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

Percentage Accurate: 96.4% → 99.1%

Time: 7.5s

Alternatives: 10

Speedup: 1.4×

• 41.7% 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: 96.4% 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: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 98.4

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

5. Applied rewrites 98.4%

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

Alternative 2: 44.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\ \mathbf{if}\;t\_1 \leq 0.99999999999995 \lor \neg \left(t\_1 \leq 9\right):\\ \;\;\;\;0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))
   (if (or (<= t_1 0.99999999999995) (not (<= t_1 9.0)))
     (* 0.5 (* (* t t) (* x (* y y))))
     x)))

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 tmp;
	if ((t_1 <= 0.99999999999995) || !(t_1 <= 9.0)) {
		tmp = 0.5 * ((t * t) * (x * (y * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
    if ((t_1 <= 0.99999999999995d0) .or. (.not. (t_1 <= 9.0d0))) then
        tmp = 0.5d0 * ((t * t) * (x * (y * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
	double tmp;
	if ((t_1 <= 0.99999999999995) || !(t_1 <= 9.0)) {
		tmp = 0.5 * ((t * t) * (x * (y * y)));
	} else {
		tmp = x;
	}
	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))))
	tmp = 0
	if (t_1 <= 0.99999999999995) or not (t_1 <= 9.0):
		tmp = 0.5 * ((t * t) * (x * (y * y)))
	else:
		tmp = x
	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))))
	tmp = 0.0
	if ((t_1 <= 0.99999999999995) || !(t_1 <= 9.0))
		tmp = Float64(0.5 * Float64(Float64(t * t) * Float64(x * Float64(y * y))));
	else
		tmp = x;
	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))));
	tmp = 0.0;
	if ((t_1 <= 0.99999999999995) || ~((t_1 <= 9.0)))
		tmp = 0.5 * ((t * t) * (x * (y * y)));
	else
		tmp = x;
	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]}, If[Or[LessEqual[t$95$1, 0.99999999999995], N[Not[LessEqual[t$95$1, 9.0]], $MachinePrecision]], N[(0.5 * N[(N[(t * t), $MachinePrecision] * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

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)))) < 0.99999999999995004 or 9 < (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 96.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right), \color{blue}{y}, x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\right) \]

   5. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, \left(y \cdot {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)}\right) \cdot {\left(\log z - t\right)}^{2}, \left({\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) \cdot x\right), y, {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + \color{blue}{x \cdot \left(\log z - t\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto x + y \cdot \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left(y \cdot {\left(\log z - t\right)}^{2}\right)}, x \cdot \left(\log z - t\right)\right) \]

   8. Applied rewrites 27.8%

      \[\leadsto x + \color{blue}{y \cdot \mathsf{fma}\left(0.5, x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right), x \cdot \left(\log z - t\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{{y}^{2}}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{\color{blue}{2}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \]

       7. lower-*.f64 36.4

          \[\leadsto 0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \]

   11. Applied rewrites 36.4%

       \[\leadsto 0.5 \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(x \cdot \left(y \cdot y\right)\right)}\right) \]

   if 0.99999999999995004 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 9

   1. Initial program 90.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right), \color{blue}{y}, x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\right) \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, \left(y \cdot {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)}\right) \cdot {\left(\log z - t\right)}^{2}, \left({\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) \cdot x\right), y, {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + \color{blue}{x \cdot \left(\log z - t\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto x + y \cdot \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left(y \cdot {\left(\log z - t\right)}^{2}\right)}, x \cdot \left(\log z - t\right)\right) \]

   8. Applied rewrites 74.6%

      \[\leadsto x + \color{blue}{y \cdot \mathsf{fma}\left(0.5, x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right), x \cdot \left(\log z - t\right)\right)} \]

