Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D

Percentage Accurate: 99.7% → 99.7%

Time: 2.2s

Alternatives: 15

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(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 - (1.0d0 / (x * 9.0d0))) - (y / (3.0d0 * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(y / Float64(3.0 * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(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 - (1.0d0 / (x * 9.0d0))) - (y / (3.0d0 * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(y / Float64(3.0 * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{3}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ (/ y 3.0) (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - ((y / 3.0) / sqrt(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 - (1.0d0 / (x * 9.0d0))) - ((y / 3.0d0) / sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - ((y / 3.0) / Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - (1.0 / (x * 9.0))) - ((y / 3.0) / math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(Float64(y / 3.0) / sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (1.0 / (x * 9.0))) - ((y / 3.0) / sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \color{blue}{\sqrt{x}}} \]

   4. associate-/r* N/A

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\color{blue}{\frac{y}{3}}}{\sqrt{x}} \]

   7. lift-sqrt.f64 99.7

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{3}}{\color{blue}{\sqrt{x}}} \]

3. Applied rewrites 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}} \]
4. Add Preprocessing

Alternative 2: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+22}:\\ \;\;\;\;\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8e+22)
   (/ (- x (fma (* (sqrt x) y) 0.3333333333333333 0.1111111111111111)) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8e+22) {
		tmp = (x - fma((sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8e+22)
		tmp = Float64(Float64(x - fma(Float64(sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8e+22], N[(N[(x - N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 0.3333333333333333 + 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e22

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{\color{blue}{x}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x} \]

      3. +-commutative N/A

         \[\leadsto \frac{x - \left(\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right) + \frac{1}{9}\right)}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{x - \left(\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}\right)}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      7. lift-sqrt.f64 99.4

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}} \]

   if 8e22 < x

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.8%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(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 - (1.0d0 / (x * 9.0d0))) - (y / (3.0d0 * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(y / Float64(3.0 * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Add Preprocessing

Alternative 4: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{x - \mathsf{fma}\left(\sqrt{x}, y \cdot 0.3333333333333333, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e+35)
   (/ (- x (fma (sqrt x) (* y 0.3333333333333333) 0.1111111111111111)) x)
   (fma (/ -0.3333333333333333 (sqrt x)) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e+35) {
		tmp = (x - fma(sqrt(x), (y * 0.3333333333333333), 0.1111111111111111)) / x;
	} else {
		tmp = fma((-0.3333333333333333 / sqrt(x)), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e+35)
		tmp = Float64(Float64(x - fma(sqrt(x), Float64(y * 0.3333333333333333), 0.1111111111111111)) / x);
	else
		tmp = fma(Float64(-0.3333333333333333 / sqrt(x)), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e+35], N[(N[(x - N[(N[Sqrt[x], $MachinePrecision] * N[(y * 0.3333333333333333), $MachinePrecision] + 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e35

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{\color{blue}{x}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x} \]

      3. +-commutative N/A

         \[\leadsto \frac{x - \left(\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right) + \frac{1}{9}\right)}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{x - \left(\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}\right)}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      7. lift-sqrt.f64 99.4

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{x - \left(\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}\right)}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x - \left(\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}\right)}{x} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{x - \left(\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}\right)}{x} \]

      4. associate-*l* N/A

         \[\leadsto \frac{x - \left(\sqrt{x} \cdot \left(y \cdot \frac{1}{3}\right) + \frac{1}{9}\right)}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x}, y \cdot \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x}, y \cdot \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      7. lower-*.f64 99.4

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x}, y \cdot 0.3333333333333333, 0.1111111111111111\right)}{x} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x}, y \cdot 0.3333333333333333, 0.1111111111111111\right)}{x} \]

   if 1.9999999999999999e35 < x

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{\color{blue}{x}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x} \]

      3. +-commutative N/A

         \[\leadsto \frac{x - \left(\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right) + \frac{1}{9}\right)}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{x - \left(\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}\right)}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      7. lift-sqrt.f64 86.1

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x} \]

