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

Percentage Accurate: 99.7% → 99.7%

Time: 11.7s

Alternatives: 19

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $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. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.8%

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

   4. pow1/2 99.8%

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

4. Applied egg-rr 99.8%

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

   1. unpow1/2 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+64}:\\ \;\;\;\;1 - y \cdot \frac{{x}^{-0.5}}{3}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+85}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{3 \cdot \frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.22e+64)
   (- 1.0 (* y (/ (pow x -0.5) 3.0)))
   (if (<= y 3.4e+85)
     (- 1.0 (pow (* x 9.0) -1.0))
     (+ 1.0 (/ -1.0 (* 3.0 (/ (sqrt x) y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.22e+64) {
		tmp = 1.0 - (y * (pow(x, -0.5) / 3.0));
	} else if (y <= 3.4e+85) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (-1.0 / (3.0 * (sqrt(x) / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.22d+64)) then
        tmp = 1.0d0 - (y * ((x ** (-0.5d0)) / 3.0d0))
    else if (y <= 3.4d+85) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 + ((-1.0d0) / (3.0d0 * (sqrt(x) / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.22e+64) {
		tmp = 1.0 - (y * (Math.pow(x, -0.5) / 3.0));
	} else if (y <= 3.4e+85) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (-1.0 / (3.0 * (Math.sqrt(x) / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.22e+64:
		tmp = 1.0 - (y * (math.pow(x, -0.5) / 3.0))
	elif y <= 3.4e+85:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 + (-1.0 / (3.0 * (math.sqrt(x) / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.22e+64)
		tmp = Float64(1.0 - Float64(y * Float64((x ^ -0.5) / 3.0)));
	elseif (y <= 3.4e+85)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(3.0 * Float64(sqrt(x) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.22e+64)
		tmp = 1.0 - (y * ((x ^ -0.5) / 3.0));
	elseif (y <= 3.4e+85)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 + (-1.0 / (3.0 * (sqrt(x) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.22e+64], N[(1.0 - N[(y * N[(N[Power[x, -0.5], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+85], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.21999999999999994e64

   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 95.6%

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

      1. metadata-eval 95.6%

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

      2. *-commutative 95.6%

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

      3. sqrt-div 95.6%

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

      4. metadata-eval 95.6%

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

      5. un-div-inv 95.6%

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

      6. times-frac 95.9%

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

      7. *-un-lft-identity 95.9%

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

      8. clear-num 95.7%

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

      9. *-un-lft-identity 95.7%

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

      10. times-frac 95.5%

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

      11. metadata-eval 95.5%

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

   5. Applied egg-rr 95.5%

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

   6. Taylor expanded in x around 0 95.4%

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

      1. associate-*r* 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*r/ 95.7%

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

      4. *-rgt-identity 95.7%

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

      5. associate-/l* 95.6%

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

   8. Simplified 95.6%

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

      1. associate-/r* 95.7%

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

      2. associate-/r/ 95.7%

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

      3. pow1/2 95.7%

         \[\leadsto 1 - \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{3} \cdot y \]

      4. pow-flip 95.9%

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

      5. metadata-eval 95.9%

         \[\leadsto 1 - \frac{{x}^{\color{blue}{-0.5}}}{3} \cdot y \]

   10. Applied egg-rr 95.9%

       \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{3} \cdot y} \]

   if -1.21999999999999994e64 < y < 3.4000000000000003e85

   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 96.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 49.7%

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

      3. frac-times 49.7%

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

      4. metadata-eval 49.7%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 49.7%

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

      6. frac-times 49.7%

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

      7. sqrt-unprod 49.8%

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

      8. add-sqr-sqrt 49.8%

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

      9. frac-2neg 49.8%

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

      10. metadata-eval 49.8%

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

      11. distribute-frac-neg2 49.8%

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

   7. Applied egg-rr 49.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 74.5%

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

      3. frac-times 74.5%

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

      4. metadata-eval 74.5%

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

      5. metadata-eval 74.5%

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

      6. frac-times 74.5%

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

      7. sqrt-unprod 96.4%

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

      8. add-sqr-sqrt 96.5%

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

      9. clear-num 96.5%

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

      10. div-inv 96.6%

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

      11. metadata-eval 96.6%

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

      12. inv-pow 96.6%

          \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   9. Applied egg-rr 96.6%

      \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   if 3.4000000000000003e85 < 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 95.8%

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

      1. metadata-eval 95.8%

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

      2. *-commutative 95.8%

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

      3. sqrt-div 95.7%

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

      4. metadata-eval 95.7%

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

      5. un-div-inv 95.8%

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

      6. times-frac 95.7%

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

      7. *-un-lft-identity 95.7%

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

      8. clear-num 95.6%

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

      9. *-un-lft-identity 95.6%

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

      10. times-frac 95.9%

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

      11. metadata-eval 95.9%

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

   5. Applied egg-rr 95.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+64}:\\ \;\;\;\;1 - y \cdot \frac{{x}^{-0.5}}{3}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+85}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{3 \cdot \frac{\sqrt{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+64} \lor \neg \left(y \leq 3.4 \cdot 10^{+85}\right):\\ \;\;\;\;1 - y \cdot \frac{{x}^{-0.5}}{3}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.3e+64) (not (<= y 3.4e+85)))
   (- 1.0 (* y (/ (pow x -0.5) 3.0)))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.3e+64) || !(y <= 3.4e+85)) {
		tmp = 1.0 - (y * (pow(x, -0.5) / 3.0));
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.3d+64)) .or. (.not. (y <= 3.4d+85))) then
        tmp = 1.0d0 - (y * ((x ** (-0.5d0)) / 3.0d0))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.3e+64) || !(y <= 3.4e+85)) {
		tmp = 1.0 - (y * (Math.pow(x, -0.5) / 3.0));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.3e+64) or not (y <= 3.4e+85):
		tmp = 1.0 - (y * (math.pow(x, -0.5) / 3.0))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.3e+64) || !(y <= 3.4e+85))
		tmp = Float64(1.0 - Float64(y * Float64((x ^ -0.5) / 3.0)));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.3e+64) || ~((y <= 3.4e+85)))
		tmp = 1.0 - (y * ((x ^ -0.5) / 3.0));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.3e+64], N[Not[LessEqual[y, 3.4e+85]], $MachinePrecision]], N[(1.0 - N[(y * N[(N[Power[x, -0.5], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e64 or 3.4000000000000003e85 < 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 95.7%

