Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Percentage Accurate: 77.5% → 99.4%

Time: 4.8s

Alternatives: 6

Speedup: 6.1×

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Initial Program: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.49 \lor \neg \left(x \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.49) (not (<= x 1.4e-8))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.49) || !(x <= 1.4e-8)) {
		tmp = exp(-y) / x;
	} else {
		tmp = 1.0 / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.49d0)) .or. (.not. (x <= 1.4d-8))) then
        tmp = exp(-y) / x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.49) || !(x <= 1.4e-8)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.49) or not (x <= 1.4e-8):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.49) || !(x <= 1.4e-8))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.49) || ~((x <= 1.4e-8)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.49], N[Not[LessEqual[x, 1.4e-8]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.48999999999999999 or 1.4e-8 < x

   1. Initial program 68.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot y}}}{x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

         \[\leadsto \frac{e^{-y}}{x} \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   if -0.48999999999999999 < x < 1.4e-8

   1. Initial program 83.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{1}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.49 \lor \neg \left(x \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)\\ \mathbf{if}\;x \leq -0.49:\\ \;\;\;\;\frac{\frac{t\_0 \cdot x}{x}}{x}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- (* 0.5 y) 1.0) y 1.0)))
   (if (<= x -0.49)
     (/ (/ (* t_0 x) x) x)
     (if (<= x 2.55e+29) (/ 1.0 x) (/ (/ (fma t_0 x (* (* y y) 0.5)) x) x)))))

C

double code(double x, double y) {
	double t_0 = fma(((0.5 * y) - 1.0), y, 1.0);
	double tmp;
	if (x <= -0.49) {
		tmp = ((t_0 * x) / x) / x;
	} else if (x <= 2.55e+29) {
		tmp = 1.0 / x;
	} else {
		tmp = (fma(t_0, x, ((y * y) * 0.5)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0)
	tmp = 0.0
	if (x <= -0.49)
		tmp = Float64(Float64(Float64(t_0 * x) / x) / x);
	elseif (x <= 2.55e+29)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(fma(t_0, x, Float64(Float64(y * y) * 0.5)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.49], N[(N[(N[(t$95$0 * x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.55e+29], N[(1.0 / x), $MachinePrecision], N[(N[(N[(t$95$0 * x + N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.48999999999999999

   1. Initial program 66.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1, y, 1\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1, y, 1\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1, y, 1\right)}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\frac{1}{2} \cdot 1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      11. lower-/.f64 63.4

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}{x} \]

   5. Applied rewrites 63.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}}{x} \]

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, {y}^{2} \cdot \frac{1}{2}\right)}{x}}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, {y}^{2} \cdot \frac{1}{2}\right)}{x}}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot \frac{1}{2}\right)}{x}}{x} \]

      13. lower-*.f64 61.4

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x} \]

   8. Applied rewrites 61.4%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{\color{blue}{x}}}{x} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lift-fma.f64 73.1

          \[\leadsto \frac{\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right) \cdot x}{x}}{x} \]

   11. Applied rewrites 73.1%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right) \cdot x}{x}}{x} \]

   if -0.48999999999999999 < x < 2.55e29

   1. Initial program 84.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.9%

      \[\leadsto \frac{\color{blue}{1}}{x} \]

   if 2.55e29 < x

   1. Initial program 67.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1, y, 1\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1, y, 1\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1, y, 1\right)}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\frac{1}{2} \cdot 1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      11. lower-/.f64 58.7

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}{x} \]

   5. Applied rewrites 58.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}}{x} \]

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, {y}^{2} \cdot \frac{1}{2}\right)}{x}}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, {y}^{2} \cdot \frac{1}{2}\right)}{x}}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot \frac{1}{2}\right)}{x}}{x} \]

      13. lower-*.f64 68.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x} \]

   8. Applied rewrites 68.0%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{\color{blue}{x}}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.49 \lor \neg \left(x \leq 2.55 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right) \cdot x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.49) (not (<= x 2.55e+29)))
   (/ (/ (* (fma (- (* 0.5 y) 1.0) y 1.0) x) x) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.49) || !(x <= 2.55e+29)) {
		tmp = ((fma(((0.5 * y) - 1.0), y, 1.0) * x) / x) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.49) || !(x <= 2.55e+29))
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0) * x) / x) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.49], N[Not[LessEqual[x, 2.55e+29]], $MachinePrecision]], N[(N[(N[(N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.48999999999999999 or 2.55e29 < x

   1. Initial program 66.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1, y, 1\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1, y, 1\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1, y, 1\right)}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\frac{1}{2} \cdot 1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      11. lower-/.f64 61.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}{x} \]

   5. Applied rewrites 61.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}}{x} \]

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, {y}^{2} \cdot \frac{1}{2}\right)}{x}}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, {y}^{2} \cdot \frac{1}{2}\right)}{x}}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot \frac{1}{2}\right)}{x}}{x} \]

      13. lower-*.f64 64.8

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x} \]

   8. Applied rewrites 64.8%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{\color{blue}{x}}}{x} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lift-fma.f64 70.5

          \[\leadsto \frac{\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right) \cdot x}{x}}{x} \]