   4. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1 - \frac{1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot y\right) \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto 1 + \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right) + \color{blue}{1} \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot y + 1 \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}, \color{blue}{y}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}, y, 1\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{1}}{\sqrt{x}} \cdot \frac{-1}{3}, y, 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}, y, 1\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{1 \cdot \frac{-1}{3}}{\sqrt{x}}, y, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{3}}{\sqrt{x}}, y, 1\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{3}}{\sqrt{x}}, y, 1\right) \]

      12. lift-sqrt.f64 99.8

          \[\leadsto \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right) \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot y, 0.3333333333333333, \frac{0.1111111111111111}{x}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (-
  1.0
  (fma (* (/ 1.0 (sqrt x)) y) 0.3333333333333333 (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	return 1.0 - fma(((1.0 / sqrt(x)) * y), 0.3333333333333333, (0.1111111111111111 / x));
}

Julia

function code(x, y)
	return Float64(1.0 - fma(Float64(Float64(1.0 / sqrt(x)) * y), 0.3333333333333333, Float64(0.1111111111111111 / x)))
end

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * 0.3333333333333333 + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1 - \left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right) + \frac{1}{9} \cdot \frac{1}{x}\right)} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right) + \frac{1}{9} \cdot \frac{1}{x}\right)} \]

   2. *-commutative N/A

      \[\leadsto 1 - \left(\left(\sqrt{\frac{1}{x}} \cdot y\right) \cdot \frac{1}{3} + \color{blue}{\frac{1}{9}} \cdot \frac{1}{x}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto 1 - \mathsf{fma}\left(\sqrt{\frac{1}{x}} \cdot y, \color{blue}{\frac{1}{3}}, \frac{1}{9} \cdot \frac{1}{x}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto 1 - \mathsf{fma}\left(\sqrt{\frac{1}{x}} \cdot y, \frac{1}{3}, \frac{1}{9} \cdot \frac{1}{x}\right) \]

   5. sqrt-div N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y, \frac{1}{3}, \frac{1}{9} \cdot \frac{1}{x}\right) \]

   6. metadata-eval N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot y, \frac{1}{3}, \frac{1}{9} \cdot \frac{1}{x}\right) \]

   7. lower-/.f64 N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot y, \frac{1}{3}, \frac{1}{9} \cdot \frac{1}{x}\right) \]

   8. lift-sqrt.f64 N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot y, \frac{1}{3}, \frac{1}{9} \cdot \frac{1}{x}\right) \]

   9. associate-*r/ N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot y, \frac{1}{3}, \frac{\frac{1}{9} \cdot 1}{x}\right) \]

   10. metadata-eval N/A

       \[\leadsto 1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot y, \frac{1}{3}, \frac{\frac{1}{9}}{x}\right) \]

   11. lower-/.f64 99.5

       \[\leadsto 1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot y, 0.3333333333333333, \frac{0.1111111111111111}{x}\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot y, 0.3333333333333333, \frac{0.1111111111111111}{x}\right)} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(0.3333333333333333 \cdot \sqrt{x}, y, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.45e-5)
   (- (/ (fma (* 0.3333333333333333 (sqrt x)) y 0.1111111111111111) x))
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.45e-5) {
		tmp = -(fma((0.3333333333333333 * sqrt(x)), y, 0.1111111111111111) / x);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.45e-5)
		tmp = Float64(-Float64(fma(Float64(0.3333333333333333 * sqrt(x)), y, 0.1111111111111111) / x));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.45e-5], (-N[(N[(N[(0.3333333333333333 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e-5

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)}{x}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)}{x}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)}{x} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)}{x} \]

      4. +-commutative N/A

         \[\leadsto -\frac{\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right) + \frac{1}{9}}{x} \]

      5. *-commutative N/A

         \[\leadsto -\frac{\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      8. lift-sqrt.f64 98.8

         \[\leadsto -\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x} \]

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto -\frac{\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto -\frac{\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}}{x} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto -\frac{\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}}{x} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right) + \frac{1}{9}}{x} \]

      5. associate-*r* N/A

         \[\leadsto -\frac{\left(\frac{1}{3} \cdot \sqrt{x}\right) \cdot y + \frac{1}{9}}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{1}{3} \cdot \sqrt{x}, y, \frac{1}{9}\right)}{x} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{1}{3} \cdot \sqrt{x}, y, \frac{1}{9}\right)}{x} \]