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

      1. metadata-eval 95.7%

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

      2. *-commutative 95.7%

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

      3. sqrt-div 95.6%

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

      4. metadata-eval 95.6%

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

      5. un-div-inv 95.7%

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

      6. times-frac 95.8%

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

      7. *-un-lft-identity 95.8%

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

      8. clear-num 95.7%

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

      9. *-un-lft-identity 95.7%

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

      10. times-frac 95.7%

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

      11. metadata-eval 95.7%

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

   5. Applied egg-rr 95.7%

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

   6. Taylor expanded in x around 0 95.5%

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

      1. associate-*r* 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*r/ 95.7%

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

      4. *-rgt-identity 95.7%

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

      5. associate-/l* 95.7%

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

   8. Simplified 95.7%

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

      1. associate-/r* 95.7%

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

      2. associate-/r/ 95.7%

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

      3. pow1/2 95.7%

         \[\leadsto 1 - \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{3} \cdot y \]

      4. pow-flip 95.9%

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

      5. metadata-eval 95.9%

         \[\leadsto 1 - \frac{{x}^{\color{blue}{-0.5}}}{3} \cdot y \]

   10. Applied egg-rr 95.9%

       \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{3} \cdot y} \]

   if -2.3e64 < y < 3.4000000000000003e85

   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 96.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 49.7%

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

      3. frac-times 49.7%

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

      4. metadata-eval 49.7%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 49.7%

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

      6. frac-times 49.7%

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

      7. sqrt-unprod 49.8%

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

      8. add-sqr-sqrt 49.8%

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

      9. frac-2neg 49.8%

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

      10. metadata-eval 49.8%

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

      11. distribute-frac-neg2 49.8%

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

   7. Applied egg-rr 49.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 74.5%

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

      3. frac-times 74.5%

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

      4. metadata-eval 74.5%

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

      5. metadata-eval 74.5%

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

      6. frac-times 74.5%

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

      7. sqrt-unprod 96.4%

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

      8. add-sqr-sqrt 96.5%

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

      9. clear-num 96.5%

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

      10. div-inv 96.6%

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

      11. metadata-eval 96.6%

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

      12. inv-pow 96.6%

          \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   9. Applied egg-rr 96.6%

      \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+64} \lor \neg \left(y \leq 3.4 \cdot 10^{+85}\right):\\ \;\;\;\;1 - y \cdot \frac{{x}^{-0.5}}{3}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+63} \lor \neg \left(y \leq 3.4 \cdot 10^{+85}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.2e+63) (not (<= y 3.4e+85)))
   (- 1.0 (* (/ y (sqrt x)) 0.3333333333333333))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.2e+63) || !(y <= 3.4e+85)) {
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.2d+63)) .or. (.not. (y <= 3.4d+85))) then
        tmp = 1.0d0 - ((y / sqrt(x)) * 0.3333333333333333d0)
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.2e+63) || !(y <= 3.4e+85)) {
		tmp = 1.0 - ((y / Math.sqrt(x)) * 0.3333333333333333);
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.2e+63) or not (y <= 3.4e+85):
		tmp = 1.0 - ((y / math.sqrt(x)) * 0.3333333333333333)
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.2e+63) || !(y <= 3.4e+85))
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) * 0.3333333333333333));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.2e+63) || ~((y <= 3.4e+85)))
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.2e+63], N[Not[LessEqual[y, 3.4e+85]], $MachinePrecision]], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.19999999999999998e63 or 3.4000000000000003e85 < 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 95.7%

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

      1. *-commutative 95.7%

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

      2. sqrt-div 95.6%

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

      3. metadata-eval 95.6%

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

      4. un-div-inv 95.7%

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

   5. Applied egg-rr 95.7%

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

   if -7.19999999999999998e63 < y < 3.4000000000000003e85

   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 96.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 49.7%

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

      3. frac-times 49.7%

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

      4. metadata-eval 49.7%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 49.7%

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

      6. frac-times 49.7%

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

      7. sqrt-unprod 49.8%

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

      8. add-sqr-sqrt 49.8%

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

      9. frac-2neg 49.8%

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

      10. metadata-eval 49.8%

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

      11. distribute-frac-neg2 49.8%

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

   7. Applied egg-rr 49.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 74.5%

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

      3. frac-times 74.5%

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

      4. metadata-eval 74.5%

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

      5. metadata-eval 74.5%

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

      6. frac-times 74.5%

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

      7. sqrt-unprod 96.4%

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

      8. add-sqr-sqrt 96.5%

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

      9. clear-num 96.5%

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

      10. div-inv 96.6%

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

      11. metadata-eval 96.6%

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

      12. inv-pow 96.6%

          \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   9. Applied egg-rr 96.6%

      \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+63} \lor \neg \left(y \leq 3.4 \cdot 10^{+85}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+64} \lor \neg \left(y \leq 6 \cdot 10^{+85}\right):\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.32e+64) (not (<= y 6e+85)))
   (- 1.0 (/ 0.3333333333333333 (/ (sqrt x) y)))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.32e+64) || !(y <= 6e+85)) {
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.32d+64)) .or. (.not. (y <= 6d+85))) then
        tmp = 1.0d0 - (0.3333333333333333d0 / (sqrt(x) / y))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.32e+64) || !(y <= 6e+85)) {
		tmp = 1.0 - (0.3333333333333333 / (Math.sqrt(x) / y));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.32e+64) or not (y <= 6e+85):
		tmp = 1.0 - (0.3333333333333333 / (math.sqrt(x) / y))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.32e+64) || !(y <= 6e+85))
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(sqrt(x) / y)));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.32e+64) || ~((y <= 6e+85)))
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.32e+64], N[Not[LessEqual[y, 6e+85]], $MachinePrecision]], N[(1.0 - N[(0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3200000000000001e64 or 6.0000000000000001e85 < 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 95.7%

