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

Percentage Accurate: 99.7% → 99.7%

Time: 9.6s

Alternatives: 17

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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 99.6%

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

   1. *-commutative 99.6%

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

   2. metadata-eval 99.6%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

3. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+52} \lor \neg \left(y \leq 1.4 \cdot 10^{+20}\right):\\ \;\;\;\;1 - \left(y \cdot 0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.8e+52) (not (<= y 1.4e+20)))
   (- 1.0 (* (* y 0.3333333333333333) (sqrt (/ 1.0 x))))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e+52) || !(y <= 1.4e+20)) {
		tmp = 1.0 - ((y * 0.3333333333333333) * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.8d+52)) .or. (.not. (y <= 1.4d+20))) then
        tmp = 1.0d0 - ((y * 0.3333333333333333d0) * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e+52) || !(y <= 1.4e+20)) {
		tmp = 1.0 - ((y * 0.3333333333333333) * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.8e+52) or not (y <= 1.4e+20):
		tmp = 1.0 - ((y * 0.3333333333333333) * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.8e+52) || !(y <= 1.4e+20))
		tmp = Float64(1.0 - Float64(Float64(y * 0.3333333333333333) * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.8e+52) || ~((y <= 1.4e+20)))
		tmp = 1.0 - ((y * 0.3333333333333333) * sqrt((1.0 / x)));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.8e+52], N[Not[LessEqual[y, 1.4e+20]], $MachinePrecision]], N[(1.0 - N[(N[(y * 0.3333333333333333), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.7999999999999999e52 or 1.4e20 < 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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-flip 99.5%

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

      6. metadata-eval 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. prod-diff 84.5%

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

      3. *-commutative 84.5%

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

      4. fma-neg 84.5%

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

      5. *-un-lft-identity 84.5%

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

      6. associate--l- 84.5%

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

      7. inv-pow 84.5%

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

      8. *-commutative 84.5%

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

      9. unpow-prod-down 84.5%

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

      10. metadata-eval 84.5%

          \[\leadsto \left(1 - \left(\color{blue}{0.1111111111111111} \cdot {x}^{-1} + {x}^{-0.5} \cdot \frac{y}{3}\right)\right) + \mathsf{fma}\left(-\frac{y}{3}, {x}^{-0.5}, \frac{y}{3} \cdot {x}^{-0.5}\right) \]

      11. inv-pow 84.5%

          \[\leadsto \left(1 - \left(0.1111111111111111 \cdot \color{blue}{\frac{1}{x}} + {x}^{-0.5} \cdot \frac{y}{3}\right)\right) + \mathsf{fma}\left(-\frac{y}{3}, {x}^{-0.5}, \frac{y}{3} \cdot {x}^{-0.5}\right) \]

      12. div-inv 84.5%

          \[\leadsto \left(1 - \left(\color{blue}{\frac{0.1111111111111111}{x}} + {x}^{-0.5} \cdot \frac{y}{3}\right)\right) + \mathsf{fma}\left(-\frac{y}{3}, {x}^{-0.5}, \frac{y}{3} \cdot {x}^{-0.5}\right) \]

      13. div-inv 84.4%

          \[\leadsto \left(1 - \left(\frac{0.1111111111111111}{x} + {x}^{-0.5} \cdot \color{blue}{\left(y \cdot \frac{1}{3}\right)}\right)\right) + \mathsf{fma}\left(-\frac{y}{3}, {x}^{-0.5}, \frac{y}{3} \cdot {x}^{-0.5}\right) \]

      14. metadata-eval 84.4%

          \[\leadsto \left(1 - \left(\frac{0.1111111111111111}{x} + {x}^{-0.5} \cdot \left(y \cdot \color{blue}{0.3333333333333333}\right)\right)\right) + \mathsf{fma}\left(-\frac{y}{3}, {x}^{-0.5}, \frac{y}{3} \cdot {x}^{-0.5}\right) \]

   5. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\left(1 - \left(\frac{0.1111111111111111}{x} + {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)\right) + \mathsf{fma}\left(-y \cdot 0.3333333333333333, {x}^{-0.5}, {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)} \]
   6. Step-by-step derivation