      8. lift-sqrt.f64 98.8

         \[\leadsto -\frac{\mathsf{fma}\left(0.3333333333333333 \cdot \sqrt{x}, y, 0.1111111111111111\right)}{x} \]

   6. Applied rewrites 98.8%

      \[\leadsto -\frac{\mathsf{fma}\left(0.3333333333333333 \cdot \sqrt{x}, y, 0.1111111111111111\right)}{x} \]

   if 1.45e-5 < x

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   3. Step-by-step derivation

   4. Applied rewrites 98.0%

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

Alternative 7: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+117}:\\ \;\;\;\;1 - \frac{1}{x} \cdot 0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y (* 3.0 (sqrt x))))))
   (if (<= y -4e+76)
     t_0
     (if (<= y 1.12e+117) (- 1.0 (* (/ 1.0 x) 0.1111111111111111)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (y / (3.0 * sqrt(x)));
	double tmp;
	if (y <= -4e+76) {
		tmp = t_0;
	} else if (y <= 1.12e+117) {
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	} 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) :: tmp
    t_0 = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    if (y <= (-4d+76)) then
        tmp = t_0
    else if (y <= 1.12d+117) then
        tmp = 1.0d0 - ((1.0d0 / x) * 0.1111111111111111d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (y / (3.0 * Math.sqrt(x)));
	double tmp;
	if (y <= -4e+76) {
		tmp = t_0;
	} else if (y <= 1.12e+117) {
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (y / (3.0 * math.sqrt(x)))
	tmp = 0
	if y <= -4e+76:
		tmp = t_0
	elif y <= 1.12e+117:
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))))
	tmp = 0.0
	if (y <= -4e+76)
		tmp = t_0;
	elseif (y <= 1.12e+117)
		tmp = Float64(1.0 - Float64(Float64(1.0 / x) * 0.1111111111111111));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (y / (3.0 * sqrt(x)));
	tmp = 0.0;
	if (y <= -4e+76)
		tmp = t_0;
	elseif (y <= 1.12e+117)
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+76], t$95$0, If[LessEqual[y, 1.12e+117], N[(1.0 - N[(N[(1.0 / x), $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000002e76 or 1.12000000000000002e117 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]

   if -4.0000000000000002e76 < y < 1.12000000000000002e117

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{1}{9} \cdot \frac{1}{x}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{\color{blue}{x}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{x} \]

      4. lower-/.f64 91.4

         \[\leadsto 1 - \frac{0.1111111111111111}{\color{blue}{x}} \]

   4. Applied rewrites 91.4%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{\color{blue}{x}} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{x} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \frac{1}{9} \cdot \color{blue}{\frac{1}{x}} \]

      4. *-commutative N/A

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

      6. lower-/.f64 91.4

         \[\leadsto 1 - \frac{1}{x} \cdot 0.1111111111111111 \]

   6. Applied rewrites 91.4%

      \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{0.1111111111111111} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+117}:\\ \;\;\;\;1 - \frac{1}{x} \cdot 0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e+76)
   (fma (/ -0.3333333333333333 (sqrt x)) y 1.0)
   (if (<= y 1.12e+117)
     (- 1.0 (* (/ 1.0 x) 0.1111111111111111))
     (* (/ y (sqrt x)) -0.3333333333333333))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e+76) {
		tmp = fma((-0.3333333333333333 / sqrt(x)), y, 1.0);
	} else if (y <= 1.12e+117) {
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	} else {
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e+76)
		tmp = fma(Float64(-0.3333333333333333 / sqrt(x)), y, 1.0);
	elseif (y <= 1.12e+117)
		tmp = Float64(1.0 - Float64(Float64(1.0 / x) * 0.1111111111111111));
	else
		tmp = Float64(Float64(y / sqrt(x)) * -0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e+76], N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 1.12e+117], N[(1.0 - N[(N[(1.0 / x), $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.0000000000000002e76

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{\color{blue}{x}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x} \]

