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

Percentage Accurate: 99.7% → 99.6%

Time: 3.9s

Alternatives: 13

Speedup: N/A×

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: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + N[(1.0 - 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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mult-flip N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. distribute-neg-frac N/A

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

   13. lower-/.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 99.6

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, 1 - \frac{1}{x \cdot 9}\right) \]

   16. lift-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-/r* N/A

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

   20. lower-/.f64 N/A

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

   21. metadata-eval 99.6

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

3. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sub-negate-rev N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. associate--r+ N/A

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

   6. sub-negate N/A

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

   7. lower--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. distribute-neg-frac N/A

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

   13. lower-/.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. add-flip-rev N/A

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

   18. lift-/.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. associate-/r* N/A

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

   22. mult-flip N/A

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

3. Applied rewrites 99.6%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -0.3333333333333333, \sqrt{x}, x - 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{x}} \cdot y, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.1e+102)
   (/ (fma (* y -0.3333333333333333) (sqrt x) (- x 0.1111111111111111)) x)
   (fma (* (sqrt (/ 1.0 x)) y) -0.3333333333333333 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e+102) {
		tmp = fma((y * -0.3333333333333333), sqrt(x), (x - 0.1111111111111111)) / x;
	} else {
		tmp = fma((sqrt((1.0 / x)) * y), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e+102)
		tmp = Float64(fma(Float64(y * -0.3333333333333333), sqrt(x), Float64(x - 0.1111111111111111)) / x);
	else
		tmp = fma(Float64(sqrt(Float64(1.0 / x)) * y), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e+102], N[(N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[Sqrt[x], $MachinePrecision] + N[(x - 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.10000000000000004e102

   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

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 99.6

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, 1 - \frac{1}{x \cdot 9}\right) \]

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-/r* N/A

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

      20. lower-/.f64 N/A

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

      21. metadata-eval 99.6

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

   3. Applied rewrites 99.6%

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

   5. Applied rewrites 93.7%

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

   if 1.10000000000000004e102 < x

   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

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

      8. mult-flip N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      11. pow-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\frac{-1}{2}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      14. pow-prod-up N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{\frac{1}{2}}\right)} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      15. inv-pow N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{\frac{1}{2}}\right) + \left(1 - \frac{1}{x \cdot 9}\right) \]

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 68.6

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

   6. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 68.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

Alternative 4: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{x}} \cdot y, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.056)
   (- (/ -0.1111111111111111 x) (/ y (* 3.0 (sqrt x))))
   (fma (* (sqrt (/ 1.0 x)) y) -0.3333333333333333 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.056) {
		tmp = (-0.1111111111111111 / x) - (y / (3.0 * sqrt(x)));
	} else {
		tmp = fma((sqrt((1.0 / x)) * y), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.056)
		tmp = Float64(Float64(-0.1111111111111111 / x) - Float64(y / Float64(3.0 * sqrt(x))));
	else
		tmp = fma(Float64(sqrt(Float64(1.0 / x)) * y), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.056], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0560000000000000012

   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

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

      2. sub-negate-rev N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--r+ N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. add-flip-rev N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/r* N/A

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

      22. mult-flip N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 67.6

         \[\leadsto \frac{-0.1111111111111111}{x} - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \]

   6. Applied rewrites 67.6%

      \[\leadsto \frac{-0.1111111111111111}{x} - \color{blue}{0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9}}{x} - \frac{y}{\sqrt{x}} \cdot \frac{1}{3} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{9}}{x} - \frac{y}{\sqrt{x}} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{9}}{x} - \frac{y}{\sqrt{x}} \cdot \frac{1}{3} \]

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{-1}{9}}{x} - y \cdot \frac{1}{3 \cdot \sqrt{x}} \]

      11. mult-flip N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{-1}{9}}{x} - \frac{y}{3 \cdot \sqrt{x}} \]