      1. associate--r+ 84.4%

         \[\leadsto \color{blue}{\left(\left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)} + \mathsf{fma}\left(-y \cdot 0.3333333333333333, {x}^{-0.5}, {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) \]

      2. unsub-neg 84.4%

         \[\leadsto \color{blue}{\left(\left(1 - \frac{0.1111111111111111}{x}\right) + \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)\right)} + \mathsf{fma}\left(-y \cdot 0.3333333333333333, {x}^{-0.5}, {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) \]

      3. +-commutative 84.4%

         \[\leadsto \color{blue}{\left(\left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \left(1 - \frac{0.1111111111111111}{x}\right)\right)} + \mathsf{fma}\left(-y \cdot 0.3333333333333333, {x}^{-0.5}, {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) \]

      4. associate-+l+ 84.4%

         \[\leadsto \color{blue}{\left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \left(\left(1 - \frac{0.1111111111111111}{x}\right) + \mathsf{fma}\left(-y \cdot 0.3333333333333333, {x}^{-0.5}, {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)\right)} \]

      5. fma-udef 84.4%

         \[\leadsto \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \left(\left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\left(\left(-y \cdot 0.3333333333333333\right) \cdot {x}^{-0.5} + {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)}\right) \]

      6. *-commutative 84.4%

         \[\leadsto \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \left(\left(1 - \frac{0.1111111111111111}{x}\right) + \left(\left(-y \cdot 0.3333333333333333\right) \cdot {x}^{-0.5} + \color{blue}{\left(y \cdot 0.3333333333333333\right) \cdot {x}^{-0.5}}\right)\right) \]

      7. distribute-rgt-out 99.5%

         \[\leadsto \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \left(\left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{{x}^{-0.5} \cdot \left(\left(-y \cdot 0.3333333333333333\right) + y \cdot 0.3333333333333333\right)}\right) \]

      8. distribute-lft-out 84.4%

         \[\leadsto \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \left(\left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\left({x}^{-0.5} \cdot \left(-y \cdot 0.3333333333333333\right) + {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)}\right) \]

      9. distribute-rgt-neg-in 84.4%

         \[\leadsto \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \left(\left(1 - \frac{0.1111111111111111}{x}\right) + \left(\color{blue}{\left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)} + {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)\right) \]

      10. +-commutative 84.4%

          \[\leadsto \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \left(\left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\left({x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right) + \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)\right)}\right) \]

      11. sub-neg 84.4%

          \[\leadsto \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \left(\left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\left({x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right) - {x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right)}\right) \]

      12. +-inverses 99.5%

          \[\leadsto \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \left(\left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{0}\right) \]

      13. +-rgt-identity 99.5%

          \[\leadsto \left(-{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)\right) + \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right)} \]

   7. Simplified 99.5%

      \[\leadsto \color{blue}{1 - \mathsf{fma}\left({x}^{-0.5}, 0.3333333333333333 \cdot y, \frac{0.1111111111111111}{x}\right)} \]

   8. Taylor expanded in x around inf 97.9%

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

      1. associate-*r* 97.9%

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

   10. Simplified 97.9%

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

   if -7.7999999999999999e52 < y < 1.4e20

   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. *-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%

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

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 98.3%

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

      1. associate-*r/ 98.3%

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

      2. metadata-eval 98.3%

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

   8. Simplified 98.3%

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

      1. div-inv 98.3%

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

      2. metadata-eval 98.3%

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

      3. inv-pow 98.3%

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

      4. unpow-prod-down 98.4%

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

      5. *-commutative 98.4%

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+52} \lor \neg \left(y \leq 1.4 \cdot 10^{+20}\right):\\ \;\;\;\;1 - \left(y \cdot 0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.5%

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

   8. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.5%

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

   8. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. associate-*r/ 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 99.6% 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 99.6%