      3. +-commutative N/A

         \[\leadsto \frac{x - \left(\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right) + \frac{1}{9}\right)}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{x - \left(\left(\sqrt{x} \cdot y\right) \cdot \frac{1}{3} + \frac{1}{9}\right)}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, \frac{1}{3}, \frac{1}{9}\right)}{x} \]

      7. lift-sqrt.f64 83.3

         \[\leadsto \frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x} \]

   4. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1 - \frac{1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot y\right) \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto 1 + \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right) + \color{blue}{1} \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot y + 1 \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}, \color{blue}{y}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}, y, 1\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{1}}{\sqrt{x}} \cdot \frac{-1}{3}, y, 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}, y, 1\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{1 \cdot \frac{-1}{3}}{\sqrt{x}}, y, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{3}}{\sqrt{x}}, y, 1\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{3}}{\sqrt{x}}, y, 1\right) \]

      12. lift-sqrt.f64 95.0

          \[\leadsto \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right) \]

   7. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)} \]

   if -4.0000000000000002e76 < y < 1.12000000000000002e117

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{1}{9} \cdot \frac{1}{x}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{\color{blue}{x}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{x} \]

      4. lower-/.f64 91.4

         \[\leadsto 1 - \frac{0.1111111111111111}{\color{blue}{x}} \]

   4. Applied rewrites 91.4%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{\color{blue}{x}} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{x} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \frac{1}{9} \cdot \color{blue}{\frac{1}{x}} \]

      4. *-commutative N/A

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

      6. lower-/.f64 91.4

         \[\leadsto 1 - \frac{1}{x} \cdot 0.1111111111111111 \]

   6. Applied rewrites 91.4%

      \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{0.1111111111111111} \]

   if 1.12000000000000002e117 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{y}\right) \]

      3. sqrt-div N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      6. lift-sqrt.f64 96.2

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot \color{blue}{y}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      6. sqrt-div N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      11. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      14. lift-/.f64 96.3

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot -0.3333333333333333\right) \cdot y \]

   6. Applied rewrites 96.3%

      \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot -0.3333333333333333\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      4. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \frac{-1}{3}}{\sqrt{x}} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      7. lift-sqrt.f64 96.4

         \[\leadsto \frac{-0.3333333333333333}{\sqrt{x}} \cdot y \]

   8. Applied rewrites 96.4%

      \[\leadsto \frac{-0.3333333333333333}{\sqrt{x}} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot \color{blue}{y} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot y}{\color{blue}{\sqrt{x}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{1 \cdot y}{\sqrt{\color{blue}{x}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \color{blue}{\frac{-1}{3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \color{blue}{\frac{-1}{3}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \frac{-1}{3} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \frac{-1}{3} \]

      11. *-lft-identity 96.3

          \[\leadsto \frac{y}{\sqrt{x}} \cdot -0.3333333333333333 \]

   10. Applied rewrites 96.3%

       \[\leadsto \frac{y}{\sqrt{x}} \cdot \color{blue}{-0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+117}:\\ \;\;\;\;1 - \frac{1}{x} \cdot 0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e+88)
   (/ (* -0.3333333333333333 y) (sqrt x))
   (if (<= y 1.12e+117)
     (- 1.0 (* (/ 1.0 x) 0.1111111111111111))
     (* (/ y (sqrt x)) -0.3333333333333333))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+88) {
		tmp = (-0.3333333333333333 * y) / sqrt(x);
	} else if (y <= 1.12e+117) {
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	} else {
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	}
	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 (y <= (-2.6d+88)) then
        tmp = ((-0.3333333333333333d0) * y) / sqrt(x)
    else if (y <= 1.12d+117) then
        tmp = 1.0d0 - ((1.0d0 / x) * 0.1111111111111111d0)
    else
        tmp = (y / sqrt(x)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+88) {
		tmp = (-0.3333333333333333 * y) / Math.sqrt(x);
	} else if (y <= 1.12e+117) {
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	} else {
		tmp = (y / Math.sqrt(x)) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.6e+88:
		tmp = (-0.3333333333333333 * y) / math.sqrt(x)
	elif y <= 1.12e+117:
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111)
	else:
		tmp = (y / math.sqrt(x)) * -0.3333333333333333
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e+88)
		tmp = Float64(Float64(-0.3333333333333333 * y) / sqrt(x));
	elseif (y <= 1.12e+117)
		tmp = Float64(1.0 - Float64(Float64(1.0 / x) * 0.1111111111111111));
	else
		tmp = Float64(Float64(y / sqrt(x)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.6e+88)
		tmp = (-0.3333333333333333 * y) / sqrt(x);
	elseif (y <= 1.12e+117)
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	else
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e+88], N[(N[(-0.3333333333333333 * y), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e+117], N[(1.0 - N[(N[(1.0 / x), $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6000000000000001e88