      14. lower-*.f64 67.7

          \[\leadsto \frac{-0.1111111111111111}{x} - \frac{y}{3 \cdot \color{blue}{\sqrt{x}}} \]

   8. Applied rewrites 67.7%

      \[\leadsto \frac{-0.1111111111111111}{x} - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

   if 0.0560000000000000012 < x

   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

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

      8. mult-flip N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      11. pow-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\frac{-1}{2}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      14. pow-prod-up N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{\frac{1}{2}}\right)} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      15. inv-pow N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{\frac{1}{2}}\right) + \left(1 - \frac{1}{x \cdot 9}\right) \]

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 68.6

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

   6. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 68.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

Alternative 5: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{x}} \cdot y, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.056)
   (- (/ -0.1111111111111111 x) (* 0.3333333333333333 (/ y (sqrt x))))
   (fma (* (sqrt (/ 1.0 x)) y) -0.3333333333333333 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.056) {
		tmp = (-0.1111111111111111 / x) - (0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = fma((sqrt((1.0 / x)) * y), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.056)
		tmp = Float64(Float64(-0.1111111111111111 / x) - Float64(0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = fma(Float64(sqrt(Float64(1.0 / x)) * y), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.056], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - N[(0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0560000000000000012

   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

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

      2. sub-negate-rev N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--r+ N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. add-flip-rev N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/r* N/A

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

      22. mult-flip N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 67.6

         \[\leadsto \frac{-0.1111111111111111}{x} - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \]

   6. Applied rewrites 67.6%

      \[\leadsto \frac{-0.1111111111111111}{x} - \color{blue}{0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   if 0.0560000000000000012 < x

   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

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

      8. mult-flip N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      11. pow-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\frac{-1}{2}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      14. pow-prod-up N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{\frac{1}{2}}\right)} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      15. inv-pow N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{\frac{1}{2}}\right) + \left(1 - \frac{1}{x \cdot 9}\right) \]

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 68.6

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

   6. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 68.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

Alternative 6: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{x}} \cdot y, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (- (* -0.3333333333333333 (* y (sqrt x))) 0.1111111111111111) x)
   (fma (* (sqrt (/ 1.0 x)) y) -0.3333333333333333 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = fma((sqrt((1.0 / x)) * y), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(Float64(Float64(-0.3333333333333333 * Float64(y * sqrt(x))) - 0.1111111111111111) / x);
	else
		tmp = fma(Float64(sqrt(Float64(1.0 / x)) * y), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(N[(-0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

      8. mult-flip N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      11. pow-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\frac{-1}{2}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      14. pow-prod-up N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{\frac{1}{2}}\right)} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      15. inv-pow N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{\frac{1}{2}}\right) + \left(1 - \frac{1}{x \cdot 9}\right) \]

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 61.7

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

   6. Applied rewrites 61.7%

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

   if 0.110000000000000001 < x

   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

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

      8. mult-flip N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      11. pow-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\frac{-1}{2}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      14. pow-prod-up N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{\frac{1}{2}}\right)} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      15. inv-pow N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{\frac{1}{2}}\right) + \left(1 - \frac{1}{x \cdot 9}\right) \]

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 68.6

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

   6. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 68.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

Alternative 7: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{\frac{1}{x}} \cdot y, -0.3333333333333333, 1\right)\\ \mathbf{if}\;y \leq -2.95 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* (sqrt (/ 1.0 x)) y) -0.3333333333333333 1.0)))
   (if (<= y -2.95e+20)
     t_0
     (if (<= y 1.5e+43) (- (/ -0.1111111111111111 x) -1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((sqrt((1.0 / x)) * y), -0.3333333333333333, 1.0);
	double tmp;
	if (y <= -2.95e+20) {
		tmp = t_0;
	} else if (y <= 1.5e+43) {
		tmp = (-0.1111111111111111 / x) - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(sqrt(Float64(1.0 / x)) * y), -0.3333333333333333, 1.0)
	tmp = 0.0
	if (y <= -2.95e+20)
		tmp = t_0;
	elseif (y <= 1.5e+43)
		tmp = Float64(Float64(-0.1111111111111111 / x) - -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]}, If[LessEqual[y, -2.95e+20], t$95$0, If[LessEqual[y, 1.5e+43], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.95e20 or 1.50000000000000008e43 < y

