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

Percentage Accurate: 99.7% → 99.7%

Time: 8.7s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Final simplification 99.7%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+48}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.1e+48)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 9.5e+26)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (+ 1.0 (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.1e+48) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 9.5e+26) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (sqrt((1.0 / x)) * (y * -0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.1d+48)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= 9.5d+26) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.1e+48) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 9.5e+26) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 / x)) * (y * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.1e+48:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 9.5e+26:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 + (math.sqrt((1.0 / x)) * (y * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.1e+48)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 9.5e+26)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.1e+48)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 9.5e+26)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 + (sqrt((1.0 / x)) * (y * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.1e+48], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e+26], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1000000000000003e48

   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. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*r/ 99.4%

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

      9. fma-def 99.4%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.5%

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

      14. distribute-neg-frac 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*l* 97.6%

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

      4. *-commutative 97.6%

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

   6. Simplified 97.6%

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

      1. expm1-log1p-u 91.2%

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

      2. expm1-udef 91.2%

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

      3. sqrt-div 91.2%

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

      4. metadata-eval 91.2%

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

      5. associate-*l/ 91.2%

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

      6. *-un-lft-identity 91.2%

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

   8. Applied egg-rr 91.2%

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

      1. expm1-def 91.2%

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

      2. expm1-log1p 97.7%

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

      3. associate-/l* 97.6%

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

   10. Simplified 97.6%

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

   if -4.1000000000000003e48 < y < 9.50000000000000054e26

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 98.6%

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

      1. div-inv 98.7%

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

      2. clear-num 98.6%

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

      3. div-inv 98.7%

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

      4. metadata-eval 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 9.50000000000000054e26 < y

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.3%

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

      9. fma-def 99.3%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 87.4%

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

      1. *-commutative 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-*l* 87.5%

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

      4. *-commutative 87.5%

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

   6. Simplified 87.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+48}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 3: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+50} \lor \neg \left(y \leq 4.8 \cdot 10^{+30}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.9e+50) (not (<= y 4.8e+30)))
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e+50) || !(y <= 4.8e+30)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.9d+50)) .or. (.not. (y <= 4.8d+30))) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e+50) || !(y <= 4.8e+30)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.9e+50) or not (y <= 4.8e+30):
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.9e+50) || !(y <= 4.8e+30))
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.9e+50) || ~((y <= 4.8e+30)))
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.9e+50], N[Not[LessEqual[y, 4.8e+30]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e50 or 4.7999999999999999e30 < y

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.3%

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

      9. fma-def 99.3%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 92.3%

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

      1. *-commutative 92.3%

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

      2. *-commutative 92.3%

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

      3. associate-*l* 92.4%

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

      4. *-commutative 92.4%

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

   6. Simplified 92.4%

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

      1. expm1-log1p-u 48.0%

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

      2. expm1-udef 48.0%

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

      3. sqrt-div 48.0%

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

      4. metadata-eval 48.0%

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

      5. associate-*l/ 48.0%

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

      6. *-un-lft-identity 48.0%

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

   8. Applied egg-rr 48.0%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 92.5%

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

      3. *-commutative 92.5%

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

      4. associate-*r/ 92.3%

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

   10. Simplified 92.3%

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

   if -2.9e50 < y < 4.7999999999999999e30

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 98.6%

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

      1. div-inv 98.7%

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

      2. clear-num 98.6%

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

      3. div-inv 98.7%

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

      4. metadata-eval 98.7%

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

   6. Applied egg-rr 98.7%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+50} \lor \neg \left(y \leq 4.8 \cdot 10^{+30}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 4: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+37} \lor \neg \left(y \leq 4 \cdot 10^{+30}\right):\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.1e+37) (not (<= y 4e+30)))
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.1e+37) || !(y <= 4e+30)) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.1d+37)) .or. (.not. (y <= 4d+30))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.1e+37) || !(y <= 4e+30)) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.1e+37) or not (y <= 4e+30):
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.1e+37) || !(y <= 4e+30))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.1e+37) || ~((y <= 4e+30)))
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.1e+37], N[Not[LessEqual[y, 4e+30]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0999999999999998e37 or 4.0000000000000001e30 < y