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{y}\right) \]

      3. sqrt-div N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      6. lift-sqrt.f64 91.8

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

   4. Applied rewrites 91.8%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot \color{blue}{y}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      6. sqrt-div N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      11. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      14. lift-/.f64 91.9

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot -0.3333333333333333\right) \cdot y \]

   6. Applied rewrites 91.9%

      \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot -0.3333333333333333\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      4. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \frac{-1}{3}}{\sqrt{x}} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      7. lift-sqrt.f64 91.9

         \[\leadsto \frac{-0.3333333333333333}{\sqrt{x}} \cdot y \]

   8. Applied rewrites 91.9%

      \[\leadsto \frac{-0.3333333333333333}{\sqrt{x}} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot \color{blue}{y} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot y}{\color{blue}{\sqrt{x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot y}{\color{blue}{\sqrt{x}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot y}{\sqrt{\color{blue}{x}}} \]

      7. lift-sqrt.f64 91.9

         \[\leadsto \frac{-0.3333333333333333 \cdot y}{\sqrt{x}} \]

   10. Applied rewrites 91.9%

       \[\leadsto \frac{-0.3333333333333333 \cdot y}{\color{blue}{\sqrt{x}}} \]

   if -2.6000000000000001e88 < y < 1.12000000000000002e117

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{1}{9} \cdot \frac{1}{x}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{\color{blue}{x}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{x} \]

      4. lower-/.f64 90.9

         \[\leadsto 1 - \frac{0.1111111111111111}{\color{blue}{x}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{\color{blue}{x}} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{x} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \frac{1}{9} \cdot \color{blue}{\frac{1}{x}} \]

      4. *-commutative N/A

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

      6. lower-/.f64 90.9

         \[\leadsto 1 - \frac{1}{x} \cdot 0.1111111111111111 \]

   6. Applied rewrites 90.9%

      \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{0.1111111111111111} \]

   if 1.12000000000000002e117 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{y}\right) \]

      3. sqrt-div N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      6. lift-sqrt.f64 96.2

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot \color{blue}{y}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      6. sqrt-div N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      11. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      14. lift-/.f64 96.3

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot -0.3333333333333333\right) \cdot y \]

   6. Applied rewrites 96.3%

      \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot -0.3333333333333333\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      4. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \frac{-1}{3}}{\sqrt{x}} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      7. lift-sqrt.f64 96.4

         \[\leadsto \frac{-0.3333333333333333}{\sqrt{x}} \cdot y \]

   8. Applied rewrites 96.4%

      \[\leadsto \frac{-0.3333333333333333}{\sqrt{x}} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot \color{blue}{y} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot y}{\color{blue}{\sqrt{x}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{1 \cdot y}{\sqrt{\color{blue}{x}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \color{blue}{\frac{-1}{3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \color{blue}{\frac{-1}{3}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \frac{-1}{3} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \frac{-1}{3} \]

      11. *-lft-identity 96.3

          \[\leadsto \frac{y}{\sqrt{x}} \cdot -0.3333333333333333 \]

   10. Applied rewrites 96.3%

       \[\leadsto \frac{y}{\sqrt{x}} \cdot \color{blue}{-0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+117}:\\ \;\;\;\;1 - \frac{1}{x} \cdot 0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ y (sqrt x)) -0.3333333333333333)))
   (if (<= y -2.6e+88)
     t_0
     (if (<= y 1.12e+117) (- 1.0 (* (/ 1.0 x) 0.1111111111111111)) t_0))))