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

   1. *-commutative 99.6%

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

   2. metadata-eval 99.6%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

3. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

5. Simplified 99.7%

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

   1. sub-neg 99.7%

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

7. Applied egg-rr 99.7%

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

   1. sub-neg 99.7%

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

   2. sub-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 6: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+81}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e+72)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 9e+81)
     (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))
     (* y (* -0.3333333333333333 (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+72) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 9e+81) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = y * (-0.3333333333333333 * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.2d+72)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 9d+81) then
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    else
        tmp = y * ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+72) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 9e+81) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = y * (-0.3333333333333333 * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e+72:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 9e+81:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	else:
		tmp = y * (-0.3333333333333333 * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e+72)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 9e+81)
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.2e+72)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 9e+81)
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	else
		tmp = y * (-0.3333333333333333 * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e+72], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e+81], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.2000000000000003e72

   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. *-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%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 91.8%

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

      1. associate-*r* 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*l* 91.8%

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

   8. Simplified 91.8%

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

      1. expm1-log1p-u 85.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

      2. expm1-udef 85.2%

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

      3. *-commutative 85.2%

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

      4. inv-pow 85.2%

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

      5. metadata-eval 85.2%

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

      6. pow-prod-up 85.2%

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

      7. sqrt-prod 85.2%

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

      8. add-sqr-sqrt 85.2%

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

   10. Applied egg-rr 85.2%

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

       1. expm1-def 85.4%

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

       2. expm1-log1p 91.8%

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

       3. associate-*r* 91.9%

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

       4. *-commutative 91.9%

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

       5. associate-*r* 91.9%

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

   12. Simplified 91.9%

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

       1. *-commutative 91.9%

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

       2. metadata-eval 91.9%

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

       3. pow-flip 91.8%

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

       4. pow1/2 91.8%

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

       5. div-inv 92.0%

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

       6. associate-/l* 92.0%

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

       7. div-inv 92.0%

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

       8. metadata-eval 92.0%

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

   14. Applied egg-rr 92.0%

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

   if -4.2000000000000003e72 < y < 9.00000000000000034e81

   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. distribute-frac-neg 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 95.2%

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

   if 9.00000000000000034e81 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      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%

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

   3. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around inf 94.6%

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

      1. associate-*r* 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-*l* 94.8%

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

   8. Simplified 94.8%

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

      1. expm1-log1p-u 46.6%

         \[\leadsto y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 1.2%

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

      3. *-commutative 1.2%

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

      4. inv-pow 1.2%

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

      5. metadata-eval 1.2%

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

      6. pow-prod-up 1.2%

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

      7. sqrt-prod 1.2%

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

      8. add-sqr-sqrt 1.2%

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

   10. Applied egg-rr 1.2%

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

       1. expm1-def 46.7%

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

       2. expm1-log1p 94.9%

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

       3. *-commutative 94.9%

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

   12. Simplified 94.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+81}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 7: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+84}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+71)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 7e+84)
     (- 1.0 (pow (* x 9.0) -1.0))
     (* y (* -0.3333333333333333 (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+71) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 7e+84) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = y * (-0.3333333333333333 * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.8d+71)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 7d+84) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = y * ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+71) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 7e+84) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = y * (-0.3333333333333333 * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+71:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 7e+84:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = y * (-0.3333333333333333 * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+71)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 7e+84)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.8e+71)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 7e+84)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = y * (-0.3333333333333333 * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+71], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+84], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.79999999999999961e71

   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. *-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%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 91.8%

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

      1. associate-*r* 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*l* 91.8%

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

   8. Simplified 91.8%

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

      1. expm1-log1p-u 85.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

      2. expm1-udef 85.2%

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

      3. *-commutative 85.2%

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

      4. inv-pow 85.2%

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

      5. metadata-eval 85.2%

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

      6. pow-prod-up 85.2%

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

      7. sqrt-prod 85.2%

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

      8. add-sqr-sqrt 85.2%

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

   10. Applied egg-rr 85.2%

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

       1. expm1-def 85.4%

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

       2. expm1-log1p 91.8%

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

       3. associate-*r* 91.9%

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

       4. *-commutative 91.9%

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

       5. associate-*r* 91.9%

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

   12. Simplified 91.9%

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

       1. *-commutative 91.9%

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

       2. metadata-eval 91.9%

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

       3. pow-flip 91.8%

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

       4. pow1/2 91.8%

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

       5. div-inv 92.0%

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

       6. associate-/l* 92.0%

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

       7. div-inv 92.0%

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

       8. metadata-eval 92.0%

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

   14. Applied egg-rr 92.0%

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

   if -4.79999999999999961e71 < y < 6.9999999999999998e84

   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. *-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%