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.3%

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

      9. fma-def 99.3%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 92.3%

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

      1. *-commutative 92.3%

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

      2. *-commutative 92.3%

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

      3. associate-*l* 92.4%

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

      4. *-commutative 92.4%

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

   6. Simplified 92.4%

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

      1. expm1-log1p-u 48.0%

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

      2. expm1-udef 48.0%

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

      3. sqrt-div 48.0%

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

      4. metadata-eval 48.0%

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

      5. associate-*l/ 48.0%

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

      6. *-un-lft-identity 48.0%

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

   8. Applied egg-rr 48.0%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 92.5%

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

      3. associate-/l* 92.4%

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

   10. Simplified 92.4%

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

   if -4.0999999999999998e37 < y < 4.0000000000000001e30

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 98.6%

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

      1. div-inv 98.7%

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

      2. clear-num 98.6%

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

      3. div-inv 98.7%

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

      4. metadata-eval 98.7%

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

   6. Applied egg-rr 98.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+37} \lor \neg \left(y \leq 4 \cdot 10^{+30}\right):\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 1.0 - ((0.1111111111111111 / x) + ((y / 3.0) / sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - ((0.1111111111111111d0 / x) + ((y / 3.0d0) / sqrt(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-/r* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 99.7%

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

5. Final simplification 99.7%

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

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) - (y / (3.0d0 * sqrt(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/r* 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 7: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+54} \lor \neg \left(y \leq 1.2 \cdot 10^{+104}\right):\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.5e+54) (not (<= y 1.2e+104)))
   (* (* y -0.3333333333333333) (pow x -0.5))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.5e+54) || !(y <= 1.2e+104)) {
		tmp = (y * -0.3333333333333333) * pow(x, -0.5);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.5d+54)) .or. (.not. (y <= 1.2d+104))) then
        tmp = (y * (-0.3333333333333333d0)) * (x ** (-0.5d0))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.5e+54) || !(y <= 1.2e+104)) {
		tmp = (y * -0.3333333333333333) * Math.pow(x, -0.5);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.5e+54) or not (y <= 1.2e+104):
		tmp = (y * -0.3333333333333333) * math.pow(x, -0.5)
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.5e+54) || !(y <= 1.2e+104))
		tmp = Float64(Float64(y * -0.3333333333333333) * (x ^ -0.5));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.5e+54) || ~((y <= 1.2e+104)))
		tmp = (y * -0.3333333333333333) * (x ^ -0.5);
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.5e+54], N[Not[LessEqual[y, 1.2e+104]], $MachinePrecision]], N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.50000000000000003e54 or 1.2e104 < y

   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. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in y around inf 96.7%

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

      1. associate-*r* 96.8%

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

   7. Simplified 96.8%

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

      1. expm1-log1p-u 95.4%

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

      2. expm1-udef 50.4%

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

      3. inv-pow 50.4%

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

      4. sqrt-pow1 50.4%

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

      5. metadata-eval 50.4%

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

   9. Applied egg-rr 50.4%

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

       1. expm1-def 95.4%

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

       2. expm1-log1p 96.9%

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

   11. Simplified 96.9%

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

   if -2.50000000000000003e54 < y < 1.2e104

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 92.9%

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

      1. div-inv 92.9%

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

      2. clear-num 92.9%

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

      3. div-inv 92.9%

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

      4. metadata-eval 92.9%

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

   6. Applied egg-rr 92.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+54} \lor \neg \left(y \leq 1.2 \cdot 10^{+104}\right):\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 8: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{-y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.3e+54)
   (/ (- y) (* 3.0 (sqrt x)))
   (if (<= y 1.2e+104)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* (* y -0.3333333333333333) (pow x -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.3e+54) {
		tmp = -y / (3.0 * sqrt(x));
	} else if (y <= 1.2e+104) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} 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 <= (-2.3d+54)) then
        tmp = -y / (3.0d0 * sqrt(x))
    else if (y <= 1.2d+104) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    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 <= -2.3e+54) {
		tmp = -y / (3.0 * Math.sqrt(x));
	} else if (y <= 1.2e+104) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.3e+54:
		tmp = -y / (3.0 * math.sqrt(x))
	elif y <= 1.2e+104:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (y * -0.3333333333333333) * math.pow(x, -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.3e+54)
		tmp = Float64(Float64(-y) / Float64(3.0 * sqrt(x)));
	elseif (y <= 1.2e+104)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.3e+54)
		tmp = -y / (3.0 * sqrt(x));
	elseif (y <= 1.2e+104)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (y * -0.3333333333333333) * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.3e+54], N[((-y) / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+104], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.29999999999999994e54

