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

Percentage Accurate: 84.8% → 99.7%

Time: 6.0s

Alternatives: 7

Speedup: 15.6×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.8 \lor \neg \left(y \leq 3.2 \cdot 10^{-14}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.8) (not (<= y 3.2e-14)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.8) || !(y <= 3.2e-14)) {
		tmp = x + (exp(-z) / y);
	} else {
		tmp = x + (1.0 / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-0.8d0)) .or. (.not. (y <= 3.2d-14))) then
        tmp = x + (exp(-z) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.8) || !(y <= 3.2e-14)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.8) or not (y <= 3.2e-14):
		tmp = x + (math.exp(-z) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.8) || !(y <= 3.2e-14))
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.8) || ~((y <= 3.2e-14)))
		tmp = x + (exp(-z) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.8], N[Not[LessEqual[y, 3.2e-14]], $MachinePrecision]], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.80000000000000004 or 3.2000000000000002e-14 < y

   1. Initial program 86.2%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -0.80000000000000004 < y < 3.2000000000000002e-14

   1. Initial program 85.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.8 \lor \neg \left(y \leq 3.2 \cdot 10^{-14}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+27}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(0.5, y, 0.5\right) \cdot \left(z \cdot z\right)}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.9e+27)
   (+ x (/ (/ (* (fma 0.5 y 0.5) (* z z)) y) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.9e+27) {
		tmp = x + (((fma(0.5, y, 0.5) * (z * z)) / y) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.9e+27)
		tmp = Float64(x + Float64(Float64(Float64(fma(0.5, y, 0.5) * Float64(z * z)) / y) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.9e+27], N[(x + N[(N[(N[(N[(0.5 * y + 0.5), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.90000000000000015e27

   1. Initial program 53.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. pow-to-exp N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 29.8

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 35.5

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

   11. Applied rewrites 35.5%

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

   12. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 61.2

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

   14. Applied rewrites 61.2%

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

   if -4.90000000000000015e27 < z

   1. Initial program 92.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.3%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+27}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(0.5, y, 0.5\right) \cdot \left(z \cdot z\right)}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.78:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(-y, z, y\right)}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.78) (+ x (/ (/ (fma (- y) z y) y) y)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.78) {
		tmp = x + ((fma(-y, z, y) / y) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.78)
		tmp = Float64(x + Float64(Float64(fma(Float64(-y), z, y) / y) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.78], N[(x + N[(N[(N[((-y) * z + y), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.78000000000000003

   1. Initial program 82.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. pow-to-exp N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 76.7

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 68.9

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

   11. Applied rewrites 68.9%

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

   12. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-neg.f64 77.3

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

   14. Applied rewrites 77.3%

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

   if -0.78000000000000003 < y

   1. Initial program 86.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.78:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(-y, z, y\right)}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.8:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.8) (+ x (/ (fma (- (* 0.5 z) 1.0) z 1.0) y)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.8) {
		tmp = x + (fma(((0.5 * z) - 1.0), z, 1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.8)
		tmp = Float64(x + Float64(fma(Float64(Float64(0.5 * z) - 1.0), z, 1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.8], N[(x + N[(N[(N[(N[(0.5 * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.80000000000000004

   1. Initial program 82.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. pow-to-exp N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 76.7

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 76.7%

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

   if -0.80000000000000004 < y

   1. Initial program 86.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.8:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e-13) x (if (<= y 9.4e-117) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e-13) {
		tmp = x;
	} else if (y <= 9.4e-117) {
		tmp = 1.0 / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.8d-13)) then
        tmp = x
    else if (y <= 9.4d-117) then
        tmp = 1.0d0 / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e-13) {
		tmp = x;
	} else if (y <= 9.4e-117) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.8e-13:
		tmp = x
	elif y <= 9.4e-117:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.8e-13)
		tmp = x;
	elseif (y <= 9.4e-117)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.8e-13)
		tmp = x;
	elseif (y <= 9.4e-117)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.8e-13], x, If[LessEqual[y, 9.4e-117], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7999999999999997e-13 or 9.40000000000000017e-117 < y

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 65.4%

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

   if -4.7999999999999997e-13 < y < 9.40000000000000017e-117

   1. Initial program 83.4%

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

   3. Taylor expanded in y around 0

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

      1. inv-pow N/A

         \[\leadsto {y}^{\color{blue}{-1}} \]

      2. lower-pow.f64 78.5

         \[\leadsto {y}^{\color{blue}{-1}} \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{{y}^{-1}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {y}^{\color{blue}{-1}} \]

      2. inv-pow N/A

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

      3. lower-/.f64 78.5

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

   7. Applied rewrites 78.5%

      \[\leadsto \frac{1}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.1% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 86.6%

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

6. Final simplification 86.6%

   \[\leadsto x + \frac{1}{y} \]
7. Add Preprocessing

Alternative 7: 49.5% accurate, 234.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 51.4%

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

Developer Target 1: 91.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< (/ y (+ z y)) 7.11541576e-315)
   (+ x (/ (exp (/ -1.0 z)) y))
   (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y / (z + y)) < 7.11541576e-315) {
		tmp = x + (exp((-1.0 / z)) / y);
	} else {
		tmp = x + (exp(log(pow((y / (y + z)), y))) / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y / (z + y)) < 7.11541576d-315) then
        tmp = x + (exp(((-1.0d0) / z)) / y)
    else
        tmp = x + (exp(log(((y / (y + z)) ** y))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y / (z + y)) < 7.11541576e-315) {
		tmp = x + (Math.exp((-1.0 / z)) / y);
	} else {
		tmp = x + (Math.exp(Math.log(Math.pow((y / (y + z)), y))) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y / (z + y)) < 7.11541576e-315:
		tmp = x + (math.exp((-1.0 / z)) / y)
	else:
		tmp = x + (math.exp(math.log(math.pow((y / (y + z)), y))) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y / Float64(z + y)) < 7.11541576e-315)
		tmp = Float64(x + Float64(exp(Float64(-1.0 / z)) / y));
	else
		tmp = Float64(x + Float64(exp(log((Float64(y / Float64(y + z)) ^ y))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y / (z + y)) < 7.11541576e-315)
		tmp = x + (exp((-1.0 / z)) / y);
	else
		tmp = x + (exp(log(((y / (y + z)) ^ y))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision], 7.11541576e-315], N[(x + N[(N[Exp[N[(-1.0 / z), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Exp[N[Log[N[Power[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025037 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ y (+ z y)) 17788539399477/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

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