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

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 95.2%

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

      1. associate-*r/ 95.2%

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

      2. metadata-eval 95.2%

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

   8. Simplified 95.2%

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

      1. div-inv 95.2%

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

      2. metadata-eval 95.2%

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

      3. inv-pow 95.2%

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

      4. unpow-prod-down 95.3%

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

      5. *-commutative 95.3%

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

   10. Applied egg-rr 95.3%

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

   if 6.9999999999999998e84 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      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%

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

   3. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around inf 94.6%

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

      1. associate-*r* 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-*l* 94.8%

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

   8. Simplified 94.8%

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

      1. expm1-log1p-u 46.6%

         \[\leadsto y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 1.2%

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

      3. *-commutative 1.2%

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

      4. inv-pow 1.2%

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

      5. metadata-eval 1.2%

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

      6. pow-prod-up 1.2%

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

      7. sqrt-prod 1.2%

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

      8. add-sqr-sqrt 1.2%

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

   10. Applied egg-rr 1.2%

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

       1. expm1-def 46.7%

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

       2. expm1-log1p 94.9%

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

       3. *-commutative 94.9%

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

   12. Simplified 94.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+84}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 8: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+71}:\\ \;\;\;\;\frac{-{x}^{-0.5}}{\frac{3}{y}}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+83}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8e+71)
   (/ (- (pow x -0.5)) (/ 3.0 y))
   (if (<= y 5.3e+83)
     (- 1.0 (pow (* x 9.0) -1.0))
     (* y (* -0.3333333333333333 (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8e+71) {
		tmp = -pow(x, -0.5) / (3.0 / y);
	} else if (y <= 5.3e+83) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = y * (-0.3333333333333333 * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8d+71)) then
        tmp = -(x ** (-0.5d0)) / (3.0d0 / y)
    else if (y <= 5.3d+83) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = y * ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8e+71) {
		tmp = -Math.pow(x, -0.5) / (3.0 / y);
	} else if (y <= 5.3e+83) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = y * (-0.3333333333333333 * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8e+71:
		tmp = -math.pow(x, -0.5) / (3.0 / y)
	elif y <= 5.3e+83:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = y * (-0.3333333333333333 * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8e+71)
		tmp = Float64(Float64(-(x ^ -0.5)) / Float64(3.0 / y));
	elseif (y <= 5.3e+83)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8e+71)
		tmp = -(x ^ -0.5) / (3.0 / y);
	elseif (y <= 5.3e+83)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = y * (-0.3333333333333333 * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8e+71], N[((-N[Power[x, -0.5], $MachinePrecision]) / N[(3.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.3e+83], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.0000000000000003e71

   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. *-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%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 91.8%

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

      1. associate-*r* 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*l* 91.8%

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

   8. Simplified 91.8%

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

      1. expm1-log1p-u 85.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

      2. expm1-udef 85.2%

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

      3. *-commutative 85.2%

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

      4. inv-pow 85.2%

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

      5. metadata-eval 85.2%

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

      6. pow-prod-up 85.2%

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

      7. sqrt-prod 85.2%

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

      8. add-sqr-sqrt 85.2%

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

   10. Applied egg-rr 85.2%

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

       1. expm1-def 85.4%

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

       2. expm1-log1p 91.8%

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

       3. associate-*r* 91.9%

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

       4. *-commutative 91.9%

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

       5. associate-*r* 91.9%

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

   12. Simplified 91.9%

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

       1. add-sqr-sqrt 91.6%

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

       2. sqrt-unprod 43.8%

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

       3. swap-sqr 43.8%

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

       4. metadata-eval 43.8%

          \[\leadsto {x}^{-0.5} \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{0.1111111111111111}} \]

       5. metadata-eval 43.8%

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

       6. swap-sqr 43.8%

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

       7. metadata-eval 43.8%

          \[\leadsto {x}^{-0.5} \cdot \sqrt{\left(y \cdot \color{blue}{\frac{1}{3}}\right) \cdot \left(y \cdot 0.3333333333333333\right)} \]

