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

Percentage Accurate: 99.7% → 99.7%

Time: 15.0s

Alternatives: 17

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

4. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{+80}:\\ \;\;\;\;1 + \frac{y \cdot -0.1111111111111111}{x \cdot y}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+75} \lor \neg \left(y \leq 4.1 \cdot 10^{+41}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* -0.3333333333333333 (/ y (sqrt x)))))
   (if (<= y -7.5e+109)
     t_0
     (if (<= y -1.55e+80)
       (+ 1.0 (/ (* y -0.1111111111111111) (* x y)))
       (if (or (<= y -5.6e+75) (not (<= y 4.1e+41)))
         t_0
         (+ 1.0 (/ -1.0 (* x 9.0))))))))

C

double code(double x, double y) {
	double t_0 = -0.3333333333333333 * (y / sqrt(x));
	double tmp;
	if (y <= -7.5e+109) {
		tmp = t_0;
	} else if (y <= -1.55e+80) {
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y));
	} else if ((y <= -5.6e+75) || !(y <= 4.1e+41)) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.3333333333333333d0) * (y / sqrt(x))
    if (y <= (-7.5d+109)) then
        tmp = t_0
    else if (y <= (-1.55d+80)) then
        tmp = 1.0d0 + ((y * (-0.1111111111111111d0)) / (x * y))
    else if ((y <= (-5.6d+75)) .or. (.not. (y <= 4.1d+41))) then
        tmp = t_0
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -0.3333333333333333 * (y / Math.sqrt(x));
	double tmp;
	if (y <= -7.5e+109) {
		tmp = t_0;
	} else if (y <= -1.55e+80) {
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y));
	} else if ((y <= -5.6e+75) || !(y <= 4.1e+41)) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -0.3333333333333333 * (y / math.sqrt(x))
	tmp = 0
	if y <= -7.5e+109:
		tmp = t_0
	elif y <= -1.55e+80:
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y))
	elif (y <= -5.6e+75) or not (y <= 4.1e+41):
		tmp = t_0
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-0.3333333333333333 * Float64(y / sqrt(x)))
	tmp = 0.0
	if (y <= -7.5e+109)
		tmp = t_0;
	elseif (y <= -1.55e+80)
		tmp = Float64(1.0 + Float64(Float64(y * -0.1111111111111111) / Float64(x * y)));
	elseif ((y <= -5.6e+75) || !(y <= 4.1e+41))
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -0.3333333333333333 * (y / sqrt(x));
	tmp = 0.0;
	if (y <= -7.5e+109)
		tmp = t_0;
	elseif (y <= -1.55e+80)
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y));
	elseif ((y <= -5.6e+75) || ~((y <= 4.1e+41)))
		tmp = t_0;
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+109], t$95$0, If[LessEqual[y, -1.55e+80], N[(1.0 + N[(N[(y * -0.1111111111111111), $MachinePrecision] / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -5.6e+75], N[Not[LessEqual[y, 4.1e+41]], $MachinePrecision]], t$95$0, N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.50000000000000018e109 or -1.54999999999999994e80 < y < -5.60000000000000023e75 or 4.1000000000000004e41 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 99.7%

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

   8. Taylor expanded in y around inf 92.9%

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

      1. associate-*r* 92.9%

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

      2. *-commutative 92.9%

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

   10. Simplified 92.9%

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

       1. *-commutative 92.9%

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

       2. *-commutative 92.9%

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

       3. sqrt-div 92.8%

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

       4. metadata-eval 92.8%

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

       5. div-inv 92.9%

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

       6. associate-*r/ 92.8%

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

   12. Applied egg-rr 92.8%

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

       1. *-commutative 92.8%

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

       2. associate-/l* 92.9%

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

   14. Simplified 92.9%

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

   if -7.50000000000000018e109 < y < -1.54999999999999994e80

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-prod 100.0%

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

      4. pow1/2 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1/2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 99.6%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in y around 0 75.1%

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

       1. associate-*r/ 75.1%

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

   12. Applied egg-rr 75.1%

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

   if -5.60000000000000023e75 < y < 4.1000000000000004e41

   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 96.7%

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

      1. expm1-log1p-u 45.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

      2. log1p-define 45.1%

         \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{-0.1111111111111111}{x}\right)}\right) \]

      3. expm1-undefine 45.1%

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

      4. add-exp-log 96.7%

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

   7. Applied egg-rr 96.7%

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

      1. associate--l+ 96.7%

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

   9. Simplified 96.7%

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

       1. associate-+r- 96.7%

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

       2. add-exp-log 45.1%

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

       3. log1p-undefine 45.1%

          \[\leadsto 1 + \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)}} - 1\right) \]