   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

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

      8. mult-flip N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      11. pow-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\frac{-1}{2}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      14. pow-prod-up N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{\frac{1}{2}}\right)} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      15. inv-pow N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{\frac{1}{2}}\right) + \left(1 - \frac{1}{x \cdot 9}\right) \]

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 68.6

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

   6. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 68.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

   if -2.95e20 < y < 1.50000000000000008e43

   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

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

      2. sub-negate-rev N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--r+ N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. add-flip-rev N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/r* N/A

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

      22. mult-flip N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 62.8%

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

Alternative 8: 92.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{if}\;y \leq -7 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y (sqrt x)) -3.0)))
   (if (<= y -7e+59)
     t_0
     (if (<= y 2.15e+49) (- (/ -0.1111111111111111 x) -1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = (y / sqrt(x)) / -3.0;
	double tmp;
	if (y <= -7e+59) {
		tmp = t_0;
	} else if (y <= 2.15e+49) {
		tmp = (-0.1111111111111111 / x) - -1.0;
	} 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)) / (-3.0d0)
    if (y <= (-7d+59)) then
        tmp = t_0
    else if (y <= 2.15d+49) then
        tmp = ((-0.1111111111111111d0) / x) - (-1.0d0)
    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)) / -3.0;
	double tmp;
	if (y <= -7e+59) {
		tmp = t_0;
	} else if (y <= 2.15e+49) {
		tmp = (-0.1111111111111111 / x) - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / math.sqrt(x)) / -3.0
	tmp = 0
	if y <= -7e+59:
		tmp = t_0
	elif y <= 2.15e+49:
		tmp = (-0.1111111111111111 / x) - -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / sqrt(x)) / -3.0)
	tmp = 0.0
	if (y <= -7e+59)
		tmp = t_0;
	elseif (y <= 2.15e+49)
		tmp = Float64(Float64(-0.1111111111111111 / x) - -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / sqrt(x)) / -3.0;
	tmp = 0.0;
	if (y <= -7e+59)
		tmp = t_0;
	elseif (y <= 2.15e+49)
		tmp = (-0.1111111111111111 / x) - -1.0;
	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] / -3.0), $MachinePrecision]}, If[LessEqual[y, -7e+59], t$95$0, If[LessEqual[y, 2.15e+49], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7e59 or 2.15e49 < y

   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

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

      8. mult-flip N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      11. pow-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\frac{-1}{2}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      14. pow-prod-up N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{\frac{1}{2}}\right)} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      15. inv-pow N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{\frac{1}{2}}\right) + \left(1 - \frac{1}{x \cdot 9}\right) \]

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 32.2

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

   6. Applied rewrites 32.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 34.9

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

   8. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. mult-flip N/A

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

      10. lift-sqrt.f64 N/A

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

      11. pow1/2 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(3\right)} \cdot \left(y \cdot \left({x}^{\frac{1}{2}} \cdot \frac{\color{blue}{1}}{x}\right)\right) \]

      12. inv-pow N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(3\right)} \cdot \left(y \cdot \left({x}^{\frac{1}{2}} \cdot {x}^{\color{blue}{-1}}\right)\right) \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(3\right)} \cdot \left(y \cdot {x}^{\color{blue}{\left(\frac{1}{2} + -1\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(3\right)} \cdot \left(y \cdot {x}^{\frac{-1}{2}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(3\right)} \cdot \left(y \cdot {x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      16. pow-flip N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(3\right)} \cdot \left(y \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}}\right) \]

