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

Percentage Accurate: 99.7% → 99.7%

Time: 9.3s

Alternatives: 14

Speedup: 1.0×

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, 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]

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+96}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+65}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.75e+96)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 7.6e+65)
     (+ 1.0 (/ 1.0 (* x -9.0)))
     (+ 1.0 (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.75e+96) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 7.6e+65) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = 1.0 + (sqrt((1.0 / x)) * (y * -0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.75d+96)) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else if (y <= 7.6d+65) then
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.75e+96) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else if (y <= 7.6e+65) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 / x)) * (y * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.75e+96:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	elif y <= 7.6e+65:
		tmp = 1.0 + (1.0 / (x * -9.0))
	else:
		tmp = 1.0 + (math.sqrt((1.0 / x)) * (y * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.75e+96)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 7.6e+65)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.75e+96)
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	elseif (y <= 7.6e+65)
		tmp = 1.0 + (1.0 / (x * -9.0));
	else
		tmp = 1.0 + (sqrt((1.0 / x)) * (y * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.75e+96], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.6e+65], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7499999999999999e96

   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 inf 99.4%

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

      1. metadata-eval 99.4%

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

      2. *-commutative 99.4%

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

      3. sqrt-div 99.3%

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

      4. metadata-eval 99.3%

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

      5. un-div-inv 99.4%

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

      6. times-frac 99.6%

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

      7. *-un-lft-identity 99.6%

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

   5. Applied egg-rr 99.6%

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

   if -1.7499999999999999e96 < y < 7.60000000000000022e65

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.9%

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

      1. add-cube-cbrt 97.2%

         \[\leadsto 1 + \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}} \]

      2. pow3 97.2%

         \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]

   7. Applied egg-rr 97.2%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 97.9%

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

      2. clear-num 97.9%

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

      3. div-inv 97.9%

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

      4. metadata-eval 97.9%

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

   9. Applied egg-rr 97.9%

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

   if 7.60000000000000022e65 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.3%

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

      10. fma-neg 99.3%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 96.5%

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

      1. associate-*r* 96.5%

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

      2. *-commutative 96.5%

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

      3. associate-*l* 96.5%

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

   7. Simplified 96.5%

      \[\leadsto 1 + \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-0.3333333333333333 \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+96}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+65}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+96}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+66}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.75e+96)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 2.4e+66)
     (+ 1.0 (/ 1.0 (* x -9.0)))
     (- 1.0 (* 0.3333333333333333 (* y (sqrt (/ 1.0 x))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.75e+96) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 2.4e+66) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = 1.0 - (0.3333333333333333 * (y * sqrt((1.0 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.75d+96)) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else if (y <= 2.4d+66) then
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    else
        tmp = 1.0d0 - (0.3333333333333333d0 * (y * sqrt((1.0d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.75e+96) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else if (y <= 2.4e+66) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = 1.0 - (0.3333333333333333 * (y * Math.sqrt((1.0 / x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.75e+96:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	elif y <= 2.4e+66:
		tmp = 1.0 + (1.0 / (x * -9.0))
	else:
		tmp = 1.0 - (0.3333333333333333 * (y * math.sqrt((1.0 / x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.75e+96)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 2.4e+66)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = Float64(1.0 - Float64(0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.75e+96)
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	elseif (y <= 2.4e+66)
		tmp = 1.0 + (1.0 / (x * -9.0));
	else
		tmp = 1.0 - (0.3333333333333333 * (y * sqrt((1.0 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.75e+96], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+66], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7499999999999999e96

   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 inf 99.4%

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

      1. metadata-eval 99.4%

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

      2. *-commutative 99.4%

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

      3. sqrt-div 99.3%

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

      4. metadata-eval 99.3%

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

      5. un-div-inv 99.4%

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

      6. times-frac 99.6%

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

      7. *-un-lft-identity 99.6%

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

   5. Applied egg-rr 99.6%

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

   if -1.7499999999999999e96 < y < 2.4000000000000002e66

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.9%

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

      1. add-cube-cbrt 97.2%

         \[\leadsto 1 + \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}} \]

      2. pow3 97.2%

         \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]