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

      1. *-commutative 95.7%

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

      2. sqrt-div 95.6%

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

      3. metadata-eval 95.6%

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

      4. un-div-inv 95.7%

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

   5. Applied egg-rr 95.7%

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

      1. clear-num 95.6%

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

      2. div-inv 95.7%

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

   7. Applied egg-rr 95.7%

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

   if -1.3200000000000001e64 < y < 6.0000000000000001e85

   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 96.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 49.7%

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

      3. frac-times 49.7%

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

      4. metadata-eval 49.7%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 49.7%

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

      6. frac-times 49.7%

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

      7. sqrt-unprod 49.8%

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

      8. add-sqr-sqrt 49.8%

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

      9. frac-2neg 49.8%

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

      10. metadata-eval 49.8%

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

      11. distribute-frac-neg2 49.8%

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

   7. Applied egg-rr 49.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 74.5%

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

      3. frac-times 74.5%

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

      4. metadata-eval 74.5%

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

      5. metadata-eval 74.5%

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

      6. frac-times 74.5%

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

      7. sqrt-unprod 96.4%

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

      8. add-sqr-sqrt 96.5%

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

      9. clear-num 96.5%

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

      10. div-inv 96.6%

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

      11. metadata-eval 96.6%

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

      12. inv-pow 96.6%

          \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   9. Applied egg-rr 96.6%

      \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+64} \lor \neg \left(y \leq 6 \cdot 10^{+85}\right):\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+91}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.8e+69)
   (/ (/ y (sqrt x)) -3.0)
   (if (<= y 5.2e+91)
     (+ 1.0 (/ -0.1111111111111111 x))
     (* (* y -0.3333333333333333) (sqrt (/ 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.8e+69) {
		tmp = (y / sqrt(x)) / -3.0;
	} else if (y <= 5.2e+91) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (y * -0.3333333333333333) * 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 <= (-8.8d+69)) then
        tmp = (y / sqrt(x)) / (-3.0d0)
    else if (y <= 5.2d+91) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (y * (-0.3333333333333333d0)) * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.8e+69) {
		tmp = (y / Math.sqrt(x)) / -3.0;
	} else if (y <= 5.2e+91) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (y * -0.3333333333333333) * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.8e+69:
		tmp = (y / math.sqrt(x)) / -3.0
	elif y <= 5.2e+91:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (y * -0.3333333333333333) * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.8e+69)
		tmp = Float64(Float64(y / sqrt(x)) / -3.0);
	elseif (y <= 5.2e+91)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.8e+69)
		tmp = (y / sqrt(x)) / -3.0;
	elseif (y <= 5.2e+91)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (y * -0.3333333333333333) * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.8e+69], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[y, 5.2e+91], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.8000000000000006e69

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 91.9%

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

      1. metadata-eval 91.9%

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

      2. distribute-lft-neg-in 91.9%

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

      3. *-commutative 91.9%

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

      4. distribute-lft-neg-in 91.9%

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

      5. unpow1/2 91.9%

         \[\leadsto \left(-\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      6. rem-exp-log 87.6%

         \[\leadsto \left(-{\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      7. exp-neg 87.6%

         \[\leadsto \left(-{\color{blue}{\left(e^{-\log x}\right)}}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      8. exp-prod 87.6%

         \[\leadsto \left(-\color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      9. distribute-lft-neg-out 87.6%

         \[\leadsto \left(-e^{\color{blue}{-\log x \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      10. exp-neg 87.6%

          \[\leadsto \left(-\color{blue}{\frac{1}{e^{\log x \cdot 0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      11. exp-to-pow 91.9%

          \[\leadsto \left(-\frac{1}{\color{blue}{{x}^{0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      12. unpow1/2 91.9%

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

      13. unpow-1 91.9%

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

      14. distribute-lft-neg-in 91.9%

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

      15. associate-*r* 91.9%

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

      16. unpow-1 91.9%

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

      17. associate-*l/ 92.1%

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

      18. associate-*r/ 92.1%

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

      19. associate-*r/ 92.0%

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

   9. Simplified 92.1%

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

       1. div-inv 91.9%

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

       2. pow1/2 91.9%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot \frac{1}{\color{blue}{{x}^{0.5}}} \]

       3. pow-flip 92.1%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot \color{blue}{{x}^{\left(-0.5\right)}} \]

       4. metadata-eval 92.1%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot {x}^{\color{blue}{-0.5}} \]

       5. associate-*l* 92.1%

          \[\leadsto \color{blue}{y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \]

       6. *-commutative 92.1%

          \[\leadsto y \cdot \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right)} \]

       7. *-commutative 92.1%

          \[\leadsto \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right) \cdot y} \]

       8. *-commutative 92.1%

          \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \cdot y \]

       9. metadata-eval 92.1%

          \[\leadsto \left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot y \]

       10. pow-flip 92.0%

           \[\leadsto \left(-0.3333333333333333 \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot y \]

       11. pow1/2 92.0%

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

       12. div-inv 92.0%

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

   11. Applied egg-rr 92.0%

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

       1. *-commutative 92.0%

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

       2. clear-num 91.9%

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

       3. un-div-inv 91.9%

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

       4. div-inv 92.1%

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

       5. metadata-eval 92.1%

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

   13. Applied egg-rr 92.1%

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

       1. associate-/r* 92.2%

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

   15. Simplified 92.2%

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

   if -8.8000000000000006e69 < y < 5.2000000000000001e91

   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 96.0%

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

   if 5.2000000000000001e91 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around inf 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*l* 92.5%