       4. expm1-undefine 45.1%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

       5. expm1-log1p-u 96.7%

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

       6. metadata-eval 96.7%

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

       7. distribute-neg-frac 96.7%

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

       8. clear-num 96.6%

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

       9. distribute-neg-frac 96.6%

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

       10. metadata-eval 96.6%

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

       11. div-inv 96.7%

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

       12. metadata-eval 96.7%

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

   11. Applied egg-rr 96.7%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{+80}:\\ \;\;\;\;1 + \frac{y \cdot -0.1111111111111111}{x \cdot y}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+75} \lor \neg \left(y \leq 4.1 \cdot 10^{+41}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{y \cdot -0.1111111111111111}{x \cdot y}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{+73}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+41}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.5e+109)
   (* (pow x -0.5) (* y -0.3333333333333333))
   (if (<= y -2.7e+81)
     (+ 1.0 (/ (* y -0.1111111111111111) (* x y)))
     (if (<= y -8.5e+73)
       (* -0.3333333333333333 (/ y (sqrt x)))
       (if (<= y 3.8e+41)
         (+ 1.0 (/ -1.0 (* x 9.0)))
         (/ y (* (sqrt x) -3.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.5e+109) {
		tmp = pow(x, -0.5) * (y * -0.3333333333333333);
	} else if (y <= -2.7e+81) {
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y));
	} else if (y <= -8.5e+73) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 3.8e+41) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 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 (y <= (-7.5d+109)) then
        tmp = (x ** (-0.5d0)) * (y * (-0.3333333333333333d0))
    else if (y <= (-2.7d+81)) then
        tmp = 1.0d0 + ((y * (-0.1111111111111111d0)) / (x * y))
    else if (y <= (-8.5d+73)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 3.8d+41) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = y / (sqrt(x) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.5e+109) {
		tmp = Math.pow(x, -0.5) * (y * -0.3333333333333333);
	} else if (y <= -2.7e+81) {
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y));
	} else if (y <= -8.5e+73) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 3.8e+41) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = y / (Math.sqrt(x) * -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.5e+109:
		tmp = math.pow(x, -0.5) * (y * -0.3333333333333333)
	elif y <= -2.7e+81:
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y))
	elif y <= -8.5e+73:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 3.8e+41:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = y / (math.sqrt(x) * -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.5e+109)
		tmp = Float64((x ^ -0.5) * Float64(y * -0.3333333333333333));
	elseif (y <= -2.7e+81)
		tmp = Float64(1.0 + Float64(Float64(y * -0.1111111111111111) / Float64(x * y)));
	elseif (y <= -8.5e+73)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 3.8e+41)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.5e+109)
		tmp = (x ^ -0.5) * (y * -0.3333333333333333);
	elseif (y <= -2.7e+81)
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y));
	elseif (y <= -8.5e+73)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 3.8e+41)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = y / (sqrt(x) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.5e+109], N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.7e+81], N[(1.0 + N[(N[(y * -0.1111111111111111), $MachinePrecision] / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -8.5e+73], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+41], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -7.50000000000000018e109

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf 99.5%

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

      1. metadata-eval 99.5%

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

      2. *-commutative 99.5%

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

      3. sqrt-div 99.5%

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

      4. metadata-eval 99.5%

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

      5. un-div-inv 99.6%

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

      6. times-frac 99.4%

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

      7. *-un-lft-identity 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in y around inf 99.5%

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

       1. associate-*r* 99.5%

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

       2. *-commutative 99.5%

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

       3. associate-*l* 99.5%

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

   12. Simplified 99.5%

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

       1. *-un-lft-identity 99.5%

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

       2. inv-pow 99.5%

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

       3. sqrt-pow1 99.6%

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

       4. metadata-eval 99.6%

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

   14. Applied egg-rr 99.6%

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

       1. *-lft-identity 99.6%

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

   16. Simplified 99.6%

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

   if -7.50000000000000018e109 < y < -2.6999999999999999e81

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-prod 100.0%

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

      4. pow1/2 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1/2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 99.6%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in y around 0 75.1%

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

       1. associate-*r/ 75.1%

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

   12. Applied egg-rr 75.1%

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

   if -2.6999999999999999e81 < y < -8.4999999999999998e73

   1. Initial program 98.4%

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

      1. *-commutative 98.4%

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

      2. metadata-eval 98.4%

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

      3. sqrt-prod 98.4%

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

      4. pow1/2 98.4%

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

   4. Applied egg-rr 98.4%

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

      1. unpow1/2 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in x around 0 98.4%

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

   8. Taylor expanded in y around inf 100.0%

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

      1. associate-*r* 98.4%

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

      2. *-commutative 98.4%

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

   10. Simplified 98.4%

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

       1. *-commutative 98.4%

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

       2. *-commutative 98.4%

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

       3. sqrt-div 98.4%

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

       4. metadata-eval 98.4%

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

       5. div-inv 100.0%

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

       6. associate-*r/ 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-/l* 100.0%