       8. div-inv 43.9%

          \[\leadsto {x}^{-0.5} \cdot \sqrt{\color{blue}{\frac{y}{3}} \cdot \left(y \cdot 0.3333333333333333\right)} \]

       9. metadata-eval 43.9%

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

       10. div-inv 43.9%

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

       11. sqrt-unprod 0.0%

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

       12. add-sqr-sqrt 0.7%

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

       13. frac-2neg 0.7%

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

       14. div-inv 0.7%

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

       15. metadata-eval 0.7%

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

       16. metadata-eval 0.7%

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

       17. distribute-lft-neg-in 0.7%

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

       18. add-sqr-sqrt 0.7%

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

       19. sqrt-unprod 0.6%

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

       20. swap-sqr 0.6%

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

       21. metadata-eval 0.6%

           \[\leadsto {x}^{-0.5} \cdot \left(-\sqrt{\left(y \cdot y\right) \cdot \color{blue}{0.1111111111111111}}\right) \]

       22. metadata-eval 0.6%

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

       23. swap-sqr 0.6%

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

   14. Applied egg-rr 92.1%

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

   if -8.0000000000000003e71 < y < 5.29999999999999964e83

   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. *-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%

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

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 95.2%

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

      1. associate-*r/ 95.2%

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

      2. metadata-eval 95.2%

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

   8. Simplified 95.2%

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

      1. div-inv 95.2%

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

      2. metadata-eval 95.2%

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

      3. inv-pow 95.2%

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

      4. unpow-prod-down 95.3%

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

      5. *-commutative 95.3%

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

   10. Applied egg-rr 95.3%

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

   if 5.29999999999999964e83 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      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%

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

   3. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around inf 94.6%

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

      1. associate-*r* 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-*l* 94.8%

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

   8. Simplified 94.8%

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

      1. expm1-log1p-u 46.6%

         \[\leadsto y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 1.2%

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

      3. *-commutative 1.2%

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

      4. inv-pow 1.2%

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

      5. metadata-eval 1.2%

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

      6. pow-prod-up 1.2%

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

      7. sqrt-prod 1.2%

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

      8. add-sqr-sqrt 1.2%

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

   10. Applied egg-rr 1.2%

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

       1. expm1-def 46.7%

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

       2. expm1-log1p 94.9%

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

       3. *-commutative 94.9%

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

   12. Simplified 94.9%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+71}:\\ \;\;\;\;\frac{-{x}^{-0.5}}{\frac{3}{y}}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+83}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 9: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+74} \lor \neg \left(y \leq 9.6 \cdot 10^{+80}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.5e+74) (not (<= y 9.6e+80)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.5e+74) || !(y <= 9.6e+80)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-5.5d+74)) .or. (.not. (y <= 9.6d+80))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5.5e+74) || !(y <= 9.6e+80)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.5e+74) or not (y <= 9.6e+80):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.5e+74) || !(y <= 9.6e+80))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.5e+74) || ~((y <= 9.6e+80)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.5e+74], N[Not[LessEqual[y, 9.6e+80]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000003e74 or 9.59999999999999916e80 < 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. *-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%

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

   3. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around inf 93.1%

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

      1. associate-*r* 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-*l* 93.2%

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

   8. Simplified 93.2%

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

      1. expm1-log1p-u 45.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

      2. expm1-udef 45.6%

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

      3. *-commutative 45.6%

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

      4. inv-pow 45.6%

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

      5. metadata-eval 45.6%

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

      6. pow-prod-up 45.6%

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

      7. sqrt-prod 45.6%

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

      8. add-sqr-sqrt 45.6%

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

   10. Applied egg-rr 45.6%

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

       1. expm1-def 45.7%

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

       2. expm1-log1p 93.3%

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

       3. associate-*r* 93.2%

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

       4. *-commutative 93.2%

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

       5. associate-*r* 93.2%

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

   12. Simplified 93.2%

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

       1. add-sqr-sqrt 49.0%

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

       2. sqrt-unprod 23.8%

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

       3. swap-sqr 23.8%

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

       4. metadata-eval 23.8%

          \[\leadsto {x}^{-0.5} \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{0.1111111111111111}} \]