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 84.3%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0.99999999999995 \lor \neg \left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 9\right):\\ \;\;\;\;0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0.99999999999995:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.5 (* (* t t) (* x (* y y))))))
   (if (<=
        (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))
        0.99999999999995)
     t_1
     (+ x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * ((t * t) * (x * (y * y)));
	double tmp;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.99999999999995) {
		tmp = t_1;
	} else {
		tmp = x + 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 = 0.5d0 * ((t * t) * (x * (y * y)))
    if (exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b)))) <= 0.99999999999995d0) then
        tmp = t_1
    else
        tmp = x + 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 = 0.5 * ((t * t) * (x * (y * y)));
	double tmp;
	if (Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b)))) <= 0.99999999999995) {
		tmp = t_1;
	} else {
		tmp = x + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.5 * ((t * t) * (x * (y * y)))
	tmp = 0
	if math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b)))) <= 0.99999999999995:
		tmp = t_1
	else:
		tmp = x + t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.5 * Float64(Float64(t * t) * Float64(x * Float64(y * y))))
	tmp = 0.0
	if (exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))) <= 0.99999999999995)
		tmp = t_1;
	else
		tmp = Float64(x + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.5 * ((t * t) * (x * (y * y)));
	tmp = 0.0;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.99999999999995)
		tmp = t_1;
	else
		tmp = x + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.5 * N[(N[(t * t), $MachinePrecision] * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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], 0.99999999999995], t$95$1, N[(x + t$95$1), $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)))) < 0.99999999999995004

   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. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right), \color{blue}{y}, x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\right) \]

   5. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, \left(y \cdot {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)}\right) \cdot {\left(\log z - t\right)}^{2}, \left({\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) \cdot x\right), y, {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + \color{blue}{x \cdot \left(\log z - t\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto x + y \cdot \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left(y \cdot {\left(\log z - t\right)}^{2}\right)}, x \cdot \left(\log z - t\right)\right) \]

   8. Applied rewrites 2.7%

      \[\leadsto x + \color{blue}{y \cdot \mathsf{fma}\left(0.5, x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right), x \cdot \left(\log z - t\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{{y}^{2}}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{\color{blue}{2}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \]

       7. lower-*.f64 30.1

          \[\leadsto 0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \]

   11. Applied rewrites 30.1%

       \[\leadsto 0.5 \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(x \cdot \left(y \cdot y\right)\right)}\right) \]

   if 0.99999999999995004 < (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 92.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right), \color{blue}{y}, x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\right) \]

   5. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, \left(y \cdot {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)}\right) \cdot {\left(\log z - t\right)}^{2}, \left({\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) \cdot x\right), y, {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + \color{blue}{x \cdot \left(\log z - t\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto x + y \cdot \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left(y \cdot {\left(\log z - t\right)}^{2}\right)}, x \cdot \left(\log z - t\right)\right) \]

   8. Applied rewrites 62.7%

      \[\leadsto x + \color{blue}{y \cdot \mathsf{fma}\left(0.5, x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right), x \cdot \left(\log z - t\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto x + \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x + \frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{{y}^{2}}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto x + \frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{\color{blue}{2}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x + \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto x + \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto x + \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x + \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \]

       7. lower-*.f64 53.5

          \[\leadsto x + 0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \]

   11. Applied rewrites 53.5%

       \[\leadsto x + 0.5 \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(x \cdot \left(y \cdot y\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 43.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -5e-14)
   (* 0.5 (* (* t t) (* x (* y y))))
   (+ x (* y (* 0.5 (* (* t t) (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -5e-14) {
		tmp = 0.5 * ((t * t) * (x * (y * y)));
	} else {
		tmp = x + (y * (0.5 * ((t * t) * (x * y))));
	}
	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 * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))) <= (-5d-14)) then
        tmp = 0.5d0 * ((t * t) * (x * (y * y)))
    else
        tmp = x + (y * (0.5d0 * ((t * t) * (x * y))))
    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 * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))) <= -5e-14) {
		tmp = 0.5 * ((t * t) * (x * (y * y)));
	} else {
		tmp = x + (y * (0.5 * ((t * t) * (x * y))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))) <= -5e-14:
		tmp = 0.5 * ((t * t) * (x * (y * y)))
	else:
		tmp = x + (y * (0.5 * ((t * t) * (x * y))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -5e-14)
		tmp = Float64(0.5 * Float64(Float64(t * t) * Float64(x * Float64(y * y))));
	else
		tmp = Float64(x + Float64(y * Float64(0.5 * Float64(Float64(t * t) * Float64(x * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -5e-14)
		tmp = 0.5 * ((t * t) * (x * (y * y)));
	else
		tmp = x + (y * (0.5 * ((t * t) * (x * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], -5e-14], N[(0.5 * N[(N[(t * t), $MachinePrecision] * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(0.5 * N[(N[(t * t), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -5.0000000000000002e-14

   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. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right), \color{blue}{y}, x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\right) \]

