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

Percentage Accurate: 99.7% → 99.7%

Time: 12.1s

Alternatives: 17

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 99.7%

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

   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. Final simplification 99.7%

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

Alternative 2: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+52}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+46}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.35e+52)
   (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x)))
   (if (<= y 2.7e+46)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* (pow x -0.5) (* y -0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.35e+52) {
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	} else if (y <= 2.7e+46) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (pow(x, -0.5) * (y * -0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.35d+52)) then
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / sqrt(x))
    else if (y <= 2.7d+46) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + ((x ** (-0.5d0)) * (y * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.35e+52) {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	} else if (y <= 2.7e+46) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (Math.pow(x, -0.5) * (y * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.35e+52:
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	elif y <= 2.7e+46:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (math.pow(x, -0.5) * (y * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.35e+52)
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	elseif (y <= 2.7e+46)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64((x ^ -0.5) * Float64(y * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.35e+52)
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	elseif (y <= 2.7e+46)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + ((x ^ -0.5) * (y * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.35e+52], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+46], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.35e52

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.3%

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

      12. metadata-eval 99.3%

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

      13. *-commutative 99.3%

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

      14. associate-/r* 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 92.8%

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

      1. *-commutative 92.8%

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

      2. associate-*l* 92.8%

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

      3. *-commutative 92.8%

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

   7. Simplified 92.8%

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

      1. *-commutative 92.8%

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

      2. sqrt-div 92.6%

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

      3. metadata-eval 92.6%

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

      4. un-div-inv 92.8%

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

      5. *-commutative 92.8%

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

   9. Applied egg-rr 92.8%

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

   if -2.35e52 < y < 2.7000000000000002e46

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. cancel-sign-sub-inv 98.6%

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

      2. metadata-eval 98.6%

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

      3. associate-*r/ 98.7%

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

      4. metadata-eval 98.7%

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

      5. +-commutative 98.7%

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

   7. Simplified 98.7%

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

   if 2.7000000000000002e46 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*l* 98.0%

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

      3. *-commutative 98.0%

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

   7. Simplified 98.0%

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

      1. inv-pow 87.6%

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

      2. sqrt-pow1 87.7%

         \[\leadsto \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      3. metadata-eval 87.7%

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

      4. expm1-log1p-u 85.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      5. expm1-udef 49.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   9. Applied egg-rr 57.4%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   10. Step-by-step derivation

       1. expm1-def 85.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       2. expm1-log1p 87.7%

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

   11. Simplified 98.0%

       \[\leadsto 1 + \color{blue}{{x}^{-0.5}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+52}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+46}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+52}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+46}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e+52)
   (+ 1.0 (* -0.3333333333333333 (* y (sqrt (/ 1.0 x)))))
   (if (<= y 5.6e+46)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* (pow x -0.5) (* y -0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e+52) {
		tmp = 1.0 + (-0.3333333333333333 * (y * sqrt((1.0 / x))));
	} else if (y <= 5.6e+46) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (pow(x, -0.5) * (y * -0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.4d+52)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y * sqrt((1.0d0 / x))))
    else if (y <= 5.6d+46) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + ((x ** (-0.5d0)) * (y * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4e+52) {
		tmp = 1.0 + (-0.3333333333333333 * (y * Math.sqrt((1.0 / x))));
	} else if (y <= 5.6e+46) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (Math.pow(x, -0.5) * (y * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e+52:
		tmp = 1.0 + (-0.3333333333333333 * (y * math.sqrt((1.0 / x))))
	elif y <= 5.6e+46:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (math.pow(x, -0.5) * (y * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e+52)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x)))));
	elseif (y <= 5.6e+46)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64((x ^ -0.5) * Float64(y * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e+52)
		tmp = 1.0 + (-0.3333333333333333 * (y * sqrt((1.0 / x))));
	elseif (y <= 5.6e+46)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + ((x ^ -0.5) * (y * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e+52], N[(1.0 + N[(-0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e+46], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e52