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

   14. Simplified 100.0%

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

   if -8.4999999999999998e73 < y < 3.8000000000000001e41

   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 96.7%

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

      1. expm1-log1p-u 45.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

      2. log1p-define 45.1%

         \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{-0.1111111111111111}{x}\right)}\right) \]

      3. expm1-undefine 45.1%

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

      4. add-exp-log 96.7%

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

   7. Applied egg-rr 96.7%

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

      1. associate--l+ 96.7%

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

   9. Simplified 96.7%

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

       1. associate-+r- 96.7%

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

       2. add-exp-log 45.1%

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

       3. log1p-undefine 45.1%

          \[\leadsto 1 + \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)}} - 1\right) \]

       4. expm1-undefine 45.1%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

       5. expm1-log1p-u 96.7%

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

       6. metadata-eval 96.7%

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

       7. distribute-neg-frac 96.7%

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

       8. clear-num 96.6%

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

       9. distribute-neg-frac 96.6%

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

       10. metadata-eval 96.6%

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

       11. div-inv 96.7%

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

       12. metadata-eval 96.7%

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

   11. Applied egg-rr 96.7%

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

   if 3.8000000000000001e41 < y

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. metadata-eval 99.7%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 99.8%

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

   8. Taylor expanded in y around inf 85.8%

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

      1. associate-*r* 85.8%

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

      2. *-commutative 85.8%

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

   10. Simplified 85.8%

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

       1. *-commutative 85.8%

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

       2. *-commutative 85.8%

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

       3. sqrt-div 85.6%

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

       4. metadata-eval 85.6%

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

       5. div-inv 85.7%

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

       6. clear-num 85.8%

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

       7. un-div-inv 85.9%

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

       8. div-inv 86.0%

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

       9. metadata-eval 86.0%

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

   12. Applied egg-rr 86.0%

       \[\leadsto \color{blue}{\frac{y}{\sqrt{x} \cdot -3}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{y \cdot -0.1111111111111111}{x \cdot y}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{+73}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+41}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+80}:\\ \;\;\;\;1 + \frac{y \cdot -0.1111111111111111}{x \cdot y}\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* -0.3333333333333333 (/ y (sqrt x)))))
   (if (<= y -7.5e+109)
     t_0
     (if (<= y -4.2e+80)
       (+ 1.0 (/ (* y -0.1111111111111111) (* x y)))
       (if (<= y -1.55e+75)
         t_0
         (if (<= y 8.6e+40)
           (+ 1.0 (/ -1.0 (* x 9.0)))
           (/ y (* (sqrt x) -3.0))))))))

C

double code(double x, double y) {
	double t_0 = -0.3333333333333333 * (y / sqrt(x));
	double tmp;
	if (y <= -7.5e+109) {
		tmp = t_0;
	} else if (y <= -4.2e+80) {
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y));
	} else if (y <= -1.55e+75) {
		tmp = t_0;
	} else if (y <= 8.6e+40) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (-0.3333333333333333d0) * (y / sqrt(x))
    if (y <= (-7.5d+109)) then
        tmp = t_0
    else if (y <= (-4.2d+80)) then
        tmp = 1.0d0 + ((y * (-0.1111111111111111d0)) / (x * y))
    else if (y <= (-1.55d+75)) then
        tmp = t_0
    else if (y <= 8.6d+40) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = y / (sqrt(x) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -0.3333333333333333 * (y / Math.sqrt(x));
	double tmp;
	if (y <= -7.5e+109) {
		tmp = t_0;
	} else if (y <= -4.2e+80) {
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y));
	} else if (y <= -1.55e+75) {
		tmp = t_0;
	} else if (y <= 8.6e+40) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = y / (Math.sqrt(x) * -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -0.3333333333333333 * (y / math.sqrt(x))
	tmp = 0
	if y <= -7.5e+109:
		tmp = t_0
	elif y <= -4.2e+80:
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y))
	elif y <= -1.55e+75:
		tmp = t_0
	elif y <= 8.6e+40:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = y / (math.sqrt(x) * -3.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-0.3333333333333333 * Float64(y / sqrt(x)))
	tmp = 0.0
	if (y <= -7.5e+109)
		tmp = t_0;
	elseif (y <= -4.2e+80)
		tmp = Float64(1.0 + Float64(Float64(y * -0.1111111111111111) / Float64(x * y)));
	elseif (y <= -1.55e+75)
		tmp = t_0;
	elseif (y <= 8.6e+40)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -0.3333333333333333 * (y / sqrt(x));
	tmp = 0.0;
	if (y <= -7.5e+109)
		tmp = t_0;
	elseif (y <= -4.2e+80)
		tmp = 1.0 + ((y * -0.1111111111111111) / (x * y));
	elseif (y <= -1.55e+75)
		tmp = t_0;
	elseif (y <= 8.6e+40)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = y / (sqrt(x) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+109], t$95$0, If[LessEqual[y, -4.2e+80], N[(1.0 + N[(N[(y * -0.1111111111111111), $MachinePrecision] / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.55e+75], t$95$0, If[LessEqual[y, 8.6e+40], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.50000000000000018e109 or -4.20000000000000003e80 < y < -1.5500000000000001e75

