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

Percentage Accurate: 99.7% → 99.7%

Time: 11.3s

Alternatives: 15

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 99.8%

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

   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. Final simplification 99.8%

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

Alternative 2: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+38}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{elif}\;y \leq 750000000:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{-0.5} \cdot \frac{y}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.9e+38)
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (if (<= y 750000000.0)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (* (pow x -0.5) (/ y 3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.9e+38) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else if (y <= 750000000.0) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - (pow(x, -0.5) * (y / 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 <= (-1.9d+38)) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else if (y <= 750000000.0d0) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - ((x ** (-0.5d0)) * (y / 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.9e+38) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else if (y <= 750000000.0) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - (Math.pow(x, -0.5) * (y / 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.9e+38:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	elif y <= 750000000.0:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - (math.pow(x, -0.5) * (y / 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.9e+38)
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	elseif (y <= 750000000.0)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64((x ^ -0.5) * Float64(y / 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.9e+38)
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	elseif (y <= 750000000.0)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - ((x ^ -0.5) * (y / 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.9e+38], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 750000000.0], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Power[x, -0.5], $MachinePrecision] * N[(y / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8999999999999999e38

   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 99.8%

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

      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 x around inf 93.5%

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

   if -1.8999999999999999e38 < y < 7.5e8

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      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-*r/ 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 99.1%

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

      1. div-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. cancel-sign-sub-inv 99.0%

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

      4. un-div-inv 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-/r* 99.2%

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

      7. *-commutative 99.2%

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

      8. add-sqr-sqrt 98.9%

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

      9. sqrt-unprod 75.0%

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

      10. *-commutative 75.0%

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

      11. associate-/r* 75.0%

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

      12. metadata-eval 75.0%

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

      13. *-commutative 75.0%

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

      14. associate-/r* 75.0%

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

      15. metadata-eval 75.0%

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

      16. frac-times 75.1%

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

      17. metadata-eval 75.1%

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

      18. metadata-eval 75.1%

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

      19. frac-times 75.0%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 43.1%

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

   7. Applied egg-rr 43.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 75.0%

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

      3. pow1 75.0%

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

      4. frac-times 75.1%

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

      5. metadata-eval 75.1%

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

      6. metadata-eval 75.1%

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

      7. frac-times 75.0%

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

      8. pow-prod-down 75.0%

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

      9. pow-prod-up 75.0%

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

      10. metadata-eval 75.0%

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

      11. associate-/r* 75.0%

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

      12. *-commutative 75.0%

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

      13. pow-plus 75.0%

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

      14. pow1 75.0%

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

      15. sqrt-unprod 98.9%

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

      16. add-sqr-sqrt 99.2%

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

      17. metadata-eval 99.2%

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

      18. div-inv 99.1%

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

      19. associate-/r/ 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-/r/ 99.1%

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

       2. div-inv 99.2%

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

       3. metadata-eval 99.2%

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

   11. Applied egg-rr 99.2%

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

   if 7.5e8 < y

   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 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. sqrt-prod 99.6%

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

      3. metadata-eval 99.6%

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

      4. times-frac 99.6%

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

      5. metadata-eval 99.6%

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

      6. sqrt-div 99.8%

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

      7. inv-pow 99.8%

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

      8. sqrt-pow1 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot \frac{y}{3} \]

      9. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around inf 90.1%

      \[\leadsto \color{blue}{1} - {x}^{-0.5} \cdot \frac{y}{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+38}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{elif}\;y \leq 750000000:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{-0.5} \cdot \frac{y}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+89} \lor \neg \left(y \leq 3.7 \cdot 10^{+90}\right):\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.8e+89) (not (<= y 3.7e+90)))
   (* y (* -0.3333333333333333 (sqrt (/ 1.0 x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.8e+89) || !(y <= 3.7e+90)) {
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8.8d+89)) .or. (.not. (y <= 3.7d+90))) then
        tmp = y * ((-0.3333333333333333d0) * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8.8e+89) || !(y <= 3.7e+90)) {
		tmp = y * (-0.3333333333333333 * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.8e+89) or not (y <= 3.7e+90):
		tmp = y * (-0.3333333333333333 * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.8e+89) || !(y <= 3.7e+90))
		tmp = Float64(y * Float64(-0.3333333333333333 * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8.8e+89) || ~((y <= 3.7e+90)))
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.8e+89], N[Not[LessEqual[y, 3.7e+90]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.8000000000000001e89 or 3.7e90 < 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 inf 96.0%

