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

Percentage Accurate: 99.7% → 99.7%

Time: 10.2s

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. Add Preprocessing

3. Final simplification 99.7%

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

Alternative 2: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+49} \lor \neg \left(y \leq 1.36 \cdot 10^{+54}\right):\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.2e+49) (not (<= y 1.36e+54)))
   (- 1.0 (/ 0.3333333333333333 (/ (sqrt x) y)))
   (+ 1.0 (/ -1.0 (/ x 0.1111111111111111)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e+49) || !(y <= 1.36e+54)) {
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	} else {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.2d+49)) .or. (.not. (y <= 1.36d+54))) then
        tmp = 1.0d0 - (0.3333333333333333d0 / (sqrt(x) / y))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x / 0.1111111111111111d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e+49) || !(y <= 1.36e+54)) {
		tmp = 1.0 - (0.3333333333333333 / (Math.sqrt(x) / y));
	} else {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.2e+49) or not (y <= 1.36e+54):
		tmp = 1.0 - (0.3333333333333333 / (math.sqrt(x) / y))
	else:
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.2e+49) || !(y <= 1.36e+54))
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(sqrt(x) / y)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x / 0.1111111111111111)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.2e+49) || ~((y <= 1.36e+54)))
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	else
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.2e+49], N[Not[LessEqual[y, 1.36e+54]], $MachinePrecision]], N[(1.0 - N[(0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x / 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e49 or 1.35999999999999999e54 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 91.9%

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

      1. *-commutative 91.9%

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

      2. metadata-eval 91.9%

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

      3. sqrt-div 91.9%

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

      4. metadata-eval 91.9%

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

      5. un-div-inv 92.0%

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

      6. times-frac 92.1%

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

      7. *-un-lft-identity 92.1%

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

      8. clear-num 92.0%

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

      9. associate-/l* 92.0%

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

   5. Applied egg-rr 92.0%

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

      1. associate-/r* 92.0%

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

      2. metadata-eval 92.0%

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

   7. Simplified 92.0%

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

   if -1.2e49 < y < 1.35999999999999999e54

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 96.4%

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

      1. un-div-inv 96.4%

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

      2. clear-num 96.5%

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

   7. Applied egg-rr 96.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+49} \lor \neg \left(y \leq 1.36 \cdot 10^{+54}\right):\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+102} \lor \neg \left(y \leq 2 \cdot 10^{+69}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.1e+102) (not (<= y 2e+69)))
   (* -0.3333333333333333 (* y (sqrt (/ 1.0 x))))
   (+ 1.0 (/ -1.0 (/ x 0.1111111111111111)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+102) || !(y <= 2e+69)) {
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.1d+102)) .or. (.not. (y <= 2d+69))) then
        tmp = (-0.3333333333333333d0) * (y * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x / 0.1111111111111111d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+102) || !(y <= 2e+69)) {
		tmp = -0.3333333333333333 * (y * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.1e+102) or not (y <= 2e+69):
		tmp = -0.3333333333333333 * (y * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.1e+102) || !(y <= 2e+69))
		tmp = Float64(-0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x / 0.1111111111111111)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.1e+102) || ~((y <= 2e+69)))
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.1e+102], N[Not[LessEqual[y, 2e+69]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x / 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.10000000000000001e102 or 2.0000000000000001e69 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in y around inf 90.5%