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.6%

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

   8. Taylor expanded in y around inf 99.5%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

   10. Simplified 99.5%

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

       1. *-commutative 99.5%

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

       2. *-commutative 99.5%

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

       3. sqrt-div 99.5%

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

       4. metadata-eval 99.5%

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

       5. div-inv 99.6%

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

       6. associate-*r/ 99.5%

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

   12. Applied egg-rr 99.5%

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

       1. *-commutative 99.5%

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

       2. associate-/l* 99.6%

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

   14. Simplified 99.6%

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

   if -7.50000000000000018e109 < y < -4.20000000000000003e80

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-prod 100.0%

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

      4. pow1/2 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1/2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 99.6%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in y around 0 75.1%

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

       1. associate-*r/ 75.1%

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

   12. Applied egg-rr 75.1%

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

   if -1.5500000000000001e75 < y < 8.6000000000000005e40

   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 96.7%

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

      1. expm1-log1p-u 45.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

      2. log1p-define 45.1%

         \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{-0.1111111111111111}{x}\right)}\right) \]

      3. expm1-undefine 45.1%

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

      4. add-exp-log 96.7%

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

   7. Applied egg-rr 96.7%

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

      1. associate--l+ 96.7%

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

   9. Simplified 96.7%

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

       1. associate-+r- 96.7%

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

       2. add-exp-log 45.1%

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

       3. log1p-undefine 45.1%

          \[\leadsto 1 + \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)}} - 1\right) \]

       4. expm1-undefine 45.1%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

       5. expm1-log1p-u 96.7%

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

       6. metadata-eval 96.7%

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

       7. distribute-neg-frac 96.7%

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

       8. clear-num 96.6%

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

       9. distribute-neg-frac 96.6%

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

       10. metadata-eval 96.6%

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

       11. div-inv 96.7%

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

       12. metadata-eval 96.7%

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

   11. Applied egg-rr 96.7%

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

   if 8.6000000000000005e40 < y

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. metadata-eval 99.7%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 99.8%

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

   8. Taylor expanded in y around inf 85.8%

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

      1. associate-*r* 85.8%

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

      2. *-commutative 85.8%

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

   10. Simplified 85.8%

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

       1. *-commutative 85.8%

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

       2. *-commutative 85.8%

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

       3. sqrt-div 85.6%

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

       4. metadata-eval 85.6%

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

       5. div-inv 85.7%

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

       6. clear-num 85.8%

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

       7. un-div-inv 85.9%

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

       8. div-inv 86.0%

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

       9. metadata-eval 86.0%

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

   12. Applied egg-rr 86.0%

       \[\leadsto \color{blue}{\frac{y}{\sqrt{x} \cdot -3}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+66} \lor \neg \left(y \leq 2250\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e+66) (not (<= y 2250.0)))
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5e+66) || !(y <= 2250.0)) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-5d+66)) .or. (.not. (y <= 2250.0d0))) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5e+66) || !(y <= 2250.0)) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5e+66) or not (y <= 2250.0):
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5e+66) || !(y <= 2250.0))
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5e+66) || ~((y <= 2250.0)))
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5e+66], N[Not[LessEqual[y, 2250.0]], $MachinePrecision]], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999991e66 or 2250 < 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 92.1%

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

      1. metadata-eval 92.1%

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

      2. *-commutative 92.1%

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

      3. sqrt-div 92.0%

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

      4. metadata-eval 92.0%

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

      5. un-div-inv 92.1%

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

      6. times-frac 92.1%

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

      7. *-un-lft-identity 92.1%

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

   5. Applied egg-rr 92.1%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   7. Applied egg-rr 92.2%

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

      1. unpow1/2 99.7%

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

   9. Simplified 92.2%

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

   if -4.99999999999999991e66 < y < 2250

   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 98.1%

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

      1. expm1-log1p-u 45.6%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

      2. log1p-define 45.6%

         \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{-0.1111111111111111}{x}\right)}\right) \]