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

   8. Taylor expanded in y around inf 94.0%

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

      1. associate-*r* 94.9%

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

   10. Simplified 94.9%

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

   if -8.8000000000000001e89 < y < 3.7e90

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      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-*r/ 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 94.0%

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

      1. div-inv 94.0%

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

      2. metadata-eval 94.0%

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

      3. cancel-sign-sub-inv 94.0%

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

      4. un-div-inv 94.0%

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

      5. metadata-eval 94.0%

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

      6. associate-/r* 94.1%

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

      7. *-commutative 94.1%

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

      8. add-sqr-sqrt 93.9%

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

      9. sqrt-unprod 69.7%

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

      10. *-commutative 69.7%

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

      11. associate-/r* 69.7%

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

      12. metadata-eval 69.7%

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

      13. *-commutative 69.7%

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

      14. associate-/r* 69.7%

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

      15. metadata-eval 69.7%

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

      16. frac-times 69.7%

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

      17. metadata-eval 69.7%

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

      18. metadata-eval 69.7%

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

      19. frac-times 69.7%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 69.7%

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

      3. pow1 69.7%

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

      4. frac-times 69.7%

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

      5. metadata-eval 69.7%

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

      6. metadata-eval 69.7%

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

      7. frac-times 69.7%

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

      8. pow-prod-down 69.7%

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

      9. pow-prod-up 69.7%

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

      10. metadata-eval 69.7%

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

      11. associate-/r* 69.7%

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

      12. *-commutative 69.7%

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

      13. pow-plus 69.7%

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

      14. pow1 69.7%

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

      15. sqrt-unprod 93.9%

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

      16. add-sqr-sqrt 94.1%

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

      17. metadata-eval 94.1%

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

      18. div-inv 94.1%

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

      19. associate-/r/ 94.0%

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

   9. Applied egg-rr 94.0%

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

       1. associate-/r/ 94.1%

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

       2. div-inv 94.1%

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

       3. metadata-eval 94.1%

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

   11. Applied egg-rr 94.1%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+89} \lor \neg \left(y \leq 3.7 \cdot 10^{+90}\right):\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+36} \lor \neg \left(y \leq 750000000\right):\\ \;\;\;\;1 - y \cdot \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.05e+36) (not (<= y 750000000.0)))
   (- 1.0 (* y (sqrt (/ 0.1111111111111111 x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.05e+36) || !(y <= 750000000.0)) {
		tmp = 1.0 - (y * sqrt((0.1111111111111111 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.05d+36)) .or. (.not. (y <= 750000000.0d0))) then
        tmp = 1.0d0 - (y * sqrt((0.1111111111111111d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.05e+36) || !(y <= 750000000.0)) {
		tmp = 1.0 - (y * Math.sqrt((0.1111111111111111 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.05e+36) or not (y <= 750000000.0):
		tmp = 1.0 - (y * math.sqrt((0.1111111111111111 / x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.05e+36) || !(y <= 750000000.0))
		tmp = Float64(1.0 - Float64(y * sqrt(Float64(0.1111111111111111 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.05e+36) || ~((y <= 750000000.0)))
		tmp = 1.0 - (y * sqrt((0.1111111111111111 / x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.05e+36], N[Not[LessEqual[y, 750000000.0]], $MachinePrecision]], N[(1.0 - N[(y * N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05000000000000002e36 or 7.5e8 < 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 inf 91.4%

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

      1. clear-num 91.3%

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

      2. associate-/r/ 91.4%

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

      3. metadata-eval 91.4%

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

      4. sqrt-div 91.4%

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

      5. *-commutative 91.4%

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

      6. associate-/r* 91.4%

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

      7. metadata-eval 91.4%

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

      8. sqrt-div 91.3%

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

      9. metadata-eval 91.3%

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

   9. Applied egg-rr 91.3%

      \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{\sqrt{x}} \cdot y} \]
   10. Step-by-step derivation

       1. metadata-eval 91.3%

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

       2. sqrt-div 91.4%

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

       3. metadata-eval 91.4%

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

       4. associate-*l/ 91.4%

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

       5. pow1/2 91.4%

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

       6. associate-*l/ 91.4%

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

       7. metadata-eval 91.4%

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

   11. Applied egg-rr 91.4%

       \[\leadsto 1 - \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \cdot y \]
   12. Step-by-step derivation

       1. unpow1/2 91.4%

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

   13. Simplified 91.4%

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

   if -1.05000000000000002e36 < y < 7.5e8

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      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-*r/ 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 99.1%