   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. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in y around inf 95.7%

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

      1. associate-*r* 95.8%

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

   7. Simplified 95.8%

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

      1. associate-*l* 95.7%

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

      2. sqrt-div 95.7%

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

      3. metadata-eval 95.7%

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

      4. div-inv 95.9%

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

      5. frac-2neg 95.9%

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

      6. associate-*r/ 95.9%

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

   9. Applied egg-rr 95.9%

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

       1. distribute-rgt-neg-out 95.9%

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

       2. *-commutative 95.9%

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

       3. distribute-frac-neg 95.9%

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

       4. metadata-eval 95.9%

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

       5. distribute-rgt-neg-in 95.9%

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

       6. distribute-lft-neg-out 95.9%

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

       7. associate-/l* 95.9%

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

       8. associate-/r/ 95.9%

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

       9. metadata-eval 95.9%

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

       10. times-frac 95.9%

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

       11. *-rgt-identity 95.9%

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

       12. associate-/r* 95.9%

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

       13. neg-mul-1 95.9%

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

       14. neg-mul-1 95.9%

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

       15. times-frac 95.9%

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

       16. metadata-eval 95.9%

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

       17. *-lft-identity 95.9%

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

       18. distribute-frac-neg 95.9%

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

       19. distribute-frac-neg 95.9%

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

       20. associate-/l/ 95.9%

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

       21. *-commutative 95.9%

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

   11. Simplified 95.9%

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

   if -2.29999999999999994e54 < y < 1.2e104

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 92.9%

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

      1. div-inv 92.9%

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

      2. clear-num 92.9%

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

      3. div-inv 92.9%

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

      4. metadata-eval 92.9%

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

   6. Applied egg-rr 92.9%

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

   if 1.2e104 < y

   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. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 99.5%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. associate-*r* 98.2%

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

   7. Simplified 98.2%

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

      1. expm1-log1p-u 96.6%

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

      2. expm1-udef 47.8%

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

      3. inv-pow 47.8%

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

      4. sqrt-pow1 47.8%

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

      5. metadata-eval 47.8%

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

   9. Applied egg-rr 47.8%

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

       1. expm1-def 96.6%

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

       2. expm1-log1p 98.3%

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

   11. Simplified 98.3%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{-y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 9: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+54} \lor \neg \left(y \leq 1.2 \cdot 10^{+104}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.5e+54) (not (<= y 1.2e+104)))
   (* y (/ -0.3333333333333333 (sqrt x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.5e+54) || !(y <= 1.2e+104)) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.5d+54)) .or. (.not. (y <= 1.2d+104))) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.5e+54) || !(y <= 1.2e+104)) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.5e+54) or not (y <= 1.2e+104):
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.5e+54) || !(y <= 1.2e+104))
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.5e+54) || ~((y <= 1.2e+104)))
		tmp = y * (-0.3333333333333333 / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.5e+54], N[Not[LessEqual[y, 1.2e+104]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.50000000000000003e54 or 1.2e104 < y

   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. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in y around inf 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-*l* 96.7%

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

      3. unpow-1 96.7%

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

      4. sqr-pow 96.7%

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

      5. metadata-eval 96.7%

         \[\leadsto y \cdot \left(\sqrt{{x}^{\color{blue}{-0.5}} \cdot {x}^{\left(\frac{-1}{2}\right)}} \cdot -0.3333333333333333\right) \]