       5. metadata-eval 23.8%

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

       6. swap-sqr 23.8%

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

       7. metadata-eval 23.8%

          \[\leadsto {x}^{-0.5} \cdot \sqrt{\left(y \cdot \color{blue}{\frac{1}{3}}\right) \cdot \left(y \cdot 0.3333333333333333\right)} \]

       8. div-inv 23.8%

          \[\leadsto {x}^{-0.5} \cdot \sqrt{\color{blue}{\frac{y}{3}} \cdot \left(y \cdot 0.3333333333333333\right)} \]

       9. metadata-eval 23.8%

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

       10. div-inv 23.8%

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

       11. sqrt-unprod 0.4%

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

       12. add-sqr-sqrt 0.8%

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

       13. div-inv 0.8%

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

       14. metadata-eval 0.8%

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

       15. metadata-eval 0.8%

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

       16. distribute-rgt-neg-in 0.8%

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

       17. add-sqr-sqrt 0.4%

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

       18. sqrt-unprod 21.3%

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

       19. swap-sqr 21.3%

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

       20. metadata-eval 21.3%

           \[\leadsto {x}^{-0.5} \cdot \left(-\sqrt{\left(y \cdot y\right) \cdot \color{blue}{0.1111111111111111}}\right) \]

       21. metadata-eval 21.3%

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

       22. swap-sqr 21.3%

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

       23. metadata-eval 21.3%

           \[\leadsto {x}^{-0.5} \cdot \left(-\sqrt{\left(y \cdot \color{blue}{\frac{1}{3}}\right) \cdot \left(y \cdot 0.3333333333333333\right)}\right) \]

       24. div-inv 21.3%

           \[\leadsto {x}^{-0.5} \cdot \left(-\sqrt{\color{blue}{\frac{y}{3}} \cdot \left(y \cdot 0.3333333333333333\right)}\right) \]

       25. metadata-eval 21.3%

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

       26. div-inv 21.3%

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

   14. Applied egg-rr 93.2%

       \[\leadsto \color{blue}{0 - y \cdot \frac{0.3333333333333333}{\sqrt{x}}} \]
   15. Step-by-step derivation

       1. neg-sub0 93.2%

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

       2. distribute-rgt-neg-in 93.2%

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

       3. distribute-neg-frac 93.2%

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

       4. metadata-eval 93.2%

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

       5. associate-*r/ 93.3%

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

       6. associate-*l/ 93.2%

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

       7. *-commutative 93.2%

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

   16. Simplified 93.2%

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

   if -5.5000000000000003e74 < y < 9.59999999999999916e80

   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. distribute-frac-neg 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 95.2%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+74} \lor \neg \left(y \leq 9.6 \cdot 10^{+80}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]

Alternative 10: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+72} \lor \neg \left(y \leq 1.6 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7e+72) (not (<= y 1.6e+89)))
   (/ y (* (sqrt x) -3.0))
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7e+72) || !(y <= 1.6e+89)) {
		tmp = y / (sqrt(x) * -3.0);
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7d+72)) .or. (.not. (y <= 1.6d+89))) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7e+72) || !(y <= 1.6e+89)) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7e+72) or not (y <= 1.6e+89):
		tmp = y / (math.sqrt(x) * -3.0)
	else:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7e+72) || !(y <= 1.6e+89))
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	else
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7e+72) || ~((y <= 1.6e+89)))
		tmp = y / (sqrt(x) * -3.0);
	else
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7e+72], N[Not[LessEqual[y, 1.6e+89]], $MachinePrecision]], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.0000000000000002e72 or 1.59999999999999994e89 < 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. *-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%