   7. Applied egg-rr 97.2%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 97.9%

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

      2. clear-num 97.9%

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

      3. div-inv 97.9%

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

      4. metadata-eval 97.9%

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

   9. Applied egg-rr 97.9%

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

   if 2.4000000000000002e66 < y

   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 inf 96.5%

      \[\leadsto \color{blue}{1 - 0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+96}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+66}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+96} \lor \neg \left(y \leq 1.22 \cdot 10^{+67}\right):\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.75e+96) (not (<= y 1.22e+67)))
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (+ 1.0 (/ 1.0 (* x -9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.75e+96) || !(y <= 1.22e+67)) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else {
		tmp = 1.0 + (1.0 / (x * -9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.75d+96)) .or. (.not. (y <= 1.22d+67))) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.75e+96) || !(y <= 1.22e+67)) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else {
		tmp = 1.0 + (1.0 / (x * -9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.75e+96) or not (y <= 1.22e+67):
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	else:
		tmp = 1.0 + (1.0 / (x * -9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.75e+96) || !(y <= 1.22e+67))
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.75e+96) || ~((y <= 1.22e+67)))
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	else
		tmp = 1.0 + (1.0 / (x * -9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.75e+96], N[Not[LessEqual[y, 1.22e+67]], $MachinePrecision]], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7499999999999999e96 or 1.22000000000000004e67 < y

   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 inf 97.9%

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

      1. metadata-eval 97.9%

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

      2. *-commutative 97.9%

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

      3. sqrt-div 97.8%

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

      4. metadata-eval 97.8%

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

      5. un-div-inv 97.8%

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

      6. times-frac 98.0%

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

      7. *-un-lft-identity 98.0%

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

   5. Applied egg-rr 98.0%

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

   if -1.7499999999999999e96 < y < 1.22000000000000004e67

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.9%

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

      1. add-cube-cbrt 97.2%

         \[\leadsto 1 + \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}} \]

      2. pow3 97.2%

         \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]

   7. Applied egg-rr 97.2%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 97.9%

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

      2. clear-num 97.9%

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

      3. div-inv 97.9%

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

      4. metadata-eval 97.9%

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

   9. Applied egg-rr 97.9%

      \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot -9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+96} \lor \neg \left(y \leq 1.22 \cdot 10^{+67}\right):\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+69}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e+96)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 2.3e+69)
     (+ 1.0 (/ 1.0 (* x -9.0)))
     (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+96) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 2.3e+69) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.7d+96)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 2.3d+69) then
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    else
        tmp = sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+96) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 2.3e+69) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = Math.sqrt((1.0 / x)) * (y * -0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.7e+96:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 2.3e+69:
		tmp = 1.0 + (1.0 / (x * -9.0))
	else:
		tmp = math.sqrt((1.0 / x)) * (y * -0.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.7e+96)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 2.3e+69)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.7e+96)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 2.3e+69)
		tmp = 1.0 + (1.0 / (x * -9.0));
	else
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.7e+96], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e+69], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.70000000000000022e96

   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 99.6%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   4. Taylor expanded in y around inf 97.4%

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

      1. *-commutative 97.4%

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

      2. *-commutative 97.4%

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

   6. Simplified 97.4%

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

      1. pow1 97.4%

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

      2. associate-*l* 97.4%

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

      3. sqrt-div 97.4%

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

      4. metadata-eval 97.4%

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

      5. associate-*l/ 97.5%

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

      6. metadata-eval 97.5%

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

   8. Applied egg-rr 97.5%

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

      1. unpow1 97.5%

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

   10. Simplified 97.5%

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

       1. clear-num 97.6%

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

       2. un-div-inv 97.6%

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

       3. div-inv 97.6%

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

       4. metadata-eval 97.6%

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

   12. Applied egg-rr 97.6%

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

   if -2.70000000000000022e96 < y < 2.30000000000000017e69

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.9%

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

      1. add-cube-cbrt 97.2%

         \[\leadsto 1 + \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}} \]

      2. pow3 97.2%

         \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]

   7. Applied egg-rr 97.2%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 97.9%

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

      2. clear-num 97.9%

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

      3. div-inv 97.9%

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

      4. metadata-eval 97.9%

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

   9. Applied egg-rr 97.9%

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

   if 2.30000000000000017e69 < y

   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 99.5%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   4. Taylor expanded in y around inf 94.6%

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

      1. *-commutative 94.6%

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

      2. *-commutative 94.6%

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

   6. Simplified 94.6%

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

   7. Taylor expanded in y around 0 94.6%

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

      1. *-commutative 94.6%

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

      2. associate-*l* 94.6%

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

      3. *-commutative 94.6%

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

   9. Simplified 94.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-0.3333333333333333 \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+69}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+96} \lor \neg \left(y \leq 8.8 \cdot 10^{+72}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4e+96) (not (<= y 8.8e+72)))
   (* y (/ -0.3333333333333333 (sqrt x)))
   (+ 1.0 (/ 1.0 (* x -9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4e+96) || !(y <= 8.8e+72)) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else {
		tmp = 1.0 + (1.0 / (x * -9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4d+96)) .or. (.not. (y <= 8.8d+72))) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4e+96) || !(y <= 8.8e+72)) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else {
		tmp = 1.0 + (1.0 / (x * -9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4e+96) or not (y <= 8.8e+72):
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	else:
		tmp = 1.0 + (1.0 / (x * -9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4e+96) || !(y <= 8.8e+72))
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4e+96) || ~((y <= 8.8e+72)))
		tmp = y * (-0.3333333333333333 / sqrt(x));
	else
		tmp = 1.0 + (1.0 / (x * -9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4e+96], N[Not[LessEqual[y, 8.8e+72]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000002e96 or 8.8e72 < y