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

      1. *-commutative 90.5%

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

      2. *-commutative 90.5%

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

   6. Simplified 90.5%

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

   if -2.10000000000000001e102 < y < 2.0000000000000001e69

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 93.1%

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

      1. un-div-inv 93.1%

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

      2. clear-num 93.2%

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

   7. Applied egg-rr 93.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+102} \lor \neg \left(y \leq 2 \cdot 10^{+69}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+46}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.15e+46)
   (- 1.0 (/ (/ y (sqrt x)) 3.0))
   (if (<= y 5.7e+53)
     (+ 1.0 (/ -1.0 (/ x 0.1111111111111111)))
     (- 1.0 (/ y (sqrt (* x 9.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+46) {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	} else if (y <= 5.7e+53) {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	} else {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.15d+46)) then
        tmp = 1.0d0 - ((y / sqrt(x)) / 3.0d0)
    else if (y <= 5.7d+53) then
        tmp = 1.0d0 + ((-1.0d0) / (x / 0.1111111111111111d0))
    else
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+46) {
		tmp = 1.0 - ((y / Math.sqrt(x)) / 3.0);
	} else if (y <= 5.7e+53) {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.15e+46:
		tmp = 1.0 - ((y / math.sqrt(x)) / 3.0)
	elif y <= 5.7e+53:
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111))
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.15e+46)
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) / 3.0));
	elseif (y <= 5.7e+53)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x / 0.1111111111111111)));
	else
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.15e+46)
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	elseif (y <= 5.7e+53)
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.15e+46], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.7e+53], N[(1.0 + N[(-1.0 / N[(x / 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15e46

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 91.6%

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

      1. *-commutative 91.6%

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

      2. metadata-eval 91.6%

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

      3. sqrt-div 91.6%

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

      4. metadata-eval 91.6%

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

      5. un-div-inv 91.7%

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

      6. times-frac 91.9%

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

      7. *-un-lft-identity 91.9%

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

      8. *-commutative 91.9%

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

      9. associate-/r* 92.1%

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

   5. Applied egg-rr 92.1%

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

   if -1.15e46 < y < 5.70000000000000017e53

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 96.4%

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

      1. un-div-inv 96.4%

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

      2. clear-num 96.5%

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

   7. Applied egg-rr 96.5%

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

   if 5.70000000000000017e53 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 92.2%

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

      1. *-commutative 92.2%

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

      2. metadata-eval 92.2%

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

      3. sqrt-div 92.2%

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

      4. metadata-eval 92.2%

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

      5. un-div-inv 92.2%

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

      6. times-frac 92.2%

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

      7. *-un-lft-identity 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. metadata-eval 92.2%

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

      3. sqrt-prod 92.5%

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

      4. pow1/2 92.5%

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

   7. Applied egg-rr 92.5%

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

      1. unpow1/2 92.5%

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

   9. Simplified 92.5%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+46}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+46}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+54}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e+46)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 1.35e+54)
     (+ 1.0 (/ -1.0 (/ x 0.1111111111111111)))
     (- 1.0 (/ y (sqrt (* x 9.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e+46) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 1.35e+54) {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	} else {
		tmp = 1.0 - (y / sqrt((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 <= (-5d+46)) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else if (y <= 1.35d+54) then
        tmp = 1.0d0 + ((-1.0d0) / (x / 0.1111111111111111d0))
    else
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5e+46) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else if (y <= 1.35e+54) {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e+46:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	elif y <= 1.35e+54:
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111))
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e+46)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 1.35e+54)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x / 0.1111111111111111)));
	else
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e+46)
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	elseif (y <= 1.35e+54)
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e+46], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+54], N[(1.0 + N[(-1.0 / N[(x / 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.0000000000000002e46

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 91.6%

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

      1. *-commutative 91.6%

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

      2. metadata-eval 91.6%

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

      3. sqrt-div 91.6%

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

      4. metadata-eval 91.6%

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

      5. un-div-inv 91.7%

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

      6. times-frac 91.9%

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

      7. *-un-lft-identity 91.9%

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

   5. Applied egg-rr 91.9%

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

   if -5.0000000000000002e46 < y < 1.35000000000000005e54

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 96.4%

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

      1. un-div-inv 96.4%

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

      2. clear-num 96.5%

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

   7. Applied egg-rr 96.5%

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

   if 1.35000000000000005e54 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 92.2%

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

      1. *-commutative 92.2%

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

      2. metadata-eval 92.2%

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

      3. sqrt-div 92.2%

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

      4. metadata-eval 92.2%

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

      5. un-div-inv 92.2%

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

      6. times-frac 92.2%

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

      7. *-un-lft-identity 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. metadata-eval 92.2%