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

   3. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around inf 93.1%

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

      1. associate-*r* 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-*l* 93.2%

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

   8. Simplified 93.2%

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

      1. expm1-log1p-u 45.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

      2. expm1-udef 45.6%

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

      3. *-commutative 45.6%

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

      4. inv-pow 45.6%

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

      5. metadata-eval 45.6%

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

      6. pow-prod-up 45.6%

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

      7. sqrt-prod 45.6%

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

      8. add-sqr-sqrt 45.6%

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

   10. Applied egg-rr 45.6%

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

       1. expm1-def 45.7%

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

       2. expm1-log1p 93.3%

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

       3. associate-*r* 93.2%

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

       4. *-commutative 93.2%

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

       5. associate-*r* 93.2%

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

   12. Simplified 93.2%

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

       1. *-commutative 93.2%

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

       2. metadata-eval 93.2%

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

       3. pow-flip 93.2%

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

       4. pow1/2 93.2%

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

       5. div-inv 93.3%

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

       6. associate-/l* 93.2%

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

       7. div-inv 93.3%

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

       8. metadata-eval 93.3%

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

   14. Applied egg-rr 93.3%

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

   if -7.0000000000000002e72 < y < 1.59999999999999994e89

   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. distribute-frac-neg 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 95.2%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+72} \lor \neg \left(y \leq 1.6 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]

Alternative 11: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+88}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e+74)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 2.1e+88)
     (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e+74) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 2.1e+88) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6d+74)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 2.1d+88) then
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6e+74) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 2.1e+88) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6e+74:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 2.1e+88:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e+74)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 2.1e+88)
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6e+74)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 2.1e+88)
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e+74], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+88], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6e74

   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. *-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%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 91.8%

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

      1. associate-*r* 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*l* 91.8%

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

   8. Simplified 91.8%

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

      1. expm1-log1p-u 85.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

      2. expm1-udef 85.2%

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

      3. *-commutative 85.2%

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

      4. inv-pow 85.2%

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

      5. metadata-eval 85.2%

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

      6. pow-prod-up 85.2%

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

      7. sqrt-prod 85.2%

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

      8. add-sqr-sqrt 85.2%

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

   10. Applied egg-rr 85.2%

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

       1. expm1-def 85.4%

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

       2. expm1-log1p 91.8%

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

       3. associate-*r* 91.9%

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

       4. *-commutative 91.9%

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

       5. associate-*r* 91.9%

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

   12. Simplified 91.9%

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

       1. *-commutative 91.9%

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

       2. metadata-eval 91.9%

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

       3. pow-flip 91.8%

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

       4. pow1/2 91.8%

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

       5. div-inv 92.0%

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

       6. associate-/l* 92.0%

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

       7. div-inv 92.0%

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

       8. metadata-eval 92.0%

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

   14. Applied egg-rr 92.0%

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

   if -6e74 < y < 2.1e88

   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. distribute-frac-neg 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 95.2%

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

   if 2.1e88 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      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%

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

   3. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around inf 94.6%

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

      1. associate-*r* 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-*l* 94.8%

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

   8. Simplified 94.8%

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

      1. expm1-log1p-u 0.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

      2. expm1-udef 0.1%

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

      3. *-commutative 0.1%

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

      4. inv-pow 0.1%

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

      5. metadata-eval 0.1%

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

      6. pow-prod-up 0.1%

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

      7. sqrt-prod 0.1%

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

      8. add-sqr-sqrt 0.1%

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

   10. Applied egg-rr 0.1%

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

       1. expm1-def 0.1%

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

       2. expm1-log1p 94.9%

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

       3. associate-*r* 94.7%

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

       4. *-commutative 94.7%

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

       5. associate-*r* 94.8%

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

   12. Simplified 94.8%

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

       1. *-commutative 94.8%

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

       2. metadata-eval 94.8%

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

       3. pow-flip 94.7%

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

       4. pow1/2 94.7%

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

       5. div-inv 94.9%

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

   14. Applied egg-rr 94.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+88}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 12: 64.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e+154)
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))
   (/
    (- 1.0 (/ 0.012345679012345678 (* x x)))
    (+ 1.0 (/ 0.1111111111111111 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.4e+154) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (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 <= 1.4d+154) then
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    else
        tmp = (1.0d0 - (0.012345679012345678d0 / (x * x))) / (1.0d0 + (0.1111111111111111d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.4e+154) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (1.0 + (0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.4e+154:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	else:
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (1.0 + (0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.4e+154)
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	else
		tmp = Float64(Float64(1.0 - Float64(0.012345679012345678 / Float64(x * x))) / Float64(1.0 + Float64(0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e+154)
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	else
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (1.0 + (0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.4e+154], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4e154

   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. distribute-frac-neg 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 69.2%