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

   9. Simplified 92.5%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+91}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+86}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+77)
   (/ (/ y (sqrt x)) -3.0)
   (if (<= y 6e+86)
     (- 1.0 (pow (* x 9.0) -1.0))
     (* (* y -0.3333333333333333) (sqrt (/ 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+77) {
		tmp = (y / sqrt(x)) / -3.0;
	} else if (y <= 6e+86) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = (y * -0.3333333333333333) * 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 <= (-1.45d+77)) then
        tmp = (y / sqrt(x)) / (-3.0d0)
    else if (y <= 6d+86) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = (y * (-0.3333333333333333d0)) * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+77) {
		tmp = (y / Math.sqrt(x)) / -3.0;
	} else if (y <= 6e+86) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = (y * -0.3333333333333333) * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+77:
		tmp = (y / math.sqrt(x)) / -3.0
	elif y <= 6e+86:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = (y * -0.3333333333333333) * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+77)
		tmp = Float64(Float64(y / sqrt(x)) / -3.0);
	elseif (y <= 6e+86)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+77)
		tmp = (y / sqrt(x)) / -3.0;
	elseif (y <= 6e+86)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = (y * -0.3333333333333333) * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+77], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[y, 6e+86], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4500000000000001e77

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 91.9%

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

      1. metadata-eval 91.9%

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

      2. distribute-lft-neg-in 91.9%

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

      3. *-commutative 91.9%

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

      4. distribute-lft-neg-in 91.9%

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

      5. unpow1/2 91.9%

         \[\leadsto \left(-\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      6. rem-exp-log 87.6%

         \[\leadsto \left(-{\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      7. exp-neg 87.6%

         \[\leadsto \left(-{\color{blue}{\left(e^{-\log x}\right)}}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      8. exp-prod 87.6%

         \[\leadsto \left(-\color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      9. distribute-lft-neg-out 87.6%

         \[\leadsto \left(-e^{\color{blue}{-\log x \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      10. exp-neg 87.6%

          \[\leadsto \left(-\color{blue}{\frac{1}{e^{\log x \cdot 0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      11. exp-to-pow 91.9%

          \[\leadsto \left(-\frac{1}{\color{blue}{{x}^{0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      12. unpow1/2 91.9%

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

      13. unpow-1 91.9%

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

      14. distribute-lft-neg-in 91.9%

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

      15. associate-*r* 91.9%

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

      16. unpow-1 91.9%

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

      17. associate-*l/ 92.1%

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

      18. associate-*r/ 92.1%

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

      19. associate-*r/ 92.0%

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

   9. Simplified 92.1%

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

       1. div-inv 91.9%

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

       2. pow1/2 91.9%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot \frac{1}{\color{blue}{{x}^{0.5}}} \]

       3. pow-flip 92.1%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot \color{blue}{{x}^{\left(-0.5\right)}} \]

       4. metadata-eval 92.1%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot {x}^{\color{blue}{-0.5}} \]

       5. associate-*l* 92.1%

          \[\leadsto \color{blue}{y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \]

       6. *-commutative 92.1%

          \[\leadsto y \cdot \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right)} \]

       7. *-commutative 92.1%

          \[\leadsto \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right) \cdot y} \]

       8. *-commutative 92.1%

          \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \cdot y \]

       9. metadata-eval 92.1%

          \[\leadsto \left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot y \]

       10. pow-flip 92.0%

           \[\leadsto \left(-0.3333333333333333 \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot y \]

       11. pow1/2 92.0%

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

       12. div-inv 92.0%

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

   11. Applied egg-rr 92.0%

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

       1. *-commutative 92.0%

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

       2. clear-num 91.9%

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

       3. un-div-inv 91.9%

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

       4. div-inv 92.1%

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

       5. metadata-eval 92.1%

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

   13. Applied egg-rr 92.1%

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

       1. associate-/r* 92.2%

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

   15. Simplified 92.2%

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

   if -1.4500000000000001e77 < y < 5.99999999999999954e86

   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 96.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 49.8%

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

      3. frac-times 49.8%

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

      4. metadata-eval 49.8%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 49.8%

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

      6. frac-times 49.8%

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

      7. sqrt-unprod 49.8%

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

      8. add-sqr-sqrt 49.8%

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

      9. frac-2neg 49.8%

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

      10. metadata-eval 49.8%

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

      11. distribute-frac-neg2 49.8%

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

   7. Applied egg-rr 49.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 74.2%

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

      3. frac-times 74.3%

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

      4. metadata-eval 74.3%

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

      5. metadata-eval 74.3%

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

      6. frac-times 74.2%

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

      7. sqrt-unprod 95.9%

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

      8. add-sqr-sqrt 96.0%

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

      9. clear-num 96.0%

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

      10. div-inv 96.1%

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

      11. metadata-eval 96.1%

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

      12. inv-pow 96.1%

          \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   9. Applied egg-rr 96.1%

      \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   if 5.99999999999999954e86 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around inf 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*l* 92.5%

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

   9. Simplified 92.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+86}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+63}:\\ \;\;\;\;1 - \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2e+63)
   (- 1.0 (/ (* y 0.3333333333333333) (sqrt x)))
   (if (<= y 1.1e+86)
     (- 1.0 (pow (* x 9.0) -1.0))
     (- 1.0 (/ 0.3333333333333333 (/ (sqrt x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+63) {
		tmp = 1.0 - ((y * 0.3333333333333333) / sqrt(x));
	} else if (y <= 1.1e+86) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.2d+63)) then
        tmp = 1.0d0 - ((y * 0.3333333333333333d0) / sqrt(x))
    else if (y <= 1.1d+86) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 - (0.3333333333333333d0 / (sqrt(x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+63) {
		tmp = 1.0 - ((y * 0.3333333333333333) / Math.sqrt(x));
	} else if (y <= 1.1e+86) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - (0.3333333333333333 / (Math.sqrt(x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e+63:
		tmp = 1.0 - ((y * 0.3333333333333333) / math.sqrt(x))
	elif y <= 1.1e+86:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 - (0.3333333333333333 / (math.sqrt(x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e+63)
		tmp = Float64(1.0 - Float64(Float64(y * 0.3333333333333333) / sqrt(x)));
	elseif (y <= 1.1e+86)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(sqrt(x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.2e+63)
		tmp = 1.0 - ((y * 0.3333333333333333) / sqrt(x));
	elseif (y <= 1.1e+86)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e+63], N[(1.0 - N[(N[(y * 0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+86], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999998e63