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.3%

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

      12. metadata-eval 99.3%

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

      13. *-commutative 99.3%

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

      14. associate-/r* 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 92.8%

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

   if -2.4e52 < y < 5.60000000000000037e46

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. cancel-sign-sub-inv 98.6%

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

      2. metadata-eval 98.6%

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

      3. associate-*r/ 98.7%

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

      4. metadata-eval 98.7%

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

      5. +-commutative 98.7%

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

   7. Simplified 98.7%

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

   if 5.60000000000000037e46 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*l* 98.0%

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

      3. *-commutative 98.0%

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

   7. Simplified 98.0%

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

      1. inv-pow 87.6%

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

      2. sqrt-pow1 87.7%

         \[\leadsto \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      3. metadata-eval 87.7%

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

      4. expm1-log1p-u 85.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      5. expm1-udef 49.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   9. Applied egg-rr 57.4%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   10. Step-by-step derivation

       1. expm1-def 85.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       2. expm1-log1p 87.7%

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

   11. Simplified 98.0%

       \[\leadsto 1 + \color{blue}{{x}^{-0.5}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+52}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+46}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+56}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+48}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e+56)
   (+ 1.0 (* y (* -0.3333333333333333 (sqrt (/ 1.0 x)))))
   (if (<= y 2.1e+48)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* (pow x -0.5) (* y -0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e+56) {
		tmp = 1.0 + (y * (-0.3333333333333333 * sqrt((1.0 / x))));
	} else if (y <= 2.1e+48) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (pow(x, -0.5) * (y * -0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9d+56)) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) * sqrt((1.0d0 / x))))
    else if (y <= 2.1d+48) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + ((x ** (-0.5d0)) * (y * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9e+56) {
		tmp = 1.0 + (y * (-0.3333333333333333 * Math.sqrt((1.0 / x))));
	} else if (y <= 2.1e+48) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (Math.pow(x, -0.5) * (y * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e+56:
		tmp = 1.0 + (y * (-0.3333333333333333 * math.sqrt((1.0 / x))))
	elif y <= 2.1e+48:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (math.pow(x, -0.5) * (y * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e+56)
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 * sqrt(Float64(1.0 / x)))));
	elseif (y <= 2.1e+48)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64((x ^ -0.5) * Float64(y * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9e+56)
		tmp = 1.0 + (y * (-0.3333333333333333 * sqrt((1.0 / x))));
	elseif (y <= 2.1e+48)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + ((x ^ -0.5) * (y * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e+56], N[(1.0 + N[(y * N[(-0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+48], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.0000000000000006e56

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.3%

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

      12. metadata-eval 99.3%

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

      13. *-commutative 99.3%

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

      14. associate-/r* 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 92.8%

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

      1. associate-*r* 92.8%

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

   7. Simplified 92.8%

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

   if -9.0000000000000006e56 < y < 2.0999999999999998e48

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. cancel-sign-sub-inv 98.6%

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

      2. metadata-eval 98.6%

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

      3. associate-*r/ 98.7%

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

      4. metadata-eval 98.7%

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

      5. +-commutative 98.7%

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

   7. Simplified 98.7%

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

   if 2.0999999999999998e48 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*l* 98.0%

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

      3. *-commutative 98.0%

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

   7. Simplified 98.0%

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

      1. inv-pow 87.6%

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

      2. sqrt-pow1 87.7%

         \[\leadsto \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      3. metadata-eval 87.7%

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

      4. expm1-log1p-u 85.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      5. expm1-udef 49.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   9. Applied egg-rr 57.4%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   10. Step-by-step derivation

       1. expm1-def 85.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       2. expm1-log1p 87.7%

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

   11. Simplified 98.0%

       \[\leadsto 1 + \color{blue}{{x}^{-0.5}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+56}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+48}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+54} \lor \neg \left(y \leq 8.8 \cdot 10^{+46}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2e+54) (not (<= y 8.8e+46)))
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2e+54) || !(y <= 8.8e+46)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / 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 <= (-2d+54)) .or. (.not. (y <= 8.8d+46))) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / 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 <= -2e+54) || !(y <= 8.8e+46)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2e+54) or not (y <= 8.8e+46):
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2e+54) || !(y <= 8.8e+46))
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / 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 <= -2e+54) || ~((y <= 8.8e+46)))
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2e+54], N[Not[LessEqual[y, 8.8e+46]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0000000000000002e54 or 8.8000000000000001e46 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 95.5%