   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 99.6%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   4. Taylor expanded in y around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. *-commutative 95.9%

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

   6. Simplified 95.9%

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

      1. pow1 95.9%

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

      2. associate-*l* 95.9%

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

      3. sqrt-div 95.8%

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

      4. metadata-eval 95.8%

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

      5. associate-*l/ 95.8%

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

      6. metadata-eval 95.8%

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

   8. Applied egg-rr 95.8%

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

      1. unpow1 95.8%

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

   10. Simplified 95.8%

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

   if -4.0000000000000002e96 < y < 8.8e72

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.9%

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

      1. add-cube-cbrt 97.2%

         \[\leadsto 1 + \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}} \]

      2. pow3 97.2%

         \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]

   7. Applied egg-rr 97.2%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 97.9%

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

      2. clear-num 97.9%

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

      3. div-inv 97.9%

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

      4. metadata-eval 97.9%

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

   9. Applied egg-rr 97.9%

      \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot -9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+96} \lor \neg \left(y \leq 8.8 \cdot 10^{+72}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+96} \lor \neg \left(y \leq 3.05 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.75e+96) (not (<= y 3.05e+68)))
   (/ y (* (sqrt x) -3.0))
   (+ 1.0 (/ 1.0 (* x -9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.75e+96) || !(y <= 3.05e+68)) {
		tmp = y / (sqrt(x) * -3.0);
	} else {
		tmp = 1.0 + (1.0 / (x * -9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.75d+96)) .or. (.not. (y <= 3.05d+68))) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.75e+96) || !(y <= 3.05e+68)) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else {
		tmp = 1.0 + (1.0 / (x * -9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.75e+96) or not (y <= 3.05e+68):
		tmp = y / (math.sqrt(x) * -3.0)
	else:
		tmp = 1.0 + (1.0 / (x * -9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.75e+96) || !(y <= 3.05e+68))
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.75e+96) || ~((y <= 3.05e+68)))
		tmp = y / (sqrt(x) * -3.0);
	else
		tmp = 1.0 + (1.0 / (x * -9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.75e+96], N[Not[LessEqual[y, 3.05e+68]], $MachinePrecision]], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7499999999999999e96 or 3.05e68 < y

   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 99.6%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   4. Taylor expanded in y around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. *-commutative 95.9%

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

   6. Simplified 95.9%

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

      1. pow1 95.9%

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

      2. associate-*l* 95.9%

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

      3. sqrt-div 95.8%

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

      4. metadata-eval 95.8%

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

      5. associate-*l/ 95.8%

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

      6. metadata-eval 95.8%

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

   8. Applied egg-rr 95.8%

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

      1. unpow1 95.8%

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

   10. Simplified 95.8%

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

       1. clear-num 95.8%

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

       2. un-div-inv 95.8%

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

       3. div-inv 95.9%

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

       4. metadata-eval 95.9%

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

   12. Applied egg-rr 95.9%

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

   if -1.7499999999999999e96 < y < 3.05e68

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.9%

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

      1. add-cube-cbrt 97.2%

         \[\leadsto 1 + \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}} \]

      2. pow3 97.2%

         \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]

   7. Applied egg-rr 97.2%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 97.9%

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

      2. clear-num 97.9%

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

      3. div-inv 97.9%

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

      4. metadata-eval 97.9%

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

   9. Applied egg-rr 97.9%

      \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot -9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+96} \lor \neg \left(y \leq 3.05 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{0.1111111111111111 + 0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (+ 0.1111111111111111 (* 0.3333333333333333 (* y (sqrt x)))) (- x))
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * sqrt(x)))) / -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 <= 0.11d0) then
        tmp = (0.1111111111111111d0 + (0.3333333333333333d0 * (y * sqrt(x)))) / -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 <= 0.11) {
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * Math.sqrt(x)))) / -x;
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * math.sqrt(x)))) / -x
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(Float64(0.1111111111111111 + Float64(0.3333333333333333 * Float64(y * sqrt(x)))) / Float64(-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 <= 0.11)
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * sqrt(x)))) / -x;
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(0.1111111111111111 + N[(0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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 97.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{0.1111111111111111 + 0.3333333333333333 \cdot \left(\sqrt{x} \cdot y\right)}{x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 97.6%