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

   if 1.4e154 < y

   1. Initial program 99.4%

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

   3. Applied egg-rr 86.4%

      \[\leadsto \color{blue}{\frac{\left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \left(\sqrt{x} \cdot 3\right) - \left(\frac{0.1111111111111111}{x} + 1\right) \cdot y}{\left(\frac{0.1111111111111111}{x} + 1\right) \cdot \left(\sqrt{x} \cdot 3\right)}} \]

   4. Taylor expanded in y around 0 25.5%

      \[\leadsto \color{blue}{\frac{1 - 0.012345679012345678 \cdot \frac{1}{{x}^{2}}}{1 + 0.1111111111111111 \cdot \frac{1}{x}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 25.5%

         \[\leadsto \frac{1 - \color{blue}{\frac{0.012345679012345678 \cdot 1}{{x}^{2}}}}{1 + 0.1111111111111111 \cdot \frac{1}{x}} \]

      2. metadata-eval 25.5%

         \[\leadsto \frac{1 - \frac{\color{blue}{0.012345679012345678}}{{x}^{2}}}{1 + 0.1111111111111111 \cdot \frac{1}{x}} \]

      3. unpow2 25.5%

         \[\leadsto \frac{1 - \frac{0.012345679012345678}{\color{blue}{x \cdot x}}}{1 + 0.1111111111111111 \cdot \frac{1}{x}} \]

      4. associate-*r/ 25.5%

         \[\leadsto \frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}}} \]

      5. metadata-eval 25.5%

         \[\leadsto \frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{\color{blue}{0.1111111111111111}}{x}} \]

   6. Simplified 25.5%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}}\\ \end{array} \]

Alternative 13: 61.0% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 34500000000000:\\ \;\;\;\;\frac{1}{x} \cdot -0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 34500000000000.0) (* (/ 1.0 x) -0.1111111111111111) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 34500000000000.0) {
		tmp = (1.0 / x) * -0.1111111111111111;
	} 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 <= 34500000000000.0d0) then
        tmp = (1.0d0 / x) * (-0.1111111111111111d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 34500000000000.0) {
		tmp = (1.0 / x) * -0.1111111111111111;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 34500000000000.0:
		tmp = (1.0 / x) * -0.1111111111111111
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 34500000000000.0)
		tmp = Float64(Float64(1.0 / x) * -0.1111111111111111);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 34500000000000.0)
		tmp = (1.0 / x) * -0.1111111111111111;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 34500000000000.0], N[(N[(1.0 / x), $MachinePrecision] * -0.1111111111111111), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.45e13

   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%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. neg-mul-1 99.4%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 58.7%

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

      1. div-inv 58.7%

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

   6. Applied egg-rr 58.7%

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

   if 3.45e13 < 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. distribute-frac-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r* 99.8%

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

      5. metadata-eval 99.8%

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

      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 99.6%

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

      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. Taylor expanded in x around inf 61.9%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 34500000000000:\\ \;\;\;\;\frac{1}{x} \cdot -0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 62.5% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.5%

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

   8. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 60.5%

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

5. Final simplification 60.5%

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

Alternative 15: 61.0% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 34500000000000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 34500000000000.0) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 34500000000000.0) {
		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 <= 34500000000000.0d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 34500000000000.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 34500000000000.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 34500000000000.0)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 34500000000000.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 34500000000000.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.45e13

   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%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. neg-mul-1 99.4%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 58.7%

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

   if 3.45e13 < 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. distribute-frac-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r* 99.8%

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

      5. metadata-eval 99.8%

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

      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 99.6%

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

      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. Taylor expanded in x around inf 61.9%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 34500000000000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16: 62.5% 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 99.6%

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

   1. sub-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.5%

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

   8. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 60.5%

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

   1. cancel-sign-sub-inv 60.5%

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

   2. metadata-eval 60.5%

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

   3. associate-*r/ 60.5%

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

   4. metadata-eval 60.5%

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

6. Simplified 60.5%

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

7. Final simplification 60.5%

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

Alternative 17: 31.5% 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 99.6%

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

   1. sub-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.5%

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

   8. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around inf 29.0%

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

5. Final simplification 29.0%

   \[\leadsto 1 \]

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 2023268 
(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)))))