      17. pow1/2 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. mult-flip N/A

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

      20. lift-sqrt.f64 N/A

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

      21. associate-*l/ N/A

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

      22. *-lft-identity N/A

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

      23. lower-/.f64 N/A

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

   10. Applied rewrites 38.2%

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

   if -7e59 < y < 2.15e49

   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

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

      2. sub-negate-rev N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--r+ N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. add-flip-rev N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/r* N/A

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

      22. mult-flip N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 62.8%

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

Alternative 9: 92.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+59}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e+59)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 2.15e+49)
     (- (/ -0.1111111111111111 x) -1.0)
     (* (/ -0.3333333333333333 (sqrt x)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e+59) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 2.15e+49) {
		tmp = (-0.1111111111111111 / x) - -1.0;
	} else {
		tmp = (-0.3333333333333333 / sqrt(x)) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7d+59)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 2.15d+49) then
        tmp = ((-0.1111111111111111d0) / x) - (-1.0d0)
    else
        tmp = ((-0.3333333333333333d0) / sqrt(x)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7e+59) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 2.15e+49) {
		tmp = (-0.1111111111111111 / x) - -1.0;
	} else {
		tmp = (-0.3333333333333333 / Math.sqrt(x)) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e+59:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 2.15e+49:
		tmp = (-0.1111111111111111 / x) - -1.0
	else:
		tmp = (-0.3333333333333333 / math.sqrt(x)) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e+59)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 2.15e+49)
		tmp = Float64(Float64(-0.1111111111111111 / x) - -1.0);
	else
		tmp = Float64(Float64(-0.3333333333333333 / sqrt(x)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7e+59)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 2.15e+49)
		tmp = (-0.1111111111111111 / x) - -1.0;
	else
		tmp = (-0.3333333333333333 / sqrt(x)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e+59], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e+49], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - -1.0), $MachinePrecision], N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7e59

   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 inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 38.2

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

   4. Applied rewrites 38.2%

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

   if -7e59 < y < 2.15e49

   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

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

      2. sub-negate-rev N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--r+ N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. add-flip-rev N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/r* N/A

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

      22. mult-flip N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 62.8%

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

   if 2.15e49 < y

   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

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

      8. mult-flip N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      11. pow-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\frac{-1}{2}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      14. pow-prod-up N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{\frac{1}{2}}\right)} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      15. inv-pow N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{\frac{1}{2}}\right) + \left(1 - \frac{1}{x \cdot 9}\right) \]

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 32.2

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

   6. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. mult-flip N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(y \cdot \left({x}^{\frac{1}{2}} \cdot \frac{\color{blue}{1}}{x}\right)\right) \]

      8. inv-pow N/A

         \[\leadsto \frac{-1}{3} \cdot \left(y \cdot \left({x}^{\frac{1}{2}} \cdot {x}^{\color{blue}{-1}}\right)\right) \]

      9. pow-prod-up N/A

         \[\leadsto \frac{-1}{3} \cdot \left(y \cdot {x}^{\color{blue}{\left(\frac{1}{2} + -1\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{3} \cdot \left(y \cdot {x}^{\frac{-1}{2}}\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{3} \cdot \left(y \cdot {x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      12. pow-flip N/A

          \[\leadsto \frac{-1}{3} \cdot \left(y \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}}\right) \]