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

      3. sqrt-prod 92.5%

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

      4. pow1/2 92.5%

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

   7. Applied egg-rr 92.5%

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

      1. unpow1/2 92.5%

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

   9. Simplified 92.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+46}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+54}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+51}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e+51)
   (- 1.0 (/ 0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 4e+53)
     (+ 1.0 (/ -1.0 (/ x 0.1111111111111111)))
     (- 1.0 (/ y (sqrt (* x 9.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+51) {
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 4e+53) {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	} else {
		tmp = 1.0 - (y / sqrt((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+51)) then
        tmp = 1.0d0 - (0.3333333333333333d0 / (sqrt(x) / y))
    else if (y <= 4d+53) then
        tmp = 1.0d0 + ((-1.0d0) / (x / 0.1111111111111111d0))
    else
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+51) {
		tmp = 1.0 - (0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 4e+53) {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e+51:
		tmp = 1.0 - (0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 4e+53:
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111))
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e+51)
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 4e+53)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x / 0.1111111111111111)));
	else
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e+51)
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 4e+53)
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e+51], N[(1.0 - N[(0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+53], N[(1.0 + N[(-1.0 / N[(x / 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5e51

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 91.6%

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

      1. *-commutative 91.6%

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

      2. metadata-eval 91.6%

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

      3. sqrt-div 91.6%

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

      4. metadata-eval 91.6%

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

      5. un-div-inv 91.7%

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

      6. times-frac 91.9%

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

      7. *-un-lft-identity 91.9%

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

      8. clear-num 91.8%

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

      9. associate-/l* 91.7%

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

   5. Applied egg-rr 91.7%

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

      1. associate-/r* 91.8%

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

      2. metadata-eval 91.8%

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

   7. Simplified 91.8%

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

   if -2.5e51 < y < 4e53

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 96.4%

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

      1. un-div-inv 96.4%

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

      2. clear-num 96.5%

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

   7. Applied egg-rr 96.5%

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

   if 4e53 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 92.2%

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

      1. *-commutative 92.2%

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

      2. metadata-eval 92.2%

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

      3. sqrt-div 92.2%

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

      4. metadata-eval 92.2%

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

      5. un-div-inv 92.2%

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

      6. times-frac 92.2%

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

      7. *-un-lft-identity 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. metadata-eval 92.2%

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

      3. sqrt-prod 92.5%

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

      4. pow1/2 92.5%

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

   7. Applied egg-rr 92.5%

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

      1. unpow1/2 92.5%

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

   9. Simplified 92.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+51}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{0.1111111111111111 + 0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112)
   (/ (+ 0.1111111111111111 (* 0.3333333333333333 (* y (sqrt x)))) (- x))
   (- 1.0 (/ (/ y (sqrt x)) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * sqrt(x)))) / -x;
	} else {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = (0.1111111111111111d0 + (0.3333333333333333d0 * (y * sqrt(x)))) / -x
    else
        tmp = 1.0d0 - ((y / sqrt(x)) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * Math.sqrt(x)))) / -x;
	} else {
		tmp = 1.0 - ((y / Math.sqrt(x)) / 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * math.sqrt(x)))) / -x
	else:
		tmp = 1.0 - ((y / math.sqrt(x)) / 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(Float64(0.1111111111111111 + Float64(0.3333333333333333 * Float64(y * sqrt(x)))) / Float64(-x));
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * sqrt(x)))) / -x;
	else
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(N[(0.1111111111111111 + N[(0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   if 0.112000000000000002 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 98.7%

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

      1. *-commutative 98.7%

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

      2. metadata-eval 98.7%

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

      3. sqrt-div 98.7%

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

      4. metadata-eval 98.7%

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

      5. un-div-inv 98.8%

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

      6. times-frac 98.8%

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

      7. *-un-lft-identity 98.8%

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

      8. *-commutative 98.8%

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

      9. associate-/r* 98.9%

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

   5. Applied egg-rr 98.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{0.1111111111111111 + 0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0 99.7%