   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 95.6%

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

      1. metadata-eval 95.6%

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

      2. *-commutative 95.6%

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

      3. sqrt-div 95.6%

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

      4. metadata-eval 95.6%

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

      5. un-div-inv 95.6%

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

      6. times-frac 95.9%

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

      7. *-un-lft-identity 95.9%

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

      8. associate-/r* 95.8%

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

      9. div-inv 95.8%

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

      10. metadata-eval 95.8%

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

   5. Applied egg-rr 95.8%

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

   if -7.19999999999999998e63 < y < 1.10000000000000002e86

   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 96.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 49.7%

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

      3. frac-times 49.7%

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

      4. metadata-eval 49.7%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 49.7%

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

      6. frac-times 49.7%

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

      7. sqrt-unprod 49.8%

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

      8. add-sqr-sqrt 49.8%

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

      9. frac-2neg 49.8%

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

      10. metadata-eval 49.8%

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

      11. distribute-frac-neg2 49.8%

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

   7. Applied egg-rr 49.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 74.5%

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

      3. frac-times 74.5%

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

      4. metadata-eval 74.5%

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

      5. metadata-eval 74.5%

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

      6. frac-times 74.5%

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

      7. sqrt-unprod 96.4%

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

      8. add-sqr-sqrt 96.5%

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

      9. clear-num 96.5%

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

      10. div-inv 96.6%

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

      11. metadata-eval 96.6%

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

      12. inv-pow 96.6%

          \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   9. Applied egg-rr 96.6%

      \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   if 1.10000000000000002e86 < 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 95.8%

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

      1. *-commutative 95.8%

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

      2. sqrt-div 95.7%

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

      3. metadata-eval 95.7%

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

      4. un-div-inv 95.8%

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

   5. Applied egg-rr 95.8%

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

      1. clear-num 95.9%

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

      2. div-inv 95.9%

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

   7. Applied egg-rr 95.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+63}:\\ \;\;\;\;1 - \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+63}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+85}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2e+63)
   (- 1.0 (/ (/ y (sqrt x)) 3.0))
   (if (<= y 3.4e+85)
     (- 1.0 (pow (* x 9.0) -1.0))
     (- 1.0 (/ 0.3333333333333333 (/ (sqrt x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+63) {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	} else if (y <= 3.4e+85) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.2d+63)) then
        tmp = 1.0d0 - ((y / sqrt(x)) / 3.0d0)
    else if (y <= 3.4d+85) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 - (0.3333333333333333d0 / (sqrt(x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+63) {
		tmp = 1.0 - ((y / Math.sqrt(x)) / 3.0);
	} else if (y <= 3.4e+85) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - (0.3333333333333333 / (Math.sqrt(x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e+63:
		tmp = 1.0 - ((y / math.sqrt(x)) / 3.0)
	elif y <= 3.4e+85:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 - (0.3333333333333333 / (math.sqrt(x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e+63)
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) / 3.0));
	elseif (y <= 3.4e+85)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(sqrt(x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.2e+63)
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	elseif (y <= 3.4e+85)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e+63], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+85], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999998e63

   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 95.6%

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

      1. *-commutative 95.6%

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

      2. sqrt-div 95.6%

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

      3. metadata-eval 95.6%

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

      4. un-div-inv 95.6%

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

   5. Applied egg-rr 95.6%

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

      1. clear-num 95.4%

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

      2. div-inv 95.6%

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

      3. metadata-eval 95.6%

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

      4. associate-/r* 95.5%

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

      5. *-commutative 95.5%

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

      6. associate-/r* 95.7%

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

      7. clear-num 95.9%

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

   7. Applied egg-rr 95.9%

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

   if -7.19999999999999998e63 < y < 3.4000000000000003e85

   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 96.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 49.7%

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

      3. frac-times 49.7%

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

      4. metadata-eval 49.7%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 49.7%

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

      6. frac-times 49.7%

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

      7. sqrt-unprod 49.8%

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

      8. add-sqr-sqrt 49.8%

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

      9. frac-2neg 49.8%

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

      10. metadata-eval 49.8%

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

      11. distribute-frac-neg2 49.8%

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

   7. Applied egg-rr 49.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 74.5%

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

      3. frac-times 74.5%

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

      4. metadata-eval 74.5%

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

      5. metadata-eval 74.5%

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

      6. frac-times 74.5%

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

      7. sqrt-unprod 96.4%

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

      8. add-sqr-sqrt 96.5%

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

      9. clear-num 96.5%

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

      10. div-inv 96.6%

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

      11. metadata-eval 96.6%

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

      12. inv-pow 96.6%

          \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   9. Applied egg-rr 96.6%

      \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]

   if 3.4000000000000003e85 < 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 95.8%

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

      1. *-commutative 95.8%

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

      2. sqrt-div 95.7%

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

      3. metadata-eval 95.7%

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

      4. un-div-inv 95.8%

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

   5. Applied egg-rr 95.8%

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

      1. clear-num 95.9%

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

      2. div-inv 95.9%

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

   7. Applied egg-rr 95.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+63}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+85}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+80} \lor \neg \left(y \leq 8.6 \cdot 10^{+85}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.8e+80) (not (<= y 8.6e+85)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.8e+80) || !(y <= 8.6e+85)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / 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 <= (-8.8d+80)) .or. (.not. (y <= 8.6d+85))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8.8e+80) || !(y <= 8.6e+85)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.8e+80) or not (y <= 8.6e+85):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.8e+80) || !(y <= 8.6e+85))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8.8e+80) || ~((y <= 8.6e+85)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.8e+80], N[Not[LessEqual[y, 8.6e+85]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.80000000000000011e80 or 8.5999999999999998e85 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 92.1%

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

      1. metadata-eval 92.1%

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

      2. distribute-lft-neg-in 92.1%

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

      3. *-commutative 92.1%

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

      4. distribute-lft-neg-in 92.1%

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

      5. unpow1/2 92.1%

         \[\leadsto \left(-\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      6. rem-exp-log 87.9%

         \[\leadsto \left(-{\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      7. exp-neg 87.9%

         \[\leadsto \left(-{\color{blue}{\left(e^{-\log x}\right)}}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      8. exp-prod 87.9%

         \[\leadsto \left(-\color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      9. distribute-lft-neg-out 87.9%

         \[\leadsto \left(-e^{\color{blue}{-\log x \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      10. exp-neg 87.9%

          \[\leadsto \left(-\color{blue}{\frac{1}{e^{\log x \cdot 0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      11. exp-to-pow 92.1%

          \[\leadsto \left(-\frac{1}{\color{blue}{{x}^{0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      12. unpow1/2 92.1%