C

double code(double x, double y) {
	double t_0 = (y / sqrt(x)) * -0.3333333333333333;
	double tmp;
	if (y <= -2.6e+88) {
		tmp = t_0;
	} else if (y <= 1.12e+117) {
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	} 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) :: tmp
    t_0 = (y / sqrt(x)) * (-0.3333333333333333d0)
    if (y <= (-2.6d+88)) then
        tmp = t_0
    else if (y <= 1.12d+117) then
        tmp = 1.0d0 - ((1.0d0 / x) * 0.1111111111111111d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y / Math.sqrt(x)) * -0.3333333333333333;
	double tmp;
	if (y <= -2.6e+88) {
		tmp = t_0;
	} else if (y <= 1.12e+117) {
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / math.sqrt(x)) * -0.3333333333333333
	tmp = 0
	if y <= -2.6e+88:
		tmp = t_0
	elif y <= 1.12e+117:
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / sqrt(x)) * -0.3333333333333333)
	tmp = 0.0
	if (y <= -2.6e+88)
		tmp = t_0;
	elseif (y <= 1.12e+117)
		tmp = Float64(1.0 - Float64(Float64(1.0 / x) * 0.1111111111111111));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / sqrt(x)) * -0.3333333333333333;
	tmp = 0.0;
	if (y <= -2.6e+88)
		tmp = t_0;
	elseif (y <= 1.12e+117)
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]}, If[LessEqual[y, -2.6e+88], t$95$0, If[LessEqual[y, 1.12e+117], N[(1.0 - N[(N[(1.0 / x), $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6000000000000001e88 or 1.12000000000000002e117 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{y}\right) \]

      3. sqrt-div N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      6. lift-sqrt.f64 93.9

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot \color{blue}{y}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      6. sqrt-div N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      11. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      14. lift-/.f64 94.0

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot -0.3333333333333333\right) \cdot y \]

   6. Applied rewrites 94.0%

      \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot -0.3333333333333333\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      4. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \frac{-1}{3}}{\sqrt{x}} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      7. lift-sqrt.f64 94.0

         \[\leadsto \frac{-0.3333333333333333}{\sqrt{x}} \cdot y \]

   8. Applied rewrites 94.0%

      \[\leadsto \frac{-0.3333333333333333}{\sqrt{x}} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot \color{blue}{y} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot y}{\color{blue}{\sqrt{x}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{1 \cdot y}{\sqrt{\color{blue}{x}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \color{blue}{\frac{-1}{3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \color{blue}{\frac{-1}{3}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \frac{-1}{3} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\sqrt{x}} \cdot \frac{-1}{3} \]

      11. *-lft-identity 93.9

          \[\leadsto \frac{y}{\sqrt{x}} \cdot -0.3333333333333333 \]

   10. Applied rewrites 93.9%

       \[\leadsto \frac{y}{\sqrt{x}} \cdot \color{blue}{-0.3333333333333333} \]

   if -2.6000000000000001e88 < y < 1.12000000000000002e117

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{1}{9} \cdot \frac{1}{x}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{\color{blue}{x}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{x} \]

      4. lower-/.f64 90.9

         \[\leadsto 1 - \frac{0.1111111111111111}{\color{blue}{x}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{\color{blue}{x}} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{x} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \frac{1}{9} \cdot \color{blue}{\frac{1}{x}} \]

      4. *-commutative N/A

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

      6. lower-/.f64 90.9

         \[\leadsto 1 - \frac{1}{x} \cdot 0.1111111111111111 \]

   6. Applied rewrites 90.9%

      \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{0.1111111111111111} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.3333333333333333}{\sqrt{x}} \cdot y\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+117}:\\ \;\;\;\;1 - \frac{1}{x} \cdot 0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ -0.3333333333333333 (sqrt x)) y)))
   (if (<= y -2.6e+88)
     t_0
     (if (<= y 1.12e+117) (- 1.0 (* (/ 1.0 x) 0.1111111111111111)) t_0))))