      6. metadata-eval 96.7%

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

      7. rem-sqrt-square 96.7%

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

      8. rem-square-sqrt 96.5%

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

      9. fabs-sqr 96.5%

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

      10. rem-square-sqrt 96.7%

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

   7. Simplified 96.7%

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

      1. expm1-log1p-u 47.3%

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

      2. expm1-udef 2.1%

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

      3. metadata-eval 2.1%

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

      4. sqrt-pow1 2.1%

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

      5. inv-pow 2.1%

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

      6. sqrt-div 2.1%

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

      7. metadata-eval 2.1%

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

      8. associate-*l/ 2.1%

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

      9. metadata-eval 2.1%

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

   9. Applied egg-rr 2.1%

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

       1. expm1-def 47.3%

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

       2. expm1-log1p 96.7%

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

   11. Simplified 96.7%

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

   if -2.50000000000000003e54 < y < 1.2e104

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 92.9%

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

      1. div-inv 92.9%

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

      2. clear-num 92.9%

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

      3. div-inv 92.9%

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

      4. metadata-eval 92.9%

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

   6. Applied egg-rr 92.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+54} \lor \neg \left(y \leq 1.2 \cdot 10^{+104}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 10: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+54} \lor \neg \left(y \leq 1.2 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2e+54) (not (<= y 1.2e+104)))
   (/ -0.3333333333333333 (/ (sqrt x) y))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2e+54) || !(y <= 1.2e+104)) {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2d+54)) .or. (.not. (y <= 1.2d+104))) then
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2e+54) || !(y <= 1.2e+104)) {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2e+54) or not (y <= 1.2e+104):
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2e+54) || !(y <= 1.2e+104))
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2e+54) || ~((y <= 1.2e+104)))
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2e+54], N[Not[LessEqual[y, 1.2e+104]], $MachinePrecision]], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0000000000000002e54 or 1.2e104 < y

   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. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in y around inf 96.7%

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

      1. associate-*r* 96.8%

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

   7. Simplified 96.8%

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

      1. associate-*l* 96.7%

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

      2. sqrt-div 96.7%

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

      3. metadata-eval 96.7%

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

      4. div-inv 96.8%

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

      5. clear-num 96.8%

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

      6. un-div-inv 96.8%

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

   9. Applied egg-rr 96.8%

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

   if -2.0000000000000002e54 < y < 1.2e104

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 92.9%

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

      1. div-inv 92.9%

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

      2. clear-num 92.9%

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

      3. div-inv 92.9%

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

      4. metadata-eval 92.9%

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

   6. Applied egg-rr 92.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+54} \lor \neg \left(y \leq 1.2 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 11: 64.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e+154)
   (+ 1.0 (/ -1.0 (* x 9.0)))
   (/
    (- 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 + (-1.0 / (x * 9.0));
	} 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 + ((-1.0d0) / (x * 9.0d0))
    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 + (-1.0 / (x * 9.0));
	} 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 + (-1.0 / (x * 9.0))
	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(-1.0 / Float64(x * 9.0)));
	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 + (-1.0 / (x * 9.0));
	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[(-1.0 / N[(x * 9.0), $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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.6%

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

      9. fma-def 99.6%

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

      10. associate-/r* 99.6%

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

      11. metadata-eval 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r* 99.6%

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

      14. distribute-neg-frac 99.6%

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

      15. metadata-eval 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 68.6%

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

      1. div-inv 68.6%

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

      2. clear-num 68.6%

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

      3. div-inv 68.6%

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

      4. metadata-eval 68.6%

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

   6. Applied egg-rr 68.6%

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

   if 1.4e154 < y

   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. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*r/ 99.1%

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

      9. fma-def 99.1%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 3.6%

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

      1. cancel-sign-sub-inv 3.6%

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

      2. metadata-eval 3.6%

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

      3. associate-*r/ 3.6%

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

      4. metadata-eval 3.6%

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

      5. +-commutative 3.6%

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

   6. Simplified 3.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.7%

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

      3. frac-times 0.7%

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

      4. metadata-eval 0.7%

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

      5. metadata-eval 0.7%

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

      6. frac-times 0.7%

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

      7. pow1 0.7%

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

      8. pow1 0.7%

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

      9. pow-prod-up 0.7%

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

      10. clear-num 0.7%

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

      11. div-inv 0.7%

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

      12. metadata-eval 0.7%

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

      13. pow-plus 0.7%

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

      14. pow1 0.7%

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

      15. sqrt-unprod 0.7%

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

      16. add-sqr-sqrt 0.7%

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

      17. inv-pow 0.7%

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

   8. Applied egg-rr 0.7%

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

      1. unpow-1 0.7%

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

   10. Simplified 0.7%

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

       1. +-commutative 0.7%

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

       2. flip-+ 0.7%

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

       3. metadata-eval 0.7%

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

       4. pow2 0.7%

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

       5. *-commutative 0.7%

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

       6. associate-/r* 0.7%

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

       7. metadata-eval 0.7%

          \[\leadsto \frac{1 - {\left(\frac{\color{blue}{0.1111111111111111}}{x}\right)}^{2}}{1 - \frac{1}{x \cdot 9}} \]