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

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. un-div-inv 99.7%

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

6. Applied egg-rr 99.7%

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

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 11: 64.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{\frac{0.1111111111111111}{x}}{x \cdot -9} + -1}{\frac{-0.1111111111111111}{x} + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3e+125)
   (/
    (+ (/ (/ 0.1111111111111111 x) (* x -9.0)) -1.0)
    (+ (/ -0.1111111111111111 x) -1.0))
   (+ 1.0 (/ -1.0 (/ x 0.1111111111111111)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3e+125) {
		tmp = (((0.1111111111111111 / x) / (x * -9.0)) + -1.0) / ((-0.1111111111111111 / x) + -1.0);
	} else {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3d+125)) then
        tmp = (((0.1111111111111111d0 / x) / (x * (-9.0d0))) + (-1.0d0)) / (((-0.1111111111111111d0) / x) + (-1.0d0))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x / 0.1111111111111111d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3e+125) {
		tmp = (((0.1111111111111111 / x) / (x * -9.0)) + -1.0) / ((-0.1111111111111111 / x) + -1.0);
	} else {
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3e+125:
		tmp = (((0.1111111111111111 / x) / (x * -9.0)) + -1.0) / ((-0.1111111111111111 / x) + -1.0)
	else:
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3e+125)
		tmp = Float64(Float64(Float64(Float64(0.1111111111111111 / x) / Float64(x * -9.0)) + -1.0) / Float64(Float64(-0.1111111111111111 / x) + -1.0));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x / 0.1111111111111111)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3e+125)
		tmp = (((0.1111111111111111 / x) / (x * -9.0)) + -1.0) / ((-0.1111111111111111 / x) + -1.0);
	else
		tmp = 1.0 + (-1.0 / (x / 0.1111111111111111));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3e+125], N[(N[(N[(N[(0.1111111111111111 / x), $MachinePrecision] / N[(x * -9.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(-0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x / 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000015e125

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. flip-+ 2.8%

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

      3. frac-times 2.8%

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

      4. metadata-eval 2.8%

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

      5. pow2 2.8%

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

      6. metadata-eval 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. metadata-eval 2.8%

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

      2. unpow2 2.8%

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

      3. frac-times 2.8%

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

      4. div-inv 2.8%

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

      5. associate-*r* 2.8%

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

   9. Applied egg-rr 2.8%

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

       1. associate-*l* 2.8%

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

       2. div-inv 2.8%

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

       3. clear-num 2.8%

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

       4. un-div-inv 2.8%

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

       5. add-sqr-sqrt 0.0%

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

       6. sqrt-prod 19.9%

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

       7. div-inv 19.9%

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

       8. associate-*l* 19.9%

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

       9. un-div-inv 19.9%

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

       10. sqrt-div 19.9%

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

       11. associate-*l/ 19.9%

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

       12. metadata-eval 19.9%

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

       13. sqrt-div 19.9%

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

       14. metadata-eval 19.9%

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

       15. associate-/r* 19.9%

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

       16. add-sqr-sqrt 19.9%

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

       17. div-inv 19.9%

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

       18. metadata-eval 19.9%

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

   11. Applied egg-rr 19.9%

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

   if -3.00000000000000015e125 < 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-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 73.7%

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

      1. un-div-inv 73.7%

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

      2. clear-num 73.7%

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

   7. Applied egg-rr 73.7%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{\frac{0.1111111111111111}{x}}{x \cdot -9} + -1}{\frac{-0.1111111111111111}{x} + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{x}{0.1111111111111111}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.6% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   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-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.6%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 61.9%

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

      1. +-commutative 61.9%

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

      2. flip-+ 31.5%

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

      3. frac-times 31.6%

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

      4. metadata-eval 31.6%

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

      5. pow2 31.6%

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

      6. metadata-eval 31.6%

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

   7. Applied egg-rr 31.6%

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

   8. Taylor expanded in x around 0 60.8%

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

   if 0.112000000000000002 < 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-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.8%

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

      15. distribute-neg-frac 99.8%

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

      16. metadata-eval 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 64.0%

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

   6. Taylor expanded in x around inf 62.9%

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

Alternative 13: 62.5% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.7%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.7%

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

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 62.9%

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

   1. un-div-inv 62.9%

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

   2. clear-num 62.9%

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

7. Applied egg-rr 62.9%

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

8. Final simplification 62.9%

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

Alternative 14: 62.5% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.7%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.7%

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

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 62.9%

   \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
6. Add Preprocessing

Alternative 15: 31.5% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.7%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.7%

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

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 62.9%

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

6. Taylor expanded in x around inf 31.0%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024086 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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