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

      1. div-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. cancel-sign-sub-inv 99.0%

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

      4. un-div-inv 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-/r* 99.2%

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

      7. *-commutative 99.2%

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

      8. add-sqr-sqrt 98.9%

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

      9. sqrt-unprod 75.0%

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

      10. *-commutative 75.0%

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

      11. associate-/r* 75.0%

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

      12. metadata-eval 75.0%

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

      13. *-commutative 75.0%

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

      14. associate-/r* 75.0%

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

      15. metadata-eval 75.0%

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

      16. frac-times 75.1%

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

      17. metadata-eval 75.1%

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

      18. metadata-eval 75.1%

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

      19. frac-times 75.0%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 43.1%

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

   7. Applied egg-rr 43.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 75.0%

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

      3. pow1 75.0%

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

      4. frac-times 75.1%

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

      5. metadata-eval 75.1%

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

      6. metadata-eval 75.1%

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

      7. frac-times 75.0%

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

      8. pow-prod-down 75.0%

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

      9. pow-prod-up 75.0%

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

      10. metadata-eval 75.0%

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

      11. associate-/r* 75.0%

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

      12. *-commutative 75.0%

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

      13. pow-plus 75.0%

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

      14. pow1 75.0%

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

      15. sqrt-unprod 98.9%

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

      16. add-sqr-sqrt 99.2%

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

      17. metadata-eval 99.2%

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

      18. div-inv 99.1%

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

      19. associate-/r/ 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-/r/ 99.1%

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

       2. div-inv 99.2%

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

       3. metadata-eval 99.2%

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

   11. Applied egg-rr 99.2%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+36} \lor \neg \left(y \leq 750000000\right):\\ \;\;\;\;1 - y \cdot \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+36}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+87}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.02e+36)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 7e+87)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* y (* -0.3333333333333333 (sqrt (/ 1.0 x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.02e+36) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 7e+87) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.02d+36)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 7d+87) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = y * ((-0.3333333333333333d0) * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.02e+36) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 7e+87) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = y * (-0.3333333333333333 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.02e+36:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 7e+87:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = y * (-0.3333333333333333 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.02e+36)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 7e+87)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.02e+36)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 7e+87)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.02e+36], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+87], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.02000000000000003e36

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 93.2%

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

   if -1.02000000000000003e36 < y < 6.99999999999999972e87

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      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-*r/ 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 95.1%

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

      1. div-inv 95.1%

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

      2. metadata-eval 95.1%

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

      3. cancel-sign-sub-inv 95.1%

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

      4. un-div-inv 95.1%

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

      5. metadata-eval 95.1%

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

      6. associate-/r* 95.2%

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

      7. *-commutative 95.2%

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

      8. add-sqr-sqrt 94.9%

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

      9. sqrt-unprod 70.4%

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

      10. *-commutative 70.4%

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

      11. associate-/r* 70.4%

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

      12. metadata-eval 70.4%

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

      13. *-commutative 70.4%

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

      14. associate-/r* 70.3%

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

      15. metadata-eval 70.3%

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

      16. frac-times 70.4%

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

      17. metadata-eval 70.4%

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

      18. metadata-eval 70.4%

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

      19. frac-times 70.3%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 41.8%

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

   7. Applied egg-rr 41.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 70.3%

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

      3. pow1 70.3%

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

      4. frac-times 70.4%

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

      5. metadata-eval 70.4%

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

      6. metadata-eval 70.4%

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

      7. frac-times 70.3%

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

      8. pow-prod-down 70.3%

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

      9. pow-prod-up 70.3%

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

      10. metadata-eval 70.3%

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

      11. associate-/r* 70.4%

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

      12. *-commutative 70.4%

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

      13. pow-plus 70.4%

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

      14. pow1 70.4%

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

      15. sqrt-unprod 94.9%

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

      16. add-sqr-sqrt 95.2%

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

      17. metadata-eval 95.2%

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

      18. div-inv 95.1%

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

      19. associate-/r/ 95.1%

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

   9. Applied egg-rr 95.1%

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

       1. associate-/r/ 95.1%

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

       2. div-inv 95.2%

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

       3. metadata-eval 95.2%

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

   11. Applied egg-rr 95.2%

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

   if 6.99999999999999972e87 < 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 inf 97.0%