      3. expm1-undefine 45.6%

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

      4. add-exp-log 98.1%

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

   7. Applied egg-rr 98.1%

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

      1. associate--l+ 98.1%

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

   9. Simplified 98.1%

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

       1. associate-+r- 98.1%

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

       2. add-exp-log 45.6%

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

       3. log1p-undefine 45.6%

          \[\leadsto 1 + \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)}} - 1\right) \]

       4. expm1-undefine 45.6%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

       5. expm1-log1p-u 98.1%

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

       6. metadata-eval 98.1%

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

       7. distribute-neg-frac 98.1%

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

       8. clear-num 98.0%

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

       9. distribute-neg-frac 98.0%

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

       10. metadata-eval 98.0%

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

       11. div-inv 98.1%

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

       12. metadata-eval 98.1%

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

   11. Applied egg-rr 98.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+66} \lor \neg \left(y \leq 2250\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+66} \lor \neg \left(y \leq 2.6 \cdot 10^{-15}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.8e+66) (not (<= y 2.6e-15)))
   (- 1.0 (* (/ y (sqrt x)) 0.3333333333333333))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.8e+66) || !(y <= 2.6e-15)) {
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.8d+66)) .or. (.not. (y <= 2.6d-15))) then
        tmp = 1.0d0 - ((y / sqrt(x)) * 0.3333333333333333d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.8e+66) || !(y <= 2.6e-15)) {
		tmp = 1.0 - ((y / Math.sqrt(x)) * 0.3333333333333333);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.8e+66) or not (y <= 2.6e-15):
		tmp = 1.0 - ((y / math.sqrt(x)) * 0.3333333333333333)
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.8e+66) || !(y <= 2.6e-15))
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) * 0.3333333333333333));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.8e+66) || ~((y <= 2.6e-15)))
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.8e+66], N[Not[LessEqual[y, 2.6e-15]], $MachinePrecision]], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000003e66 or 2.60000000000000004e-15 < 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 91.5%

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

      1. *-commutative 91.5%

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

      2. sqrt-div 91.5%

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

      3. metadata-eval 91.5%

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

      4. un-div-inv 91.6%

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

   5. Applied egg-rr 91.6%

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

   if -4.8000000000000003e66 < y < 2.60000000000000004e-15

   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 98.0%

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

      1. expm1-log1p-u 45.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

      2. log1p-define 45.1%

         \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{-0.1111111111111111}{x}\right)}\right) \]

      3. expm1-undefine 45.1%

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

      4. add-exp-log 98.0%

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

   7. Applied egg-rr 98.0%

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

      1. associate--l+ 98.0%

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

   9. Simplified 98.0%

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

       1. associate-+r- 98.0%

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

       2. add-exp-log 45.1%

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

       3. log1p-undefine 45.1%

          \[\leadsto 1 + \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)}} - 1\right) \]

       4. expm1-undefine 45.1%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

       5. expm1-log1p-u 98.0%

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

       6. metadata-eval 98.0%

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

       7. distribute-neg-frac 98.0%

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

       8. clear-num 98.0%

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

       9. distribute-neg-frac 98.0%

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

       10. metadata-eval 98.0%

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

       11. div-inv 98.1%

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

       12. metadata-eval 98.1%

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

   11. Applied egg-rr 98.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+66} \lor \neg \left(y \leq 2.6 \cdot 10^{-15}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+67}:\\ \;\;\;\;1 - 0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 2250:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.25e+67)
   (- 1.0 (* 0.3333333333333333 (* y (pow x -0.5))))
   (if (<= y 2250.0)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (/ y (sqrt (* x 9.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.25e+67) {
		tmp = 1.0 - (0.3333333333333333 * (y * pow(x, -0.5)));
	} else if (y <= 2250.0) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} 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 <= (-1.25d+67)) then
        tmp = 1.0d0 - (0.3333333333333333d0 * (y * (x ** (-0.5d0))))
    else if (y <= 2250.0d0) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    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 <= -1.25e+67) {
		tmp = 1.0 - (0.3333333333333333 * (y * Math.pow(x, -0.5)));
	} else if (y <= 2250.0) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.25e+67:
		tmp = 1.0 - (0.3333333333333333 * (y * math.pow(x, -0.5)))
	elif y <= 2250.0:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.25e+67)
		tmp = Float64(1.0 - Float64(0.3333333333333333 * Float64(y * (x ^ -0.5))));
	elseif (y <= 2250.0)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	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 <= -1.25e+67)
		tmp = 1.0 - (0.3333333333333333 * (y * (x ^ -0.5)));
	elseif (y <= 2250.0)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.25e+67], N[(1.0 - N[(0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2250.0], N[(1.0 + N[(-1.0 / N[(x * 9.0), $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 < -1.24999999999999994e67