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

      1. *-commutative 95.5%

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

      2. associate-*l* 95.5%

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

      3. *-commutative 95.5%

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

   7. Simplified 95.5%

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

      1. associate-*r* 86.3%

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

      2. *-commutative 86.3%

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

      3. inv-pow 86.2%

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

      4. sqrt-pow1 86.3%

         \[\leadsto \left(-0.3333333333333333 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot y \]

      5. metadata-eval 86.3%

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

      6. expm1-log1p-u 36.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot {x}^{-0.5}\right) \cdot y\right)\right)} \]

      7. associate-*r* 36.8%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)}\right)\right) \]

      8. expm1-udef 36.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)} - 1} \]

   9. Applied egg-rr 45.7%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 36.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \]

       2. expm1-log1p 86.2%

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

   11. Simplified 95.5%

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

   if -2.0000000000000002e54 < y < 8.8000000000000001e46

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. cancel-sign-sub-inv 98.6%

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

      2. metadata-eval 98.6%

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

      3. associate-*r/ 98.7%

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

      4. metadata-eval 98.7%

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

      5. +-commutative 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+54} \lor \neg \left(y \leq 8.8 \cdot 10^{+46}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+53} \lor \neg \left(y \leq 2.95 \cdot 10^{+47}\right):\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.6e+53) (not (<= y 2.95e+47)))
   (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.6e+53) || !(y <= 2.95e+47)) {
		tmp = 1.0 + ((y * -0.3333333333333333) / 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 <= (-3.6d+53)) .or. (.not. (y <= 2.95d+47))) then
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / 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 <= -3.6e+53) || !(y <= 2.95e+47)) {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.6e+53) or not (y <= 2.95e+47):
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.6e+53) || !(y <= 2.95e+47))
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / 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 <= -3.6e+53) || ~((y <= 2.95e+47)))
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.6e+53], N[Not[LessEqual[y, 2.95e+47]], $MachinePrecision]], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / 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 < -3.6e53 or 2.95000000000000017e47 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 95.5%

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

      1. *-commutative 95.5%

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

      2. associate-*l* 95.5%

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

      3. *-commutative 95.5%

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

   7. Simplified 95.5%

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

      1. *-commutative 95.5%

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

      2. sqrt-div 95.5%

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

      3. metadata-eval 95.5%

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

      4. un-div-inv 95.5%

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

      5. *-commutative 95.5%

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

   9. Applied egg-rr 95.5%

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

   if -3.6e53 < y < 2.95000000000000017e47

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. cancel-sign-sub-inv 98.6%

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

      2. metadata-eval 98.6%

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

      3. associate-*r/ 98.7%

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

      4. metadata-eval 98.7%

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

      5. +-commutative 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+53} \lor \neg \left(y \leq 2.95 \cdot 10^{+47}\right):\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+82} \lor \neg \left(y \leq 1.06 \cdot 10^{+51}\right):\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3e+82) (not (<= y 1.06e+51)))
   (* y (* -0.3333333333333333 (pow x -0.5)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3e+82) || !(y <= 1.06e+51)) {
		tmp = y * (-0.3333333333333333 * pow(x, -0.5));
	} 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 <= (-3d+82)) .or. (.not. (y <= 1.06d+51))) then
        tmp = y * ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    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 <= -3e+82) || !(y <= 1.06e+51)) {
		tmp = y * (-0.3333333333333333 * Math.pow(x, -0.5));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3e+82) or not (y <= 1.06e+51):
		tmp = y * (-0.3333333333333333 * math.pow(x, -0.5))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3e+82) || !(y <= 1.06e+51))
		tmp = Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5)));
	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 <= -3e+82) || ~((y <= 1.06e+51)))
		tmp = y * (-0.3333333333333333 * (x ^ -0.5));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3e+82], N[Not[LessEqual[y, 1.06e+51]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.99999999999999989e82 or 1.06000000000000004e51 < 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 90.5%