C

double code(double x, double y) {
	double t_0 = (-0.3333333333333333 / sqrt(x)) * y;
	double tmp;
	if (y <= -2.6e+88) {
		tmp = t_0;
	} else if (y <= 1.12e+117) {
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	} 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) :: tmp
    t_0 = ((-0.3333333333333333d0) / sqrt(x)) * y
    if (y <= (-2.6d+88)) then
        tmp = t_0
    else if (y <= 1.12d+117) then
        tmp = 1.0d0 - ((1.0d0 / x) * 0.1111111111111111d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (-0.3333333333333333 / Math.sqrt(x)) * y;
	double tmp;
	if (y <= -2.6e+88) {
		tmp = t_0;
	} else if (y <= 1.12e+117) {
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (-0.3333333333333333 / math.sqrt(x)) * y
	tmp = 0
	if y <= -2.6e+88:
		tmp = t_0
	elif y <= 1.12e+117:
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-0.3333333333333333 / sqrt(x)) * y)
	tmp = 0.0
	if (y <= -2.6e+88)
		tmp = t_0;
	elseif (y <= 1.12e+117)
		tmp = Float64(1.0 - Float64(Float64(1.0 / x) * 0.1111111111111111));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (-0.3333333333333333 / sqrt(x)) * y;
	tmp = 0.0;
	if (y <= -2.6e+88)
		tmp = t_0;
	elseif (y <= 1.12e+117)
		tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.6e+88], t$95$0, If[LessEqual[y, 1.12e+117], N[(1.0 - N[(N[(1.0 / x), $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6000000000000001e88 or 1.12000000000000002e117 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{y}\right) \]

      3. sqrt-div N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      6. lift-sqrt.f64 93.9

         \[\leadsto -0.3333333333333333 \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot \color{blue}{y}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{1}{\sqrt{x}} \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot y\right) \]

      6. sqrt-div N/A

         \[\leadsto \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      11. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{1}}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      14. lift-/.f64 94.0

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot -0.3333333333333333\right) \cdot y \]

   6. Applied rewrites 94.0%

      \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot -0.3333333333333333\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{-1}{3}\right) \cdot y \]

      4. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \frac{-1}{3}}{\sqrt{x}} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{\sqrt{x}} \cdot y \]

      7. lift-sqrt.f64 94.0

         \[\leadsto \frac{-0.3333333333333333}{\sqrt{x}} \cdot y \]

   8. Applied rewrites 94.0%

      \[\leadsto \frac{-0.3333333333333333}{\sqrt{x}} \cdot \color{blue}{y} \]

   if -2.6000000000000001e88 < y < 1.12000000000000002e117

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{1}{9} \cdot \frac{1}{x}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{\color{blue}{x}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{x} \]

      4. lower-/.f64 90.9

         \[\leadsto 1 - \frac{0.1111111111111111}{\color{blue}{x}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{\color{blue}{x}} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{x} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \frac{1}{9} \cdot \color{blue}{\frac{1}{x}} \]

      4. *-commutative N/A

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

      6. lower-/.f64 90.9

         \[\leadsto 1 - \frac{1}{x} \cdot 0.1111111111111111 \]

   6. Applied rewrites 90.9%

      \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{0.1111111111111111} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{1}{x} \cdot 0.1111111111111111 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (* (/ 1.0 x) 0.1111111111111111)))

C

double code(double x, double y) {
	return 1.0 - ((1.0 / x) * 0.1111111111111111);
}

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 - ((1.0d0 / x) * 0.1111111111111111d0)
end function

Java

public static double code(double x, double y) {
	return 1.0 - ((1.0 / x) * 0.1111111111111111);
}

Python

def code(x, y):
	return 1.0 - ((1.0 / x) * 0.1111111111111111)

Julia

function code(x, y)
	return Float64(1.0 - Float64(Float64(1.0 / x) * 0.1111111111111111))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - ((1.0 / x) * 0.1111111111111111);
end

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(1.0 / x), $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto 1 - \color{blue}{\frac{1}{9} \cdot \frac{1}{x}} \]

   2. associate-*r/ N/A

      \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{\color{blue}{x}} \]

   3. metadata-eval N/A

      \[\leadsto 1 - \frac{\frac{1}{9}}{x} \]