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 95.2%

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

      1. *-un-lft-identity 95.2%

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

      2. inv-pow 95.2%

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

      3. sqrt-pow1 95.3%

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

      4. metadata-eval 95.3%

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

   5. Applied egg-rr 95.3%

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

      1. *-lft-identity 95.3%

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

   7. Simplified 95.3%

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

   if -1.24999999999999994e67 < y < 2250

   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 98.1%

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

      1. expm1-log1p-u 45.6%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

      2. log1p-define 45.6%

         \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{-0.1111111111111111}{x}\right)}\right) \]

      3. expm1-undefine 45.6%

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

      4. add-exp-log 98.1%

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

   7. Applied egg-rr 98.1%

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

      1. associate--l+ 98.1%

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

   9. Simplified 98.1%

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

       1. associate-+r- 98.1%

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

       2. add-exp-log 45.6%

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

       3. log1p-undefine 45.6%

          \[\leadsto 1 + \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)}} - 1\right) \]

       4. expm1-undefine 45.6%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

       5. expm1-log1p-u 98.1%

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

       6. metadata-eval 98.1%

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

       7. distribute-neg-frac 98.1%

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

       8. clear-num 98.0%

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

       9. distribute-neg-frac 98.0%

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

       10. metadata-eval 98.0%

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

       11. div-inv 98.1%

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

       12. metadata-eval 98.1%

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

   11. Applied egg-rr 98.1%

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

   if 2250 < y

   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 inf 88.7%

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

      1. metadata-eval 88.7%

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

      2. *-commutative 88.7%

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

      3. sqrt-div 88.6%

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

      4. metadata-eval 88.6%

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

      5. un-div-inv 88.7%

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

      6. times-frac 88.9%

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

      7. *-un-lft-identity 88.9%

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

   5. Applied egg-rr 88.9%

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

      1. *-commutative 99.7%

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

      2. metadata-eval 99.7%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   7. Applied egg-rr 88.9%

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

      1. unpow1/2 99.8%

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

   9. Simplified 88.9%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+67}:\\ \;\;\;\;1 - 0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 2250:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{0.1111111111111111 + 0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (+ 0.1111111111111111 (* 0.3333333333333333 (* y (sqrt x)))) (- x))
   (- 1.0 (/ y (sqrt (* x 9.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * sqrt(x)))) / -x;
	} 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 (x <= 0.11d0) then
        tmp = (0.1111111111111111d0 + (0.3333333333333333d0 * (y * sqrt(x)))) / -x
    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 (x <= 0.11) {
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * Math.sqrt(x)))) / -x;
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * math.sqrt(x)))) / -x
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(Float64(0.1111111111111111 + Float64(0.3333333333333333 * Float64(y * sqrt(x)))) / Float64(-x));
	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 (x <= 0.11)
		tmp = (0.1111111111111111 + (0.3333333333333333 * (y * sqrt(x)))) / -x;
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(0.1111111111111111 + N[(0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. *-commutative 98.3%

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

   5. Simplified 98.3%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. metadata-eval 99.4%

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

      2. *-commutative 99.4%

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

      3. sqrt-div 99.3%

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

      4. metadata-eval 99.3%

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

      5. un-div-inv 99.4%

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

      6. times-frac 99.5%

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

      7. *-un-lft-identity 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   7. Applied egg-rr 99.5%

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

      1. unpow1/2 99.8%

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

   9. Simplified 99.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{0.1111111111111111 + 0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) - (y / sqrt((x * 9.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

4. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 99.7%

   \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]
8. 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 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

Alternative 11: 66.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+206}:\\ \;\;\;\;1 + \frac{\frac{1}{y}}{x \cdot \frac{9}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{1}{y}}{\frac{x}{y \cdot -0.1111111111111111}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e+206)
   (+ 1.0 (/ (/ 1.0 y) (* x (/ 9.0 y))))
   (+ 1.0 (/ (/ 1.0 y) (/ x (* y -0.1111111111111111))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e+206) {
		tmp = 1.0 + ((1.0 / y) / (x * (9.0 / y)));
	} else {
		tmp = 1.0 + ((1.0 / y) / (x / (y * -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 <= (-1d+206)) then
        tmp = 1.0d0 + ((1.0d0 / y) / (x * (9.0d0 / y)))
    else
        tmp = 1.0d0 + ((1.0d0 / y) / (x / (y * (-0.1111111111111111d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1e+206) {
		tmp = 1.0 + ((1.0 / y) / (x * (9.0 / y)));
	} else {
		tmp = 1.0 + ((1.0 / y) / (x / (y * -0.1111111111111111)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1e+206:
		tmp = 1.0 + ((1.0 / y) / (x * (9.0 / y)))
	else:
		tmp = 1.0 + ((1.0 / y) / (x / (y * -0.1111111111111111)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e+206)
		tmp = Float64(1.0 + Float64(Float64(1.0 / y) / Float64(x * Float64(9.0 / y))));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 / y) / Float64(x / Float64(y * -0.1111111111111111))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1e+206)
		tmp = 1.0 + ((1.0 / y) / (x * (9.0 / y)));
	else
		tmp = 1.0 + ((1.0 / y) / (x / (y * -0.1111111111111111)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e+206], N[(1.0 + N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[(9.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 / y), $MachinePrecision] / N[(x / N[(y * -0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e206