   5. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, \left(y \cdot {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)}\right) \cdot {\left(\log z - t\right)}^{2}, \left({\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) \cdot x\right), y, {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + \color{blue}{x \cdot \left(\log z - t\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto x + y \cdot \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left(y \cdot {\left(\log z - t\right)}^{2}\right)}, x \cdot \left(\log z - t\right)\right) \]

   8. Applied rewrites 2.7%

      \[\leadsto x + \color{blue}{y \cdot \mathsf{fma}\left(0.5, x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right), x \cdot \left(\log z - t\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{{y}^{2}}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{\color{blue}{2}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot {y}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \]

       7. lower-*.f64 30.1

          \[\leadsto 0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \]

   11. Applied rewrites 30.1%

       \[\leadsto 0.5 \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(x \cdot \left(y \cdot y\right)\right)}\right) \]

   if -5.0000000000000002e-14 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 92.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right), \color{blue}{y}, x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\right) \]

   5. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, \left(y \cdot {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)}\right) \cdot {\left(\log z - t\right)}^{2}, \left({\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) \cdot x\right), y, {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + \color{blue}{x \cdot \left(\log z - t\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto x + y \cdot \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left(y \cdot {\left(\log z - t\right)}^{2}\right)}, x \cdot \left(\log z - t\right)\right) \]

   8. Applied rewrites 62.7%

      \[\leadsto x + \color{blue}{y \cdot \mathsf{fma}\left(0.5, x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right), x \cdot \left(\log z - t\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{y}\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot y\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot y\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot y\right)\right)\right) \]

       5. lower-*.f64 56.7

          \[\leadsto x + y \cdot \left(0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot y\right)\right)\right) \]

   11. Applied rewrites 56.7%

       \[\leadsto x + y \cdot \left(0.5 \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -840 \lor \neg \left(y \leq 11.6\right):\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(y, -t, a \cdot \left(\left(-z\right) - b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -840.0) (not (<= y 11.6)))
   (* x (exp (* (- (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 ((y <= -840.0) || !(y <= 11.6)) {
		tmp = x * exp(((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 ((y <= -840.0) || !(y <= 11.6))
		tmp = Float64(x * exp(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[Or[LessEqual[y, -840.0], N[Not[LessEqual[y, 11.6]], $MachinePrecision]], N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $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 y < -840 or 11.5999999999999996 < 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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 93.3

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

   5. Applied rewrites 93.3%

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

   if -840 < y < 11.5999999999999996

   1. Initial program 94.2%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 99.3

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

   8. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.3

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

   10. Applied rewrites 99.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -840 \lor \neg \left(y \leq 11.6\right):\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(y, -t, a \cdot \left(\left(-z\right) - b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -135 \lor \neg \left(y \leq 11.6\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -135.0) (not (<= y 11.6)))
   (* x (pow z y))
   (* x (exp (* (- a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -135.0) || !(y <= 11.6)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((-a * b));
	}
	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 <= (-135.0d0)) .or. (.not. (y <= 11.6d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((-a * b))
    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 <= -135.0) || !(y <= 11.6)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((-a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -135.0) or not (y <= 11.6):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((-a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -135.0) || !(y <= 11.6))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(Float64(-a) * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -135.0) || ~((y <= 11.6)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((-a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -135.0], N[Not[LessEqual[y, 11.6]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -135 or 11.5999999999999996 < 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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. *-commutative N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in a around 0

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

      1. lift-pow.f64 65.8

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

   8. Applied rewrites 65.8%

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

   if -135 < y < 11.5999999999999996

   1. Initial program 94.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 73.0

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

   5. Applied rewrites 73.0%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -135 \lor \neg \left(y \leq 11.6\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{elif}\;y \leq 11.6:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.75e-127)
   (* x (exp (* (- t) y)))
   (if (<= y 11.6) (* x (exp (* (- a) b))) (* x (pow z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.75e-127) {
		tmp = x * exp((-t * y));
	} else if (y <= 11.6) {
		tmp = x * exp((-a * b));
	} else {
		tmp = x * pow(z, y);
	}
	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 <= (-1.75d-127)) then
        tmp = x * exp((-t * y))
    else if (y <= 11.6d0) then
        tmp = x * exp((-a * b))
    else
        tmp = x * (z ** y)
    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 <= -1.75e-127) {
		tmp = x * Math.exp((-t * y));
	} else if (y <= 11.6) {
		tmp = x * Math.exp((-a * b));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.75e-127:
		tmp = x * math.exp((-t * y))
	elif y <= 11.6:
		tmp = x * math.exp((-a * b))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.75e-127)
		tmp = Float64(x * exp(Float64(Float64(-t) * y)));
	elseif (y <= 11.6)
		tmp = Float64(x * exp(Float64(Float64(-a) * b)));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.75e-127)
		tmp = x * exp((-t * y));
	elseif (y <= 11.6)
		tmp = x * exp((-a * b));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.75e-127], N[(x * N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 11.6], N[(x * N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.74999999999999995e-127