   4. lower-/.f64 62.8

      \[\leadsto 1 - \frac{0.1111111111111111}{\color{blue}{x}} \]

4. Applied rewrites 62.8%

   \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 1 - \frac{\frac{1}{9}}{\color{blue}{x}} \]

   2. metadata-eval N/A

      \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{x} \]

   3. associate-*r/ N/A

      \[\leadsto 1 - \frac{1}{9} \cdot \color{blue}{\frac{1}{x}} \]

   4. *-commutative N/A

      \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

   5. lower-*.f64 N/A

      \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\frac{1}{9}} \]

   6. lower-/.f64 62.8

      \[\leadsto 1 - \frac{1}{x} \cdot 0.1111111111111111 \]

6. Applied rewrites 62.8%

   \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{0.1111111111111111} \]
7. Add Preprocessing

Alternative 13: 62.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ 0.1111111111111111 x)))

C

double code(double x, double y) {
	return 1.0 - (0.1111111111111111 / 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 - (0.1111111111111111d0 / x)
end function

Java

public static double code(double x, double y) {
	return 1.0 - (0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 - (0.1111111111111111 / x)

Julia

function code(x, y)
	return Float64(1.0 - Float64(0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - (0.1111111111111111 / x);
end

Wolfram

code[x_, y_] := N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto 1 - \color{blue}{\frac{1}{9} \cdot \frac{1}{x}} \]

   2. associate-*r/ N/A

      \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{\color{blue}{x}} \]

   3. metadata-eval N/A

      \[\leadsto 1 - \frac{\frac{1}{9}}{x} \]

   4. lower-/.f64 62.8

      \[\leadsto 1 - \frac{0.1111111111111111}{\color{blue}{x}} \]

4. Applied rewrites 62.8%

   \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
5. Add Preprocessing

Alternative 14: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \leq -200:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))) -200.0)
   (/ -0.1111111111111111 x)
   1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)))) <= -200.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	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) :: tmp
    if (((1.0d0 - (1.0d0 / (x * 9.0d0))) - (y / (3.0d0 * sqrt(x)))) <= (-200.0d0)) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * Math.sqrt(x)))) <= -200.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * math.sqrt(x)))) <= -200.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(y / Float64(3.0 * sqrt(x)))) <= -200.0)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)))) <= -200.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -200.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) (/.f64 y (*.f64 #s(literal 3 binary64) (sqrt.f64 x)))) < -200

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{1}{9} \cdot \frac{1}{x}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{\color{blue}{x}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{x} \]

      4. lower-/.f64 62.6

         \[\leadsto 1 - \frac{0.1111111111111111}{\color{blue}{x}} \]

   4. Applied rewrites 62.6%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{-1}{9}}{\color{blue}{x}} \]
   6. Step-by-step derivation

      1. lift-/.f64 62.3

         \[\leadsto \frac{-0.1111111111111111}{x} \]

   7. Applied rewrites 62.3%

      \[\leadsto \frac{-0.1111111111111111}{\color{blue}{x}} \]

   if -200 < (-.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) (/.f64 y (*.f64 #s(literal 3 binary64) (sqrt.f64 x))))

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{1}{9} \cdot \frac{1}{x}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{\color{blue}{x}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{\frac{1}{9}}{x} \]

      4. lower-/.f64 63.0

         \[\leadsto 1 - \frac{0.1111111111111111}{\color{blue}{x}} \]

   4. Applied rewrites 63.0%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 62.2%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 31.6% accurate, 49.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

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
end function

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto 1 - \color{blue}{\frac{1}{9} \cdot \frac{1}{x}} \]

   2. associate-*r/ N/A

      \[\leadsto 1 - \frac{\frac{1}{9} \cdot 1}{\color{blue}{x}} \]

   3. metadata-eval N/A

      \[\leadsto 1 - \frac{\frac{1}{9}}{x} \]

   4. lower-/.f64 62.8

      \[\leadsto 1 - \frac{0.1111111111111111}{\color{blue}{x}} \]

4. Applied rewrites 62.8%

   \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]

5. Taylor expanded in x around inf

   \[\leadsto 1 \]
6. Step-by-step derivation

7. Applied rewrites 31.6%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64
  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))