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. metadata-eval 99.7%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow1/2 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in y around inf 99.6%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in y around 0 2.3%

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

       1. associate-/r* 2.3%

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

       2. div-inv 2.3%

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

   12. Applied egg-rr 2.3%

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

       1. associate-*r* 2.2%

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

       2. associate-/l* 2.2%

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

       3. clear-num 2.2%

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

       4. associate-*l/ 2.2%

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

       5. *-un-lft-identity 2.2%

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

   14. Applied egg-rr 2.2%

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

       1. clear-num 2.2%

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

       2. associate-/l* 2.2%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 49.3%

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

       5. sqr-neg 49.3%

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

       6. sqrt-unprod 49.3%

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

       7. add-sqr-sqrt 49.3%

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

       8. distribute-neg-frac 49.3%

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

       9. distribute-neg-frac 49.3%

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

       10. *-un-lft-identity 49.3%

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

       11. *-inverses 49.3%

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

       12. times-frac 49.3%

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

       13. *-commutative 49.3%

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

       14. clear-num 49.3%

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

       15. distribute-rgt-neg-out 49.3%

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

       16. *-commutative 49.3%

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

       17. times-frac 49.3%

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

       18. *-inverses 49.3%

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

       19. *-un-lft-identity 49.3%

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

       20. clear-num 49.3%

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

       21. associate-/l* 49.3%

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

   16. Applied egg-rr 2.3%

       \[\leadsto 1 + \frac{\frac{1}{y}}{\color{blue}{0 - x \cdot \frac{-9}{y}}} \]
   17. Step-by-step derivation

       1. neg-sub0 49.3%

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

       2. distribute-rgt-neg-in 49.3%

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

       3. distribute-neg-frac 49.3%

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

       4. metadata-eval 49.3%

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

   18. Simplified 49.3%

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

   if -1e206 < y

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. metadata-eval 99.7%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 86.7%

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

   9. Simplified 86.8%

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

   10. Taylor expanded in y around 0 50.2%

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

       1. associate-/r* 50.3%

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

       2. div-inv 50.2%

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

   12. Applied egg-rr 50.2%

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

       1. associate-*r* 65.9%

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

       2. associate-/l* 65.9%

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

       3. clear-num 65.8%

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

       4. associate-*l/ 66.3%

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

       5. *-un-lft-identity 66.3%

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

   14. Applied egg-rr 66.3%

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

Alternative 12: 66.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+206}:\\ \;\;\;\;1 + \frac{\frac{1}{y}}{x \cdot \frac{9}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111 \cdot \frac{y}{x}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e+206)
   (+ 1.0 (/ (/ 1.0 y) (* x (/ 9.0 y))))
   (+ 1.0 (/ (* -0.1111111111111111 (/ y x)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e+206) {
		tmp = 1.0 + ((1.0 / y) / (x * (9.0 / y)));
	} else {
		tmp = 1.0 + ((-0.1111111111111111 * (y / x)) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1d+206)) then
        tmp = 1.0d0 + ((1.0d0 / y) / (x * (9.0d0 / y)))
    else
        tmp = 1.0d0 + (((-0.1111111111111111d0) * (y / x)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1e+206) {
		tmp = 1.0 + ((1.0 / y) / (x * (9.0 / y)));
	} else {
		tmp = 1.0 + ((-0.1111111111111111 * (y / x)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1e+206:
		tmp = 1.0 + ((1.0 / y) / (x * (9.0 / y)))
	else:
		tmp = 1.0 + ((-0.1111111111111111 * (y / x)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e+206)
		tmp = Float64(1.0 + Float64(Float64(1.0 / y) / Float64(x * Float64(9.0 / y))));
	else
		tmp = Float64(1.0 + Float64(Float64(-0.1111111111111111 * Float64(y / x)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1e+206)
		tmp = 1.0 + ((1.0 / y) / (x * (9.0 / y)));
	else
		tmp = 1.0 + ((-0.1111111111111111 * (y / x)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e+206], N[(1.0 + N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[(9.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-0.1111111111111111 * N[(y / x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e206

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. metadata-eval 99.7%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow1/2 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in y around inf 99.6%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in y around 0 2.3%

