66)
		tmp = exp((-t * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[a, -6.6e-11], t$95$1, If[LessEqual[a, 9.5e+66], N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.6000000000000005e-11 or 9.50000000000000051e66 < a

   1. Initial program 93.1%

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

   2. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.3

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

   6. Applied rewrites 69.3%

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

   if -6.6000000000000005e-11 < a < 9.50000000000000051e66

   1. Initial program 99.9%

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

   2. 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. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 68.6

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

   4. Applied rewrites 68.6%

      \[\leadsto x \cdot e^{\color{blue}{\left(-t\right) \cdot y}} \]
   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 68.6

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

   6. Applied rewrites 68.6%

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

Alternative 8: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+150}:\\ \;\;\;\;\left(\log z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-a\right) \cdot b} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.6e+150) (* (* (log z) y) x) (* (exp (* (- a) b)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.6e+150) {
		tmp = (log(z) * y) * x;
	} else {
		tmp = exp((-a * b)) * 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) :: tmp
    if (y <= (-3.6d+150)) then
        tmp = (log(z) * y) * x
    else
        tmp = exp((-a * b)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.6e+150) {
		tmp = (Math.log(z) * y) * x;
	} else {
		tmp = Math.exp((-a * b)) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.6e+150:
		tmp = (math.log(z) * y) * x
	else:
		tmp = math.exp((-a * b)) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.6e+150)
		tmp = Float64(Float64(log(z) * y) * x);
	else
		tmp = Float64(exp(Float64(Float64(-a) * b)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.6e+150)
		tmp = (log(z) * y) * x;
	else
		tmp = exp((-a * b)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.6e+150], N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision], N[(N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.59999999999999986e150

   1. Initial program 97.5%

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

   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 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-pow.f64 71.5

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

   7. Applied rewrites 71.5%

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

   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. lower-fma.f64 N/A

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

      3. lift-log.f64 29.9

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

   10. Applied rewrites 29.9%

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

   11. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-log.f64 29.9

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

   13. Applied rewrites 29.9%

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

   if -3.59999999999999986e150 < 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 b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 61.5

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

   4. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.5

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

   6. Applied rewrites 61.5%

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

Alternative 9: 34.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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))) < -1

   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 67.3

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

   4. Applied rewrites 67.3%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 16.1

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

   10. Applied rewrites 16.1%

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

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

   1. Initial program 95.7%

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

   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 75.6

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

   4. Applied rewrites 75.6%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 47.6

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

   7. Applied rewrites 47.6%

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

Alternative 10: 34.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z - t\\ t_2 := y \cdot t\_1 + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\left(-t\right) \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot y\right) \cdot x\\ \end{array} \end{array} \]

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 -1.0)
     (* (- t) (* y x))
     (if (<= t_2 2e+128) (fma (* (- t) y) x 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 <= -1.0) {
		tmp = -t * (y * x);
	} else if (t_2 <= 2e+128) {
		tmp = fma((-t * y), x, 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 <= -1.0)
		tmp = Float64(Float64(-t) * Float64(y * x));
	elseif (t_2 <= 2e+128)
		tmp = fma(Float64(Float64(-t) * y), x, 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, -1.0], N[((-t) * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+128], N[(N[((-t) * y), $MachinePrecision] * x + 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))) < -1

   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 67.3

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

   4. Applied rewrites 67.3%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 16.1

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

   10. Applied rewrites 16.1%

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

   if -1 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000002e128

   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 81.7

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

   4. Applied rewrites 81.7%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 58.1

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

   10. Applied rewrites 58.1%

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

   if 2.0000000000000002e128 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.4%

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

   2. Taylor expanded in 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 70.0

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

   4. Applied rewrites 70.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 36.3

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

   7. Applied rewrites 36.3%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 35.9

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

   10. Applied rewrites 35.9%

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

Alternative 11: 32.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;\left(-t\right) \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\log z \cdot y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -1000000000000.0)
     (* (- t) (* y x))
     (if (<= t_1 2e+307) (fma (* (log z) y) x x) (fma (* (- t) y) x x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = -t * (y * x);
	} else if (t_1 <= 2e+307) {
		tmp = fma((log(z) * y), x, x);
	} else {
		tmp = fma((-t * y), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = Float64(Float64(-t) * Float64(y * x));
	elseif (t_1 <= 2e+307)
		tmp = fma(Float64(log(z) * y), x, x);
	else
		tmp = fma(Float64(Float64(-t) * y), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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$1, -1000000000000.0], N[((-t) * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+307], N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] * x + x), $MachinePrecision], N[(N[((-t) * y), $MachinePrecision] * x + 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))) < -1e12

   1. Initial program 98.4%

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

   2. 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 15.9

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

   10. Applied rewrites 15.9%

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

   if -1e12 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.99999999999999997e307

   1. Initial program 96.0%

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

   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 78.5

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

   4. Applied rewrites 78.5%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 46.1

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

   7. Applied rewrites 46.1%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 41.3

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

   10. Applied rewrites 41.3%

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

   if 1.99999999999999997e307 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.3%

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

   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 63.1

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

   4. Applied rewrites 63.1%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 50.8

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

   7. Applied rewrites 50.8%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

Alternative 12: 32.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\ t_2 := \left(-t\right) \cdot \left(y \cdot x\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.0000002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
        (t_2 (* (- t) (* y x))))
   (if (<= t_1 0.0) t_2 (if (<= t_1 1.0000002) x t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
	double t_2 = -t * (y * x);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 1.0000002) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	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 = exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
    t_2 = -t * (y * x)
    if (t_1 <= 0.0d0) then
        tmp = t_2
    else if (t_1 <= 1.0000002d0) then
        tmp = x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
	double t_2 = -t * (y * x);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 1.0000002) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))
	t_2 = -t * (y * x)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 1.0000002:
		tmp = x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))))
	t_2 = Float64(Float64(-t) * Float64(y * x))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 1.0000002)
		tmp = x;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
	t_2 = -t * (y * x);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 1.0000002)
		tmp = x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[((-t) * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 1.0000002], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 0.0 or 1.00000019999999989 < (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 97.5%

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

   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 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 16.9

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

   7. Applied rewrites 16.9%

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

   8. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 18.9

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

   10. Applied rewrites 18.9%

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

   if 0.0 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 1.00000019999999989

   1. Initial program 93.0%

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

   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 91.2

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 89.0%

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

Alternative 13: 31.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0.4:\\ \;\;\;\;\left(-t\right) \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))) 0.4)
   (* (- t) (* y x))
   (fma (* (- t) y) x x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.4) {
		tmp = -t * (y * x);
	} else {
		tmp = fma((-t * y), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))) <= 0.4)
		tmp = Float64(Float64(-t) * Float64(y * x));
	else
		tmp = fma(Float64(Float64(-t) * y), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 0.4], N[((-t) * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[((-t) * y), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 0.40000000000000002

   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 67.3

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

   4. Applied rewrites 67.3%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 16.1

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

   10. Applied rewrites 16.1%

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

   if 0.40000000000000002 < (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 95.7%

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

   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 75.6

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

   4. Applied rewrites 75.6%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 47.6

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

   7. Applied rewrites 47.6%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 43.3

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

   10. Applied rewrites 43.3%

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

Alternative 14: 18.7% 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 96.7%

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

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 72.2

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

4. Applied rewrites 72.2%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 18.7%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025119 
(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))))))