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

      1. associate-*r* 90.5%

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

      2. *-commutative 90.5%

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

      3. rem-exp-log 85.7%

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

      4. exp-neg 85.7%

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

      5. unpow1/2 85.7%

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

      6. exp-prod 85.7%

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

      7. distribute-lft-neg-out 85.7%

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

      8. distribute-rgt-neg-in 85.7%

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

      9. metadata-eval 85.7%

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

      10. exp-to-pow 90.6%

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

   9. Simplified 90.6%

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

   if -2.99999999999999989e82 < y < 1.06000000000000004e51

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. metadata-eval 98.0%

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

      3. associate-*r/ 98.0%

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

      4. metadata-eval 98.0%

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

      5. +-commutative 98.0%

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

   7. Simplified 98.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+82} \lor \neg \left(y \leq 1.06 \cdot 10^{+51}\right):\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+82}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+50}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3e+82)
   (* y (* -0.3333333333333333 (pow x -0.5)))
   (if (<= y 8.2e+50)
     (+ 1.0 (/ -0.1111111111111111 x))
     (* (pow x -0.5) (* y -0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3e+82) {
		tmp = y * (-0.3333333333333333 * pow(x, -0.5));
	} else if (y <= 8.2e+50) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = pow(x, -0.5) * (y * -0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3d+82)) then
        tmp = y * ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    else if (y <= 8.2d+50) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (x ** (-0.5d0)) * (y * (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3e+82) {
		tmp = y * (-0.3333333333333333 * Math.pow(x, -0.5));
	} else if (y <= 8.2e+50) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = Math.pow(x, -0.5) * (y * -0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3e+82:
		tmp = y * (-0.3333333333333333 * math.pow(x, -0.5))
	elif y <= 8.2e+50:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = math.pow(x, -0.5) * (y * -0.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3e+82)
		tmp = Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5)));
	elseif (y <= 8.2e+50)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64((x ^ -0.5) * Float64(y * -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3e+82)
		tmp = y * (-0.3333333333333333 * (x ^ -0.5));
	elseif (y <= 8.2e+50)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (x ^ -0.5) * (y * -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3e+82], N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+50], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.99999999999999989e82

   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 92.3%

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

      1. associate-*r* 92.3%

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

      2. *-commutative 92.3%

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

      3. rem-exp-log 86.9%

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

      4. exp-neg 86.9%

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

      5. unpow1/2 86.9%

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

      6. exp-prod 86.9%

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

      7. distribute-lft-neg-out 86.9%

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

      8. distribute-rgt-neg-in 86.9%

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

      9. metadata-eval 86.9%

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

      10. exp-to-pow 92.3%

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

   9. Simplified 92.3%

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

   if -2.99999999999999989e82 < y < 8.2000000000000002e50

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. metadata-eval 98.0%

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

      3. associate-*r/ 98.0%

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

      4. metadata-eval 98.0%

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

      5. +-commutative 98.0%

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

   7. Simplified 98.0%

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

   if 8.2000000000000002e50 < 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 99.5%

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

      4. pow1/2 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 99.5%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   8. Taylor expanded in y around inf 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*l* 89.0%

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

      3. *-commutative 89.0%

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

   10. Simplified 89.0%

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

       1. inv-pow 89.0%

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

       2. sqrt-pow1 89.1%

          \[\leadsto \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       3. metadata-eval 89.1%

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

       4. expm1-log1p-u 87.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       5. expm1-udef 50.3%

          \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   12. Applied egg-rr 50.3%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   13. Step-by-step derivation

       1. expm1-def 87.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       2. expm1-log1p 89.1%

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

   14. Simplified 89.1%

       \[\leadsto \color{blue}{{x}^{-0.5}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+82}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+50}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+82}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+51}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.85e+82)
   (* -0.3333333333333333 (* y (sqrt (/ 1.0 x))))
   (if (<= y 1.06e+51)
     (+ 1.0 (/ -0.1111111111111111 x))
     (* (pow x -0.5) (* y -0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.85e+82) {
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	} else if (y <= 1.06e+51) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = pow(x, -0.5) * (y * -0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.85d+82)) then
        tmp = (-0.3333333333333333d0) * (y * sqrt((1.0d0 / x)))
    else if (y <= 1.06d+51) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (x ** (-0.5d0)) * (y * (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.85e+82) {
		tmp = -0.3333333333333333 * (y * Math.sqrt((1.0 / x)));
	} else if (y <= 1.06e+51) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = Math.pow(x, -0.5) * (y * -0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.85e+82:
		tmp = -0.3333333333333333 * (y * math.sqrt((1.0 / x)))
	elif y <= 1.06e+51:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = math.pow(x, -0.5) * (y * -0.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.85e+82)
		tmp = Float64(-0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x))));
	elseif (y <= 1.06e+51)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64((x ^ -0.5) * Float64(y * -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.85e+82)
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	elseif (y <= 1.06e+51)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (x ^ -0.5) * (y * -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.85e+82], N[(-0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e+51], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.85000000000000008e82