   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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if -1.74999999999999995e-127 < y < 11.5999999999999996

   1. Initial program 94.6%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if 11.5999999999999996 < y

   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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. *-commutative N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in a around 0

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

      1. lift-pow.f64 69.5

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

   8. Applied rewrites 69.5%

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

Alternative 8: 87.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 89.0

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

8. Applied rewrites 89.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 90.1

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

10. Applied rewrites 90.1%

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

Alternative 9: 55.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-107}:\\ \;\;\;\;x + y \cdot \left(0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.85e-107)
   (+ x (* y (* 0.5 (* (* t t) (* x y)))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.85e-107) {
		tmp = x + (y * (0.5 * ((t * t) * (x * y))));
	} else {
		tmp = x * pow(z, y);
	}
	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 (t <= (-1.85d-107)) then
        tmp = x + (y * (0.5d0 * ((t * t) * (x * y))))
    else
        tmp = x * (z ** y)
    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 (t <= -1.85e-107) {
		tmp = x + (y * (0.5 * ((t * t) * (x * y))));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.85e-107:
		tmp = x + (y * (0.5 * ((t * t) * (x * y))))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.85e-107)
		tmp = Float64(x + Float64(y * Float64(0.5 * Float64(Float64(t * t) * Float64(x * y)))));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.85e-107)
		tmp = x + (y * (0.5 * ((t * t) * (x * y))));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.85e-107], N[(x + N[(y * N[(0.5 * N[(N[(t * t), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8500000000000001e-107

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right), \color{blue}{y}, x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\right) \]

   5. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, \left(y \cdot {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)}\right) \cdot {\left(\log z - t\right)}^{2}, \left({\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) \cdot x\right), y, {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + \color{blue}{x \cdot \left(\log z - t\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto x + y \cdot \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left(y \cdot {\left(\log z - t\right)}^{2}\right)}, x \cdot \left(\log z - t\right)\right) \]

   8. Applied rewrites 35.9%

      \[\leadsto x + \color{blue}{y \cdot \mathsf{fma}\left(0.5, x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right), x \cdot \left(\log z - t\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{y}\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot y\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot y\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot y\right)\right)\right) \]

       5. lower-*.f64 35.1

          \[\leadsto x + y \cdot \left(0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot y\right)\right)\right) \]

   11. Applied rewrites 35.1%

       \[\leadsto x + y \cdot \left(0.5 \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

   if -1.8500000000000001e-107 < t

   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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. *-commutative N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in a around 0

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

      1. lift-pow.f64 61.9

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

   8. Applied rewrites 61.9%

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 19.4% accurate, 328.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right), \color{blue}{y}, x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\right) \]

5. Applied rewrites 47.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, \left(y \cdot {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)}\right) \cdot {\left(\log z - t\right)}^{2}, \left({\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) \cdot x\right), y, {\left(e^{a}\right)}^{\left(\log \left(1 - z\right) - b\right)} \cdot x\right)} \]

6. Taylor expanded in a around 0

   \[\leadsto x + \color{blue}{y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + x \cdot \left(\log z - t\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto x + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) + \color{blue}{x \cdot \left(\log z - t\right)}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto x + y \cdot \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left(y \cdot {\left(\log z - t\right)}^{2}\right)}, x \cdot \left(\log z - t\right)\right) \]

8. Applied rewrites 35.3%

   \[\leadsto x + \color{blue}{y \cdot \mathsf{fma}\left(0.5, x \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right), x \cdot \left(\log z - t\right)\right)} \]

9. Taylor expanded in y around 0

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

11. Applied rewrites 17.3%

    \[\leadsto x \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025051 
(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))))))