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

Percentage Accurate: 99.7% → 99.7%

Time: 7.1s

Alternatives: 11

Speedup: 0.2×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    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

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (1.0d0 / (x * 9.0d0))) - (y / (3.0d0 * sqrt(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(1 - {\left(x \cdot 9\right)}^{-1}\right) - \frac{{x}^{-0.5} \cdot y}{3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (pow (* x 9.0) -1.0)) (/ (* (pow x -0.5) y) 3.0)))

C

double code(double x, double y) {
	return (1.0 - pow((x * 9.0), -1.0)) - ((pow(x, -0.5) * y) / 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - ((x * 9.0d0) ** (-1.0d0))) - (((x ** (-0.5d0)) * y) / 3.0d0)
end function

Java

public static double code(double x, double y) {
	return (1.0 - Math.pow((x * 9.0), -1.0)) - ((Math.pow(x, -0.5) * y) / 3.0);
}

Python

def code(x, y):
	return (1.0 - math.pow((x * 9.0), -1.0)) - ((math.pow(x, -0.5) * y) / 3.0)

Julia

function code(x, y)
	return Float64(Float64(1.0 - (Float64(x * 9.0) ^ -1.0)) - Float64(Float64((x ^ -0.5) * y) / 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - ((x * 9.0) ^ -1.0)) - (((x ^ -0.5) * y) / 3.0);
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[x, -0.5], $MachinePrecision] * y), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow1/2 N/A

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

   11. pow-flip N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval 99.7

       \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{{x}^{\color{blue}{-0.5}} \cdot y}{3} \]

4. Applied rewrites 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]

5. Final simplification 99.7%

   \[\leadsto \left(1 - {\left(x \cdot 9\right)}^{-1}\right) - \frac{{x}^{-0.5} \cdot y}{3} \]
6. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 - pow((x * 9.0), -1.0)) - (y / (3.0 * sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - ((x * 9.0d0) ** (-1.0d0))) - (y / (3.0d0 * sqrt(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.8e+32)
   (/ (- x (fma (* (sqrt x) y) 0.3333333333333333 0.1111111111111111)) x)
   (- 1.0 (/ (* y 0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.8e+32) {
		tmp = (x - fma((sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x;
	} else {
		tmp = 1.0 - ((y * 0.3333333333333333) / sqrt(x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.8e+32)
		tmp = Float64(Float64(x - fma(Float64(sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x);
	else
		tmp = Float64(1.0 - Float64(Float64(y * 0.3333333333333333) / sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.8e+32], N[(N[(x - N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 0.3333333333333333 + 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(N[(y * 0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999983e32

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 4.79999999999999983e32 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 99.8

         \[\leadsto 1 - \frac{\color{blue}{y \cdot 0.3333333333333333}}{\sqrt{x}} \]

   7. Applied rewrites 99.8%

      \[\leadsto 1 - \color{blue}{\frac{y \cdot 0.3333333333333333}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow1/2 N/A

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

   11. pow-flip N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval 99.7

       \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{{x}^{\color{blue}{-0.5}} \cdot y}{3} \]

4. Applied rewrites 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-flip N/A

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

   6. pow1/2 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. associate-/r/ N/A

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

   9. clear-num N/A

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

   10. associate-/l/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 99.6

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

   13. lower--.f64 N/A

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

   14. sub-neg N/A

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

   15. +-commutative N/A

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

6. Applied rewrites 99.6%

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

Alternative 5: 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 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Add Preprocessing
3. 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-neg 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. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. distribute-lft-neg-in N/A

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

   8. distribute-frac-neg2 N/A

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

   9. lower-fma.f64 N/A

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

   10. distribute-frac-neg2 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. distribute-neg-frac N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.6

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

   17. lift-/.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. associate-/r* 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 N/A

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

   22. metadata-eval N/A

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

   23. lower-/.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 6: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x} \cdot -0.3333333333333333, y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.2e-8)
   (/ (fma (* (sqrt x) -0.3333333333333333) y -0.1111111111111111) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.2e-8) {
		tmp = fma((sqrt(x) * -0.3333333333333333), y, -0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.2e-8)
		tmp = Float64(fma(Float64(sqrt(x) * -0.3333333333333333), y, -0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.2e-8], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * y + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002e-8

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval 99.5

          \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{{x}^{\color{blue}{-0.5}} \cdot y}{3} \]

   4. Applied rewrites 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]

   5. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}}{x} \]

      5. associate--r+ N/A

         \[\leadsto \frac{\color{blue}{\left(0 - \frac{1}{9}\right) - \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)}}{x} \]

      6. metadata-eval N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 98.8

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

   10. Applied rewrites 98.8%

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

   if 3.2000000000000002e-8 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.2%

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

Alternative 7: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.2e-8)
   (/ (fma -0.3333333333333333 (* (sqrt x) y) -0.1111111111111111) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.2e-8) {
		tmp = fma(-0.3333333333333333, (sqrt(x) * y), -0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.2e-8)
		tmp = Float64(fma(-0.3333333333333333, Float64(sqrt(x) * y), -0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.2e-8], N[(N[(-0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002e-8

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 3.2000000000000002e-8 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.2%

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

Alternative 8: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-122}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e-122)
   (/ -0.1111111111111111 x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e-122) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.15d-122) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.15e-122) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.15e-122:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e-122)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.15e-122)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e-122], N[(-0.1111111111111111 / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15000000000000003e-122

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval 99.5

          \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{{x}^{\color{blue}{-0.5}} \cdot y}{3} \]

   4. Applied rewrites 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]

   5. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 74.7%

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

   if 1.15000000000000003e-122 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 88.1%

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

Alternative 9: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-122}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e-122)
   (/ -0.1111111111111111 x)
   (fma (/ y (sqrt x)) -0.3333333333333333 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e-122) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e-122)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e-122], N[(-0.1111111111111111 / x), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15000000000000003e-122

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval 99.5

          \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{{x}^{\color{blue}{-0.5}} \cdot y}{3} \]

   4. Applied rewrites 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]

   5. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 74.7%

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

   if 1.15000000000000003e-122 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 88.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval 88.1

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

   7. Applied rewrites 88.1%

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

Alternative 10: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-122}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e-122)
   (/ -0.1111111111111111 x)
   (fma y (/ -0.3333333333333333 (sqrt x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e-122) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = fma(y, (-0.3333333333333333 / sqrt(x)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e-122)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = fma(y, Float64(-0.3333333333333333 / sqrt(x)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e-122], N[(-0.1111111111111111 / x), $MachinePrecision], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15000000000000003e-122

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval 99.5

          \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{{x}^{\color{blue}{-0.5}} \cdot y}{3} \]

   4. Applied rewrites 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]

   5. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 74.7%

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

   if 1.15000000000000003e-122 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 88.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 88.0%

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

Alternative 11: 30.7% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.1111111111111111d0) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow1/2 N/A

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

   11. pow-flip N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval 99.7

       \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{{x}^{\color{blue}{-0.5}} \cdot y}{3} \]

4. Applied rewrites 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]

5. Taylor expanded in x around 0

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

   1. neg-mul-1 N/A

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

   2. distribute-neg-frac N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-neg-in N/A

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

   6. metadata-eval N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 62.1

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

7. Applied rewrites 62.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 29.9%

    \[\leadsto \frac{-0.1111111111111111}{x} \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

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

real(8) function code(x, y)
    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(Float64(1.0 / 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[(N[(1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024324 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x)))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))