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

      13. unpow-1 92.1%

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

      14. distribute-lft-neg-in 92.1%

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

      15. associate-*r* 92.1%

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

      16. unpow-1 92.1%

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

      17. associate-*l/ 92.1%

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

      18. associate-*r/ 92.1%

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

      19. associate-*r/ 92.1%

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

   9. Simplified 92.1%

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

       1. *-commutative 92.1%

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

       2. associate-/l* 92.1%

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

   11. Applied egg-rr 92.1%

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

   if -8.80000000000000011e80 < y < 8.5999999999999998e85

   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 96.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+80} \lor \neg \left(y \leq 8.6 \cdot 10^{+85}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+79} \lor \neg \left(y \leq 5.4 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.5e+79) (not (<= y 5.4e+85)))
   (/ (/ y (sqrt x)) -3.0)
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.5e+79) || !(y <= 5.4e+85)) {
		tmp = (y / sqrt(x)) / -3.0;
	} else {
		tmp = 1.0 + (-0.1111111111111111 / 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 <= (-9.5d+79)) .or. (.not. (y <= 5.4d+85))) then
        tmp = (y / sqrt(x)) / (-3.0d0)
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9.5e+79) || !(y <= 5.4e+85)) {
		tmp = (y / Math.sqrt(x)) / -3.0;
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.5e+79) or not (y <= 5.4e+85):
		tmp = (y / math.sqrt(x)) / -3.0
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.5e+79) || !(y <= 5.4e+85))
		tmp = Float64(Float64(y / sqrt(x)) / -3.0);
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9.5e+79) || ~((y <= 5.4e+85)))
		tmp = (y / sqrt(x)) / -3.0;
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.5e+79], N[Not[LessEqual[y, 5.4e+85]], $MachinePrecision]], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.49999999999999994e79 or 5.39999999999999966e85 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 92.1%

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

      1. metadata-eval 92.1%

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

      2. distribute-lft-neg-in 92.1%

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

      3. *-commutative 92.1%

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

      4. distribute-lft-neg-in 92.1%

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

      5. unpow1/2 92.1%

         \[\leadsto \left(-\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      6. rem-exp-log 87.9%

         \[\leadsto \left(-{\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      7. exp-neg 87.9%

         \[\leadsto \left(-{\color{blue}{\left(e^{-\log x}\right)}}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      8. exp-prod 87.9%

         \[\leadsto \left(-\color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      9. distribute-lft-neg-out 87.9%

         \[\leadsto \left(-e^{\color{blue}{-\log x \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      10. exp-neg 87.9%

          \[\leadsto \left(-\color{blue}{\frac{1}{e^{\log x \cdot 0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      11. exp-to-pow 92.1%

          \[\leadsto \left(-\frac{1}{\color{blue}{{x}^{0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      12. unpow1/2 92.1%

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

      13. unpow-1 92.1%

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

      14. distribute-lft-neg-in 92.1%

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

      15. associate-*r* 92.1%

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

      16. unpow-1 92.1%

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

      17. associate-*l/ 92.1%

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

      18. associate-*r/ 92.1%

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

      19. associate-*r/ 92.1%

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

   9. Simplified 92.1%

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

       1. div-inv 92.1%

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

       2. pow1/2 92.1%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot \frac{1}{\color{blue}{{x}^{0.5}}} \]

       3. pow-flip 92.2%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot \color{blue}{{x}^{\left(-0.5\right)}} \]

       4. metadata-eval 92.2%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot {x}^{\color{blue}{-0.5}} \]

       5. associate-*l* 92.2%

          \[\leadsto \color{blue}{y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \]

       6. *-commutative 92.2%

          \[\leadsto y \cdot \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right)} \]

       7. *-commutative 92.2%

          \[\leadsto \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right) \cdot y} \]

       8. *-commutative 92.2%

          \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \cdot y \]

       9. metadata-eval 92.2%

          \[\leadsto \left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot y \]

       10. pow-flip 92.1%

           \[\leadsto \left(-0.3333333333333333 \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot y \]

       11. pow1/2 92.1%

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

       12. div-inv 92.1%

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

   11. Applied egg-rr 92.1%

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

       1. *-commutative 92.1%

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

       2. clear-num 92.0%

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

       3. un-div-inv 92.1%

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

       4. div-inv 92.2%

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

       5. metadata-eval 92.2%

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

   13. Applied egg-rr 92.2%

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

       1. associate-/r* 92.3%

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

   15. Simplified 92.3%

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

   if -9.49999999999999994e79 < y < 5.39999999999999966e85

   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 96.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+79} \lor \neg \left(y \leq 5.4 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+79}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+91}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.6e+79)
   (* y (/ -0.3333333333333333 (sqrt x)))
   (if (<= y 1.9e+91)
     (+ 1.0 (/ -0.1111111111111111 x))
     (* -0.3333333333333333 (/ y (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.6e+79) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else if (y <= 1.9e+91) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (y / 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 (y <= (-6.6d+79)) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else if (y <= 1.9d+91) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.6e+79) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else if (y <= 1.9e+91) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.6e+79:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	elif y <= 1.9e+91:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.6e+79)
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	elseif (y <= 1.9e+91)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.6e+79)
		tmp = y * (-0.3333333333333333 / sqrt(x));
	elseif (y <= 1.9e+91)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = -0.3333333333333333 * (y / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.6e+79], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+91], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.6000000000000003e79