       8. metadata-eval 0.7%

          \[\leadsto \frac{1 - {\left(\frac{\color{blue}{--0.1111111111111111}}{x}\right)}^{2}}{1 - \frac{1}{x \cdot 9}} \]

       9. remove-double-neg 0.7%

          \[\leadsto \frac{1 - {\left(\frac{--0.1111111111111111}{\color{blue}{-\left(-x\right)}}\right)}^{2}}{1 - \frac{1}{x \cdot 9}} \]

       10. frac-2neg 0.7%

           \[\leadsto \frac{1 - {\color{blue}{\left(\frac{-0.1111111111111111}{-x}\right)}}^{2}}{1 - \frac{1}{x \cdot 9}} \]

       11. pow2 0.7%

           \[\leadsto \frac{1 - \color{blue}{\frac{-0.1111111111111111}{-x} \cdot \frac{-0.1111111111111111}{-x}}}{1 - \frac{1}{x \cdot 9}} \]

       12. frac-times 0.7%

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

       13. metadata-eval 0.7%

           \[\leadsto \frac{1 - \frac{\color{blue}{0.012345679012345678}}{\left(-x\right) \cdot \left(-x\right)}}{1 - \frac{1}{x \cdot 9}} \]

       14. sqr-neg 0.7%

           \[\leadsto \frac{1 - \frac{0.012345679012345678}{\color{blue}{x \cdot x}}}{1 - \frac{1}{x \cdot 9}} \]

       15. *-commutative 0.7%

           \[\leadsto \frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 - \frac{1}{\color{blue}{9 \cdot x}}} \]

       16. associate-/r* 0.7%

           \[\leadsto \frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 - \color{blue}{\frac{\frac{1}{9}}{x}}} \]

       17. metadata-eval 0.7%

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

       18. metadata-eval 0.7%

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

       19. associate-/r* 0.7%

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

       20. neg-mul-1 0.7%

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

       21. add-sqr-sqrt 0.0%

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

       22. sqrt-unprod 1.5%

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

       23. sqr-neg 1.5%

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

       24. sqrt-unprod 25.5%

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

       25. add-sqr-sqrt 25.5%

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

   12. Applied egg-rr 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 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]

Alternative 12: 62.7% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. neg-mul-1 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-*r/ 99.6%

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

   9. fma-def 99.6%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

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

   12. *-commutative 99.6%

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

   13. associate-/r* 99.6%

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

   14. distribute-neg-frac 99.6%

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

   15. metadata-eval 99.6%

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

   16. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 61.0%

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

   1. div-inv 61.0%

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

   2. clear-num 61.0%

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

   3. div-inv 61.0%

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

   4. metadata-eval 61.0%

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

6. Applied egg-rr 61.0%

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

7. Final simplification 61.0%

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

Alternative 13: 61.5% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 10.0) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 10.0) {
		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 <= 10.0d0) 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 <= 10.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 10.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 10.0)
		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 <= 10.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 10.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 10

   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. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 60.2%

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

   if 10 < 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. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 59.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 62.6% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. neg-mul-1 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-*r/ 99.6%

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

   9. fma-def 99.6%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

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

   12. *-commutative 99.6%

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

   13. associate-/r* 99.6%

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

   14. distribute-neg-frac 99.6%

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

   15. metadata-eval 99.6%

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

   16. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 61.0%

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

   1. cancel-sign-sub-inv 61.0%

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

   2. metadata-eval 61.0%

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

   3. associate-*r/ 61.0%

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

   4. metadata-eval 61.0%

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

   5. +-commutative 61.0%

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

6. Simplified 61.0%

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

7. Final simplification 61.0%

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

Alternative 15: 31.1% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. neg-mul-1 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-*r/ 99.6%

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

   9. fma-def 99.6%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

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

   12. *-commutative 99.6%

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

   13. associate-/r* 99.6%

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

   14. distribute-neg-frac 99.6%

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

   15. metadata-eval 99.6%

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

   16. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around inf 29.2%

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

5. Final simplification 29.2%

   \[\leadsto 1 \]

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 2023224 
(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)))))