   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 92.3%

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

   if -2.85000000000000008e82 < y < 1.06000000000000004e51

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. metadata-eval 98.0%

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

      3. associate-*r/ 98.0%

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

      4. metadata-eval 98.0%

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

      5. +-commutative 98.0%

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

   7. Simplified 98.0%

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

   if 1.06000000000000004e51 < 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 99.5%

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

      4. pow1/2 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 99.5%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   8. Taylor expanded in y around inf 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*l* 89.0%

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

      3. *-commutative 89.0%

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

   10. Simplified 89.0%

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

       1. inv-pow 89.0%

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

       2. sqrt-pow1 89.1%

          \[\leadsto \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       3. metadata-eval 89.1%

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

       4. expm1-log1p-u 87.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       5. expm1-udef 50.3%

          \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   12. Applied egg-rr 50.3%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   13. Step-by-step derivation

       1. expm1-def 87.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       2. expm1-log1p 89.1%

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

   14. Simplified 89.1%

       \[\leadsto \color{blue}{{x}^{-0.5}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+82}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+51}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+84} \lor \neg \left(y \leq 3.1 \cdot 10^{+50}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.05e+84) (not (<= y 3.1e+50)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.05e+84) || !(y <= 3.1e+50)) {
		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 <= (-1.05d+84)) .or. (.not. (y <= 3.1d+50))) 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 <= -1.05e+84) || !(y <= 3.1e+50)) {
		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 <= -1.05e+84) or not (y <= 3.1e+50):
		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 <= -1.05e+84) || !(y <= 3.1e+50))
		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 <= -1.05e+84) || ~((y <= 3.1e+50)))
		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, -1.05e+84], N[Not[LessEqual[y, 3.1e+50]], $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 < -1.05000000000000009e84 or 3.10000000000000003e50 < 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 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-*r* 90.5%

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

      3. inv-pow 90.5%

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

      4. sqrt-pow1 90.5%

         \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \]

      5. metadata-eval 90.5%

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

      6. expm1-log1p-u 38.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot y\right) \cdot {x}^{-0.5}\right)\right)} \]

      7. *-commutative 38.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{x}^{-0.5} \cdot \left(-0.3333333333333333 \cdot y\right)}\right)\right) \]

      8. expm1-udef 38.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-0.5} \cdot \left(-0.3333333333333333 \cdot y\right)\right)} - 1} \]

      9. associate-*r* 38.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right) \cdot y}\right)} - 1 \]

      10. *-commutative 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right)}\right)} - 1 \]

      11. metadata-eval 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \left({x}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot -0.3333333333333333\right)\right)} - 1 \]

      12. sqrt-pow1 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(\color{blue}{\sqrt{{x}^{-1}}} \cdot -0.3333333333333333\right)\right)} - 1 \]

      13. inv-pow 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot -0.3333333333333333\right)\right)} - 1 \]

      14. *-commutative 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)}\right)} - 1 \]

      15. sqrt-div 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)\right)} - 1 \]

      16. metadata-eval 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)\right)} - 1 \]

      17. un-div-inv 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}\right)} - 1 \]

   9. Applied egg-rr 38.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 38.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \]