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 91.9%

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

      1. metadata-eval 91.9%

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

      2. distribute-lft-neg-in 91.9%

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

      3. *-commutative 91.9%

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

      4. distribute-lft-neg-in 91.9%

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

      5. unpow1/2 91.9%

         \[\leadsto \left(-\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      6. rem-exp-log 87.6%

         \[\leadsto \left(-{\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      7. exp-neg 87.6%

         \[\leadsto \left(-{\color{blue}{\left(e^{-\log x}\right)}}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      8. exp-prod 87.6%

         \[\leadsto \left(-\color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      9. distribute-lft-neg-out 87.6%

         \[\leadsto \left(-e^{\color{blue}{-\log x \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      10. exp-neg 87.6%

          \[\leadsto \left(-\color{blue}{\frac{1}{e^{\log x \cdot 0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      11. exp-to-pow 91.9%

          \[\leadsto \left(-\frac{1}{\color{blue}{{x}^{0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      12. unpow1/2 91.9%

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

      13. unpow-1 91.9%

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

      14. distribute-lft-neg-in 91.9%

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

      15. associate-*r* 91.9%

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

      16. unpow-1 91.9%

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

      17. associate-*l/ 92.1%

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

      18. associate-*r/ 92.1%

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

      19. associate-*r/ 92.0%

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

   9. Simplified 92.1%

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

       1. div-inv 91.9%

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

       2. pow1/2 91.9%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot \frac{1}{\color{blue}{{x}^{0.5}}} \]

       3. pow-flip 92.1%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot \color{blue}{{x}^{\left(-0.5\right)}} \]

       4. metadata-eval 92.1%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot {x}^{\color{blue}{-0.5}} \]

       5. associate-*l* 92.1%

          \[\leadsto \color{blue}{y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \]

       6. *-commutative 92.1%

          \[\leadsto y \cdot \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right)} \]

       7. *-commutative 92.1%

          \[\leadsto \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right) \cdot y} \]

       8. *-commutative 92.1%

          \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \cdot y \]

       9. metadata-eval 92.1%

          \[\leadsto \left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot y \]

       10. pow-flip 92.0%

           \[\leadsto \left(-0.3333333333333333 \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot y \]

       11. pow1/2 92.0%

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

       12. div-inv 92.0%

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

   11. Applied egg-rr 92.0%

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

   if -6.6000000000000003e79 < y < 1.8999999999999999e91

   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 96.0%

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

   if 1.8999999999999999e91 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around inf 92.4%

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

      1. metadata-eval 92.4%

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

      2. distribute-lft-neg-in 92.4%

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

      3. *-commutative 92.4%

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

      4. distribute-lft-neg-in 92.4%

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

      5. unpow1/2 92.4%

         \[\leadsto \left(-\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      6. rem-exp-log 88.3%

         \[\leadsto \left(-{\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      7. exp-neg 88.3%

         \[\leadsto \left(-{\color{blue}{\left(e^{-\log x}\right)}}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      8. exp-prod 88.3%

         \[\leadsto \left(-\color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      9. distribute-lft-neg-out 88.3%

         \[\leadsto \left(-e^{\color{blue}{-\log x \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      10. exp-neg 88.3%

          \[\leadsto \left(-\color{blue}{\frac{1}{e^{\log x \cdot 0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      11. exp-to-pow 92.3%

          \[\leadsto \left(-\frac{1}{\color{blue}{{x}^{0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      12. unpow1/2 92.3%

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

      13. unpow-1 92.3%

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

      14. distribute-lft-neg-in 92.3%

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

      15. associate-*r* 92.3%

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

      16. unpow-1 92.3%

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

      17. associate-*l/ 92.2%

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

      18. associate-*r/ 92.2%

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

      19. associate-*r/ 92.2%

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

   9. Simplified 92.2%

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

       1. *-commutative 92.2%

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

       2. associate-/l* 92.4%

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

   11. Applied egg-rr 92.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+79}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+91}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 10^{+89}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.5e+75)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 1e+89)
     (+ 1.0 (/ -0.1111111111111111 x))
     (* -0.3333333333333333 (/ y (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.5e+75) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 1e+89) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (y / 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 (y <= (-7.5d+75)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 1d+89) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.5e+75) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 1e+89) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.5e+75:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 1e+89:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.5e+75)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 1e+89)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.5e+75)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 1e+89)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = -0.3333333333333333 * (y / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.5e+75], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+89], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.4999999999999995e75

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 91.9%

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

      1. metadata-eval 91.9%

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

      2. distribute-lft-neg-in 91.9%

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

      3. *-commutative 91.9%

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

      4. distribute-lft-neg-in 91.9%

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

      5. unpow1/2 91.9%

         \[\leadsto \left(-\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      6. rem-exp-log 87.6%

         \[\leadsto \left(-{\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      7. exp-neg 87.6%

         \[\leadsto \left(-{\color{blue}{\left(e^{-\log x}\right)}}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      8. exp-prod 87.6%

         \[\leadsto \left(-\color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      9. distribute-lft-neg-out 87.6%

         \[\leadsto \left(-e^{\color{blue}{-\log x \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      10. exp-neg 87.6%

          \[\leadsto \left(-\color{blue}{\frac{1}{e^{\log x \cdot 0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      11. exp-to-pow 91.9%

          \[\leadsto \left(-\frac{1}{\color{blue}{{x}^{0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      12. unpow1/2 91.9%

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

      13. unpow-1 91.9%

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

      14. distribute-lft-neg-in 91.9%

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

      15. associate-*r* 91.9%

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

      16. unpow-1 91.9%

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

      17. associate-*l/ 92.1%

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

      18. associate-*r/ 92.1%

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

      19. associate-*r/ 92.0%

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

   9. Simplified 92.1%

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

       1. div-inv 91.9%

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

       2. pow1/2 91.9%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot \frac{1}{\color{blue}{{x}^{0.5}}} \]

       3. pow-flip 92.1%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot \color{blue}{{x}^{\left(-0.5\right)}} \]

       4. metadata-eval 92.1%

          \[\leadsto \left(y \cdot -0.3333333333333333\right) \cdot {x}^{\color{blue}{-0.5}} \]

       5. associate-*l* 92.1%

          \[\leadsto \color{blue}{y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \]

       6. *-commutative 92.1%

          \[\leadsto y \cdot \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right)} \]