      13. pow1/2 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. mult-flip N/A

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

      16. associate-/l* N/A

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

      17. associate-*l/ N/A

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

      18. lift-/.f64 N/A

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

      19. lower-*.f64 38.2

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

   8. Applied rewrites 38.2%

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

Alternative 10: 92.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.3333333333333333}{\sqrt{x}} \cdot y\\ \mathbf{if}\;y \leq -7 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ -0.3333333333333333 (sqrt x)) y)))
   (if (<= y -7e+59)
     t_0
     (if (<= y 2.15e+49) (- (/ -0.1111111111111111 x) -1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = (-0.3333333333333333 / sqrt(x)) * y;
	double tmp;
	if (y <= -7e+59) {
		tmp = t_0;
	} else if (y <= 2.15e+49) {
		tmp = (-0.1111111111111111 / x) - -1.0;
	} 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 <= (-7d+59)) then
        tmp = t_0
    else if (y <= 2.15d+49) then
        tmp = ((-0.1111111111111111d0) / x) - (-1.0d0)
    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 <= -7e+59) {
		tmp = t_0;
	} else if (y <= 2.15e+49) {
		tmp = (-0.1111111111111111 / x) - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (-0.3333333333333333 / math.sqrt(x)) * y
	tmp = 0
	if y <= -7e+59:
		tmp = t_0
	elif y <= 2.15e+49:
		tmp = (-0.1111111111111111 / x) - -1.0
	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 <= -7e+59)
		tmp = t_0;
	elseif (y <= 2.15e+49)
		tmp = Float64(Float64(-0.1111111111111111 / x) - -1.0);
	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 <= -7e+59)
		tmp = t_0;
	elseif (y <= 2.15e+49)
		tmp = (-0.1111111111111111 / x) - -1.0;
	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, -7e+59], t$95$0, If[LessEqual[y, 2.15e+49], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7e59 or 2.15e49 < y

   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

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

      8. mult-flip N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      11. pow-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\frac{-1}{2}}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      14. pow-prod-up N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{\frac{1}{2}}\right)} + \left(1 - \frac{1}{x \cdot 9}\right) \]

      15. inv-pow N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{y}{3}\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{\frac{1}{2}}\right) + \left(1 - \frac{1}{x \cdot 9}\right) \]

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 32.2

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

   6. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. mult-flip N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

         \[\leadsto \frac{-1}{3} \cdot \left(y \cdot \left({x}^{\frac{1}{2}} \cdot \frac{\color{blue}{1}}{x}\right)\right) \]

      8. inv-pow N/A

         \[\leadsto \frac{-1}{3} \cdot \left(y \cdot \left({x}^{\frac{1}{2}} \cdot {x}^{\color{blue}{-1}}\right)\right) \]

      9. pow-prod-up N/A

         \[\leadsto \frac{-1}{3} \cdot \left(y \cdot {x}^{\color{blue}{\left(\frac{1}{2} + -1\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{3} \cdot \left(y \cdot {x}^{\frac{-1}{2}}\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{3} \cdot \left(y \cdot {x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      12. pow-flip N/A

          \[\leadsto \frac{-1}{3} \cdot \left(y \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}}\right) \]

      13. pow1/2 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. mult-flip N/A

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

      16. associate-/l* N/A

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

      17. associate-*l/ N/A

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

      18. lift-/.f64 N/A

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

      19. lower-*.f64 38.2

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

   8. Applied rewrites 38.2%

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

   if -7e59 < y < 2.15e49

   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

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

      2. sub-negate-rev N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--r+ N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. add-flip-rev N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/r* N/A

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

      22. mult-flip N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 62.8%

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

Alternative 11: 62.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. sub-negate-rev N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. associate--r+ N/A

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

   6. sub-negate N/A

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

   7. lower--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. distribute-neg-frac N/A

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

   13. lower-/.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. add-flip-rev N/A

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

   18. lift-/.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. associate-/r* N/A

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

   22. mult-flip N/A

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 62.8%

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

Alternative 12: 62.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \leq -50:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))) -50.0)
   (/ -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)))) <= -50.0) {
		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)))) <= (-50.0d0)) 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)))) <= -50.0) {
		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)))) <= -50.0:
		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)))) <= -50.0)
		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)))) <= -50.0)
		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], -50.0], 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)))) < -50

   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 0

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

      1. lower-/.f64 31.6

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

   4. Applied rewrites 31.6%

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

   if -50 < (-.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 99.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.3%

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

Alternative 13: 32.3% accurate, 20.6× 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 x around inf

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

4. Applied rewrites 32.3%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(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)))))