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

       1. associate-/r* 2.3%

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

       2. div-inv 2.3%

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

   12. Applied egg-rr 2.3%

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

       1. associate-*r* 2.2%

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

       2. associate-/l* 2.2%

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

       3. clear-num 2.2%

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

       4. associate-*l/ 2.2%

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

       5. *-un-lft-identity 2.2%

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

   14. Applied egg-rr 2.2%

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

       1. clear-num 2.2%

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

       2. associate-/l* 2.2%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 49.3%

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

       5. sqr-neg 49.3%

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

       6. sqrt-unprod 49.3%

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

       7. add-sqr-sqrt 49.3%

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

       8. distribute-neg-frac 49.3%

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

       9. distribute-neg-frac 49.3%

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

       10. *-un-lft-identity 49.3%

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

       11. *-inverses 49.3%

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

       12. times-frac 49.3%

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

       13. *-commutative 49.3%

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

       14. clear-num 49.3%

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

       15. distribute-rgt-neg-out 49.3%

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

       16. *-commutative 49.3%

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

       17. times-frac 49.3%

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

       18. *-inverses 49.3%

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

       19. *-un-lft-identity 49.3%

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

       20. clear-num 49.3%

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

       21. associate-/l* 49.3%

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

   16. Applied egg-rr 2.3%

       \[\leadsto 1 + \frac{\frac{1}{y}}{\color{blue}{0 - x \cdot \frac{-9}{y}}} \]
   17. Step-by-step derivation

       1. neg-sub0 49.3%

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

       2. distribute-rgt-neg-in 49.3%

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

       3. distribute-neg-frac 49.3%

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

       4. metadata-eval 49.3%

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

   18. Simplified 49.3%

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

   if -1e206 < y

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. metadata-eval 99.7%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 86.7%

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

   9. Simplified 86.8%

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

   10. Taylor expanded in y around 0 50.2%

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

       1. associate-*r/ 50.8%

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

       2. associate-/r* 65.9%

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

   12. Applied egg-rr 65.9%

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

   13. Taylor expanded in y around 0 65.9%

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

Alternative 13: 61.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{0.1111111111111111}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.7e-14) (/ 0.1111111111111111 (- x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.7e-14) {
		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 <= 4.7d-14) 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 <= 4.7e-14) {
		tmp = 0.1111111111111111 / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.7e-14:
		tmp = 0.1111111111111111 / -x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.7e-14)
		tmp = Float64(0.1111111111111111 / Float64(-x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.7e-14)
		tmp = 0.1111111111111111 / -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.7e-14], N[(0.1111111111111111 / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.7000000000000002e-14

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.6%

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

      1. mul-1-neg 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in y around 0 58.7%

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

   if 4.7000000000000002e-14 < 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 97.2%

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

   4. Taylor expanded in y around 0 51.0%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{0.1111111111111111}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.0% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

4. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in y around inf 88.2%

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

9. Simplified 88.3%

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

10. Taylor expanded in y around 0 44.6%

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

    1. associate-*r/ 45.2%

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

    2. associate-/r* 58.4%

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

12. Applied egg-rr 58.4%

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

13. Taylor expanded in y around 0 58.5%

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

Alternative 15: 62.6% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-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* 99.6%

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

   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* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 55.9%

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

   1. expm1-log1p-u 26.8%

      \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

   2. log1p-define 26.8%

      \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{-0.1111111111111111}{x}\right)}\right) \]

   3. expm1-undefine 26.8%

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

   4. add-exp-log 55.9%

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

7. Applied egg-rr 55.9%

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

   1. associate--l+ 55.9%

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

9. Simplified 55.9%

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

    1. associate-+r- 55.9%

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

    2. add-exp-log 26.8%

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

    3. log1p-undefine 26.8%

       \[\leadsto 1 + \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)}} - 1\right) \]

    4. expm1-undefine 26.8%

       \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

    5. expm1-log1p-u 55.9%

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

    6. metadata-eval 55.9%

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

    7. distribute-neg-frac 55.9%

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

    8. clear-num 55.9%

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

    9. distribute-neg-frac 55.9%

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

    10. metadata-eval 55.9%

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

    11. div-inv 55.9%

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

    12. metadata-eval 55.9%

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

11. Applied egg-rr 55.9%

    \[\leadsto 1 + \color{blue}{\frac{-1}{x \cdot 9}} \]
12. Add Preprocessing

Alternative 16: 62.5% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   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* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 55.9%

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

Alternative 17: 32.0% 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. Add Preprocessing

3. Taylor expanded in x around inf 70.7%

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

4. Taylor expanded in y around 0 27.5%

   \[\leadsto \color{blue}{1} \]
5. 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 2024107 
(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)))))