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

      2. *-commutative 97.6%

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

   5. Simplified 97.6%

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

   if 0.110000000000000001 < 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 98.4%

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

      1. metadata-eval 98.4%

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

      2. *-commutative 98.4%

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

      3. sqrt-div 98.4%

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

      4. metadata-eval 98.4%

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

      5. un-div-inv 98.4%

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

      6. times-frac 98.5%

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

      7. *-un-lft-identity 98.5%

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

   5. Applied egg-rr 98.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{0.1111111111111111 + 0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) + ((-0.3333333333333333d0) * (y / sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / math.sqrt(x)))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / sqrt(x)));
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right) \]

   5. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   6. neg-mul-1 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

   7. times-frac 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \]
6. Add Preprocessing

Alternative 10: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 0.1111111111111111 x)) (/ y (* 3.0 (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (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 - (0.1111111111111111d0 / x)) - (y / (3.0d0 * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / (3.0 * Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) - (y / (3.0 * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) - Float64(y / Float64(3.0 * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) - (y / (3.0 * sqrt(x)));
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0 99.7%

   \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

4. Final simplification 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
5. Add Preprocessing

Alternative 11: 62.7% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{1}{x \cdot -9} \end{array} \]

FPCore

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

C

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

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)))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (1.0 / (x * -9.0));
}

Python

def code(x, y):
	return 1.0 + (1.0 / (x * -9.0))

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 + N[(1.0 / N[(x * -9.0), $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. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.6%

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

   10. fma-neg 99.6%

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

   11. associate-/r* 99.7%

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

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 64.1%

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

   1. add-cube-cbrt 63.6%

      \[\leadsto 1 + \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}} \]

   2. pow3 63.6%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]

7. Applied egg-rr 63.6%

   \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \]
8. Step-by-step derivation

   1. rem-cube-cbrt 64.1%

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

   2. clear-num 64.1%

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

   3. div-inv 64.1%

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

   4. metadata-eval 64.1%

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

9. Applied egg-rr 64.1%

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

10. Final simplification 64.1%

    \[\leadsto 1 + \frac{1}{x \cdot -9} \]
11. Add Preprocessing

Alternative 12: 62.6% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-0.1111111111111111d0) / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.6%

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

   10. fma-neg 99.6%

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

   11. associate-/r* 99.7%

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

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 64.1%

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

6. Final simplification 64.1%

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

Alternative 13: 31.1% accurate, 28.3× 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 / Float64(-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 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right) \]

   5. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   6. neg-mul-1 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

   7. times-frac 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. associate-+r- 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l/ 99.6%

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

   5. associate-*r/ 99.6%

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

   6. add-sqr-sqrt 99.5%

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

   7. sqrt-unprod 78.3%

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

   8. frac-times 78.4%

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

   9. metadata-eval 78.4%

      \[\leadsto \left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}} + 1\right) - \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

   10. metadata-eval 78.4%

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

   11. frac-times 78.3%

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

   12. sqrt-unprod 0.0%

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

   13. add-sqr-sqrt 66.3%

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

6. Applied egg-rr 66.3%

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

7. Taylor expanded in x around 0 31.3%

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

8. Taylor expanded in y around 0 3.0%

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

   1. metadata-eval 3.0%

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

   2. distribute-neg-frac 3.0%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 20.4%

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

   5. frac-times 20.5%

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

   6. metadata-eval 20.5%

      \[\leadsto -\sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

   7. metadata-eval 20.5%

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

   8. frac-times 20.4%

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

   9. sqrt-unprod 32.7%

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

   10. add-sqr-sqrt 32.8%

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

10. Applied egg-rr 32.8%

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

11. Final simplification 32.8%

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

Alternative 14: 3.0% accurate, 37.7× 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 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right) \]

   5. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   6. neg-mul-1 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

   7. times-frac 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. associate-+r- 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l/ 99.6%

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

   5. associate-*r/ 99.6%

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

   6. add-sqr-sqrt 99.5%

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

   7. sqrt-unprod 78.3%

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

   8. frac-times 78.4%

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

   9. metadata-eval 78.4%

      \[\leadsto \left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}} + 1\right) - \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

   10. metadata-eval 78.4%

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

   11. frac-times 78.3%

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

   12. sqrt-unprod 0.0%

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

   13. add-sqr-sqrt 66.3%

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

6. Applied egg-rr 66.3%

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

7. Taylor expanded in x around 0 31.3%

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

8. Taylor expanded in y around 0 3.0%

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

9. Final simplification 3.0%

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

Developer target: 99.7% accurate, 1.0× 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 2024078 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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