       7. *-commutative 92.1%

          \[\leadsto \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right) \cdot y} \]

       8. *-commutative 92.1%

          \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \cdot y \]

       9. metadata-eval 92.1%

          \[\leadsto \left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot y \]

       10. pow-flip 92.0%

           \[\leadsto \left(-0.3333333333333333 \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot y \]

       11. pow1/2 92.0%

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

       12. div-inv 92.0%

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

   11. Applied egg-rr 92.0%

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

       1. *-commutative 92.0%

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

       2. clear-num 91.9%

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

       3. un-div-inv 91.9%

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

       4. div-inv 92.1%

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

       5. metadata-eval 92.1%

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

   13. Applied egg-rr 92.1%

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

   if -7.4999999999999995e75 < y < 9.99999999999999995e88

   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 96.0%

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

   if 9.99999999999999995e88 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around inf 92.4%

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

      1. metadata-eval 92.4%

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

      2. distribute-lft-neg-in 92.4%

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

      3. *-commutative 92.4%

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

      4. distribute-lft-neg-in 92.4%

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

      5. unpow1/2 92.4%

         \[\leadsto \left(-\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      6. rem-exp-log 88.3%

         \[\leadsto \left(-{\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      7. exp-neg 88.3%

         \[\leadsto \left(-{\color{blue}{\left(e^{-\log x}\right)}}^{0.5} \cdot y\right) \cdot 0.3333333333333333 \]

      8. exp-prod 88.3%

         \[\leadsto \left(-\color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      9. distribute-lft-neg-out 88.3%

         \[\leadsto \left(-e^{\color{blue}{-\log x \cdot 0.5}} \cdot y\right) \cdot 0.3333333333333333 \]

      10. exp-neg 88.3%

          \[\leadsto \left(-\color{blue}{\frac{1}{e^{\log x \cdot 0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      11. exp-to-pow 92.3%

          \[\leadsto \left(-\frac{1}{\color{blue}{{x}^{0.5}}} \cdot y\right) \cdot 0.3333333333333333 \]

      12. unpow1/2 92.3%

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

      13. unpow-1 92.3%

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

      14. distribute-lft-neg-in 92.3%

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

      15. associate-*r* 92.3%

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

      16. unpow-1 92.3%

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

      17. associate-*l/ 92.2%

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

      18. associate-*r/ 92.2%

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

      19. associate-*r/ 92.2%

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

   9. Simplified 92.2%

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

       1. *-commutative 92.2%

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

       2. associate-/l* 92.4%

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

   11. Applied egg-rr 92.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 10^{+89}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 25000:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(y \cdot \left(-\sqrt{x}\right)\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \frac{{x}^{-0.5}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 25000.0)
   (/ (- (* 0.3333333333333333 (* y (- (sqrt x)))) 0.1111111111111111) x)
   (- 1.0 (* y (/ (pow x -0.5) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 25000.0) {
		tmp = ((0.3333333333333333 * (y * -sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y * (pow(x, -0.5) / 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 25000.0d0) then
        tmp = ((0.3333333333333333d0 * (y * -sqrt(x))) - 0.1111111111111111d0) / x
    else
        tmp = 1.0d0 - (y * ((x ** (-0.5d0)) / 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 25000.0) {
		tmp = ((0.3333333333333333 * (y * -Math.sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y * (Math.pow(x, -0.5) / 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 25000.0:
		tmp = ((0.3333333333333333 * (y * -math.sqrt(x))) - 0.1111111111111111) / x
	else:
		tmp = 1.0 - (y * (math.pow(x, -0.5) / 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 25000.0)
		tmp = Float64(Float64(Float64(0.3333333333333333 * Float64(y * Float64(-sqrt(x)))) - 0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y * Float64((x ^ -0.5) / 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 25000.0)
		tmp = ((0.3333333333333333 * (y * -sqrt(x))) - 0.1111111111111111) / x;
	else
		tmp = 1.0 - (y * ((x ^ -0.5) / 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 25000.0], N[(N[(N[(0.3333333333333333 * N[(y * (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y * N[(N[Power[x, -0.5], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 25000

   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 99.0%

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

      1. mul-1-neg 99.0%

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

      2. *-commutative 99.0%

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

   5. Simplified 99.0%

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

   if 25000 < 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 98.3%

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

      1. metadata-eval 98.3%

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

      2. *-commutative 98.3%

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

      3. sqrt-div 98.3%

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

      4. metadata-eval 98.3%

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

      5. un-div-inv 98.3%

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

      6. times-frac 98.4%

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

      7. *-un-lft-identity 98.4%

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

      8. clear-num 98.4%

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

      9. *-un-lft-identity 98.4%

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

      10. times-frac 98.3%

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

      11. metadata-eval 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in x around 0 98.2%

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

      1. associate-*r* 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*r/ 98.4%

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

      4. *-rgt-identity 98.4%

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

      5. associate-/l* 98.3%

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

   8. Simplified 98.3%

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

      1. associate-/r* 98.3%

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

      2. associate-/r/ 98.4%

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

      3. pow1/2 98.4%

         \[\leadsto 1 - \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{3} \cdot y \]

      4. pow-flip 98.4%

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

      5. metadata-eval 98.4%

         \[\leadsto 1 - \frac{{x}^{\color{blue}{-0.5}}}{3} \cdot y \]

   10. Applied egg-rr 98.4%

       \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{3} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 25000:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(y \cdot \left(-\sqrt{x}\right)\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \frac{{x}^{-0.5}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 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* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   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 16: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + ((y * -0.3333333333333333) / 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 * (-0.3333333333333333d0)) / sqrt(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   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. clear-num 99.6%

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

   2. un-div-inv 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/r/ 99.6%

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

   2. associate-*l/ 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 17: 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 99.6%

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

4. Final simplification 99.6%

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

Alternative 18: 62.0% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   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. Taylor expanded in y around 0 63.2%

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

6. Final simplification 63.2%

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

Alternative 19: 62.0% 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* 99.7%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   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 63.2%

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

6. Final simplification 63.2%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]
7. 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 2024067 
(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)))))