       2. expm1-log1p 90.5%

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

       3. associate-*r/ 90.5%

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

       4. associate-*l/ 90.4%

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

       5. *-commutative 90.4%

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

   11. Simplified 90.4%

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

   if -1.05000000000000009e84 < y < 3.10000000000000003e50

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. metadata-eval 98.0%

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

      3. associate-*r/ 98.0%

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

      4. metadata-eval 98.0%

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

      5. +-commutative 98.0%

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

   7. Simplified 98.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+84} \lor \neg \left(y \leq 3.1 \cdot 10^{+50}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+82} \lor \neg \left(y \leq 1.06 \cdot 10^{+51}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.8e+82) (not (<= y 1.06e+51)))
   (* y (/ -0.3333333333333333 (sqrt x)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.8e+82) || !(y <= 1.06e+51)) {
		tmp = y * (-0.3333333333333333 / 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 <= (-3.8d+82)) .or. (.not. (y <= 1.06d+51))) then
        tmp = y * ((-0.3333333333333333d0) / 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 <= -3.8e+82) || !(y <= 1.06e+51)) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.8e+82) or not (y <= 1.06e+51):
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.8e+82) || !(y <= 1.06e+51))
		tmp = Float64(y * Float64(-0.3333333333333333 / 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 <= -3.8e+82) || ~((y <= 1.06e+51)))
		tmp = y * (-0.3333333333333333 / sqrt(x));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.8e+82], N[Not[LessEqual[y, 1.06e+51]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 / 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 < -3.80000000000000033e82 or 1.06000000000000004e51 < 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 x around 0 99.6%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   8. Taylor expanded in y around inf 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-*l* 90.5%

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

      3. *-commutative 90.5%

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

   10. Simplified 90.5%

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

       1. associate-*r* 90.5%

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

       2. *-commutative 90.5%

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

       3. inv-pow 90.5%

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

       4. sqrt-pow1 90.6%

          \[\leadsto \left(-0.3333333333333333 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot y \]

       5. metadata-eval 90.6%

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

       6. expm1-log1p-u 38.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot {x}^{-0.5}\right) \cdot y\right)\right)} \]

       7. associate-*r* 38.4%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)}\right)\right) \]

       8. expm1-udef 38.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)} - 1} \]

   12. Applied egg-rr 38.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 38.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \]

       2. expm1-log1p 90.5%

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

   14. Simplified 90.5%

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

   if -3.80000000000000033e82 < y < 1.06000000000000004e51

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. metadata-eval 98.0%

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

      3. associate-*r/ 98.0%

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

      4. metadata-eval 98.0%

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

      5. +-commutative 98.0%

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

   7. Simplified 98.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+82} \lor \neg \left(y \leq 1.06 \cdot 10^{+51}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+83} \lor \neg \left(y \leq 7.5 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.8e+83) (not (<= y 7.5e+50)))
   (/ (* y -0.3333333333333333) (sqrt x))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.8e+83) || !(y <= 7.5e+50)) {
		tmp = (y * -0.3333333333333333) / 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 <= (-3.8d+83)) .or. (.not. (y <= 7.5d+50))) then
        tmp = (y * (-0.3333333333333333d0)) / 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 <= -3.8e+83) || !(y <= 7.5e+50)) {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.8e+83) or not (y <= 7.5e+50):
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.8e+83) || !(y <= 7.5e+50))
		tmp = Float64(Float64(y * -0.3333333333333333) / 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 <= -3.8e+83) || ~((y <= 7.5e+50)))
		tmp = (y * -0.3333333333333333) / sqrt(x);
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.8e+83], N[Not[LessEqual[y, 7.5e+50]], $MachinePrecision]], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8000000000000002e83 or 7.4999999999999999e50 < 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 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-*r* 90.5%

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

      3. inv-pow 90.5%

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

      4. sqrt-pow1 90.5%

         \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \]

      5. metadata-eval 90.5%

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

      6. expm1-log1p-u 38.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot y\right) \cdot {x}^{-0.5}\right)\right)} \]

      7. *-commutative 38.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{x}^{-0.5} \cdot \left(-0.3333333333333333 \cdot y\right)}\right)\right) \]

      8. expm1-udef 38.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-0.5} \cdot \left(-0.3333333333333333 \cdot y\right)\right)} - 1} \]

      9. associate-*r* 38.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right) \cdot y}\right)} - 1 \]

      10. *-commutative 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right)}\right)} - 1 \]

      11. metadata-eval 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \left({x}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot -0.3333333333333333\right)\right)} - 1 \]

      12. sqrt-pow1 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(\color{blue}{\sqrt{{x}^{-1}}} \cdot -0.3333333333333333\right)\right)} - 1 \]