   11. Applied rewrites 70.5%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right) \cdot x}{x}}{x} \]

   if -0.48999999999999999 < x < 2.55e29

   1. Initial program 84.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.9%

      \[\leadsto \frac{\color{blue}{1}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.49 \lor \neg \left(x \leq 2.55 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right) \cdot x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.45 \lor \neg \left(x \leq 3.3 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-x, y, x\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.45) (not (<= x 3.3e+73)))
   (/ (/ (fma (- x) y x) x) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.45) || !(x <= 3.3e+73)) {
		tmp = (fma(-x, y, x) / x) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.45) || !(x <= 3.3e+73))
		tmp = Float64(Float64(fma(Float64(-x), y, x) / x) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.45], N[Not[LessEqual[x, 3.3e+73]], $MachinePrecision]], N[(N[(N[((-x) * y + x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.450000000000000011 or 3.3000000000000003e73 < x

   1. Initial program 64.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1, y, 1\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1, y, 1\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1, y, 1\right)}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\frac{1}{2} \cdot 1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      11. lower-/.f64 62.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}{x} \]

   5. Applied rewrites 62.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}}{x} \]

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \frac{1}{2} \cdot {y}^{2}\right)}{x}}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, {y}^{2} \cdot \frac{1}{2}\right)}{x}}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, {y}^{2} \cdot \frac{1}{2}\right)}{x}}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot \frac{1}{2}\right)}{x}}{x} \]

      13. lower-*.f64 66.1

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x} \]

   8. Applied rewrites 66.1%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{\color{blue}{x}}}{x} \]

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-neg.f64 68.3

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

   11. Applied rewrites 68.3%

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

   if -0.450000000000000011 < x < 3.3000000000000003e73

   1. Initial program 85.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.6%

      \[\leadsto \frac{\color{blue}{1}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.45 \lor \neg \left(x \leq 3.3 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-x, y, x\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.49 \lor \neg \left(x \leq 2.55 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.49) (not (<= x 2.55e+29)))
   (/ (fma (- (* 0.5 y) 1.0) y 1.0) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.49) || !(x <= 2.55e+29)) {
		tmp = fma(((0.5 * y) - 1.0), y, 1.0) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.49) || !(x <= 2.55e+29))
		tmp = Float64(fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.49], N[Not[LessEqual[x, 2.55e+29]], $MachinePrecision]], N[(N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.48999999999999999 or 2.55e29 < x

   1. Initial program 66.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1, y, 1\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1, y, 1\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1, y, 1\right)}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\frac{1}{2} \cdot 1}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{x} + \frac{1}{2}\right) \cdot y - 1, y, 1\right)}{x} \]

      11. lower-/.f64 61.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}{x} \]

   5. Applied rewrites 61.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)}{x} \]

   if -0.48999999999999999 < x < 2.55e29

   1. Initial program 84.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.9%

      \[\leadsto \frac{\color{blue}{1}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.49 \lor \neg \left(x \leq 2.55 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 19.3× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 74.8%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{x} \]
4. Step-by-step derivation

5. Applied rewrites 72.8%

   \[\leadsto \frac{\color{blue}{1}}{x} \]
6. Add Preprocessing

Developer Target 1: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{-1}{y}}}{x}\\ t_1 := \frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (/ -1.0 y)) x)) (t_1 (/ (pow (/ x (+ y x)) x) x)))
   (if (< y -3.7311844206647956e+94)
     t_0
     (if (< y 2.817959242728288e+37)
       t_1
       (if (< y 2.347387415166998e+178) (log (exp t_1)) t_0)))))

C

double code(double x, double y) {
	double t_0 = exp((-1.0 / y)) / x;
	double t_1 = pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = log(exp(t_1));
	} else {
		tmp = t_0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-1.0d0) / y)) / x
    t_1 = ((x / (y + x)) ** x) / x
    if (y < (-3.7311844206647956d+94)) then
        tmp = t_0
    else if (y < 2.817959242728288d+37) then
        tmp = t_1
    else if (y < 2.347387415166998d+178) then
        tmp = log(exp(t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-1.0 / y)) / x;
	double t_1 = Math.pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = Math.log(Math.exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-1.0 / y)) / x
	t_1 = math.pow((x / (y + x)), x) / x
	tmp = 0
	if y < -3.7311844206647956e+94:
		tmp = t_0
	elif y < 2.817959242728288e+37:
		tmp = t_1
	elif y < 2.347387415166998e+178:
		tmp = math.log(math.exp(t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-1.0 / y)) / x)
	t_1 = Float64((Float64(x / Float64(y + x)) ^ x) / x)
	tmp = 0.0
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-1.0 / y)) / x;
	t_1 = ((x / (y + x)) ^ x) / x;
	tmp = 0.0;
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision]}, If[Less[y, -3.7311844206647956e+94], t$95$0, If[Less[y, 2.817959242728288e+37], t$95$1, If[Less[y, 2.347387415166998e+178], N[Log[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Reproduce

herbie shell --seed 2025064 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -37311844206647956000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (exp (/ -1 y)) x) (if (< y 28179592427282880000000000000000000000) (/ (pow (/ x (+ y x)) x) x) (if (< y 23473874151669980000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x)))))

  (/ (exp (* x (log (/ x (+ x y))))) x))