      13. inv-pow 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot -0.3333333333333333\right)\right)} - 1 \]

      14. *-commutative 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)}\right)} - 1 \]

      15. sqrt-div 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)\right)} - 1 \]

      16. metadata-eval 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)\right)} - 1 \]

      17. un-div-inv 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}\right)} - 1 \]

   9. Applied egg-rr 38.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 38.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \]

       2. expm1-log1p 90.5%

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

       3. associate-*r/ 90.5%

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

       4. *-commutative 90.5%

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

   11. Simplified 90.5%

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

   if -3.8000000000000002e83 < y < 7.4999999999999999e50

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. metadata-eval 98.0%

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

      3. associate-*r/ 98.0%

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

      4. metadata-eval 98.0%

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

      5. +-commutative 98.0%

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

   7. Simplified 98.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+83} \lor \neg \left(y \leq 7.5 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(-0.1111111111111111 / x), $MachinePrecision] - N[(y * N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $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 99.7%

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

   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. Step-by-step derivation

   1. sub-neg 99.7%

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

   2. associate--l+ 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

      \[\leadsto 1 + \left(\left(-\frac{\color{blue}{0.1111111111111111}}{x}\right) - \frac{y}{\sqrt{x \cdot 9}}\right) \]

   6. distribute-neg-frac 99.7%

      \[\leadsto 1 + \left(\color{blue}{\frac{-0.1111111111111111}{x}} - \frac{y}{\sqrt{x \cdot 9}}\right) \]

   7. metadata-eval 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-0.1111111111111111}}{x} - \frac{y}{\sqrt{x \cdot 9}}\right) \]

   8. div-inv 99.6%

      \[\leadsto 1 + \left(\frac{-0.1111111111111111}{x} - \color{blue}{y \cdot \frac{1}{\sqrt{x \cdot 9}}}\right) \]

   9. metadata-eval 99.6%

      \[\leadsto 1 + \left(\frac{-0.1111111111111111}{x} - y \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{x \cdot 9}}\right) \]

   10. sqrt-div 99.7%

       \[\leadsto 1 + \left(\frac{-0.1111111111111111}{x} - y \cdot \color{blue}{\sqrt{\frac{1}{x \cdot 9}}}\right) \]

   11. inv-pow 99.7%

       \[\leadsto 1 + \left(\frac{-0.1111111111111111}{x} - y \cdot \sqrt{\color{blue}{{\left(x \cdot 9\right)}^{-1}}}\right) \]

   12. *-commutative 99.7%

       \[\leadsto 1 + \left(\frac{-0.1111111111111111}{x} - y \cdot \sqrt{{\color{blue}{\left(9 \cdot x\right)}}^{-1}}\right) \]

   13. unpow-prod-down 99.7%

       \[\leadsto 1 + \left(\frac{-0.1111111111111111}{x} - y \cdot \sqrt{\color{blue}{{9}^{-1} \cdot {x}^{-1}}}\right) \]

   14. metadata-eval 99.7%

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

   15. inv-pow 99.7%

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

   16. div-inv 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 14: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 0.1111111111111111 x)) (/ y (sqrt (* x 9.0)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (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 - (0.1111111111111111d0 / x)) - (y / sqrt((x * 9.0d0)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / Math.sqrt((x * 9.0)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) - (y / math.sqrt((x * 9.0)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) - Float64(y / sqrt(Float64(x * 9.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $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 99.7%

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

   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 x around 0 99.7%

   \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

8. Final simplification 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \]
9. Add Preprocessing

Alternative 15: 60.8% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 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 (x <= 0.11d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Add Preprocessing
   3. 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 99.6%

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

      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 x around 0 58.3%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 59.2%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.8% 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. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 60.0%

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

   1. cancel-sign-sub-inv 60.0%

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

   2. metadata-eval 60.0%

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

   3. associate-*r/ 60.0%

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

   4. metadata-eval 60.0%

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

   5. +-commutative 60.0%

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

7. Simplified 60.0%

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

8. Final simplification 60.0%

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

Alternative 17: 31.6% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around inf 34.1%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 34.1%

   \[\leadsto 1 \]
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 2024027 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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