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

   8. Taylor expanded in y around inf 95.3%

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

      1. associate-*r* 96.9%

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

   10. Simplified 96.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+36}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+87}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+38}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 750000000:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2e+38)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 750000000.0)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2e+38) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 750000000.0) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2d+38)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 750000000.0d0) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2e+38) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 750000000.0) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2e+38:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 750000000.0:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2e+38)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 750000000.0)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2e+38)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 750000000.0)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2e+38], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 750000000.0], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.99999999999999995e38

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 93.2%

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

   if -1.99999999999999995e38 < y < 7.5e8

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      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-*r/ 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 99.1%

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

      1. div-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. cancel-sign-sub-inv 99.0%

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

      4. un-div-inv 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-/r* 99.2%

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

      7. *-commutative 99.2%

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

      8. add-sqr-sqrt 98.9%

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

      9. sqrt-unprod 75.0%

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

      10. *-commutative 75.0%

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

      11. associate-/r* 75.0%

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

      12. metadata-eval 75.0%

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

      13. *-commutative 75.0%

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

      14. associate-/r* 75.0%

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

      15. metadata-eval 75.0%

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

      16. frac-times 75.1%

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

      17. metadata-eval 75.1%

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

      18. metadata-eval 75.1%

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

      19. frac-times 75.0%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 43.1%

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

   7. Applied egg-rr 43.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 75.0%

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

      3. pow1 75.0%

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

      4. frac-times 75.1%

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

      5. metadata-eval 75.1%

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

      6. metadata-eval 75.1%

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

      7. frac-times 75.0%

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

      8. pow-prod-down 75.0%

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

      9. pow-prod-up 75.0%

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

      10. metadata-eval 75.0%

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

      11. associate-/r* 75.0%

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

      12. *-commutative 75.0%

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

      13. pow-plus 75.0%

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

      14. pow1 75.0%

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

      15. sqrt-unprod 98.9%

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

      16. add-sqr-sqrt 99.2%

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

      17. metadata-eval 99.2%

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

      18. div-inv 99.1%

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

      19. associate-/r/ 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-/r/ 99.1%

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

       2. div-inv 99.2%

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

       3. metadata-eval 99.2%

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

   11. Applied egg-rr 99.2%

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

   if 7.5e8 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.6%

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

      10. fma-neg 99.6%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 88.8%

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

      1. *-commutative 88.8%

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

      2. associate-*l* 89.9%

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

      3. *-commutative 89.9%

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

   7. Simplified 89.9%

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

      1. *-commutative 89.9%

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

      2. sqrt-div 89.8%

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

      3. metadata-eval 89.8%

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

      4. un-div-inv 89.9%

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

      5. *-commutative 89.9%

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

   9. Applied egg-rr 89.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+38}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 750000000:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+36}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{elif}\;y \leq 750000000:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e+36)
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (if (<= y 750000000.0)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (* y (sqrt (/ 0.1111111111111111 x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+36) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else if (y <= 750000000.0) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - (y * sqrt((0.1111111111111111 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.05d+36)) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else if (y <= 750000000.0d0) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - (y * sqrt((0.1111111111111111d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+36) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else if (y <= 750000000.0) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - (y * Math.sqrt((0.1111111111111111 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05e+36:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	elif y <= 750000000.0:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - (y * math.sqrt((0.1111111111111111 / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e+36)
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	elseif (y <= 750000000.0)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(y * sqrt(Float64(0.1111111111111111 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.05e+36)
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	elseif (y <= 750000000.0)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - (y * sqrt((0.1111111111111111 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e+36], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 750000000.0], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y * N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000002e36

   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 99.8%

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

      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 x around inf 93.5%

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

   if -1.05000000000000002e36 < y < 7.5e8

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      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-*r/ 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 99.1%

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

      1. div-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. cancel-sign-sub-inv 99.0%

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

      4. un-div-inv 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-/r* 99.2%

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

      7. *-commutative 99.2%

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

      8. add-sqr-sqrt 98.9%

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

      9. sqrt-unprod 75.0%

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

      10. *-commutative 75.0%

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

      11. associate-/r* 75.0%

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

      12. metadata-eval 75.0%

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

      13. *-commutative 75.0%

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

      14. associate-/r* 75.0%

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

      15. metadata-eval 75.0%

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

      16. frac-times 75.1%

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

      17. metadata-eval 75.1%

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

      18. metadata-eval 75.1%

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

      19. frac-times 75.0%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 43.1%

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

   7. Applied egg-rr 43.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 75.0%

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

      3. pow1 75.0%

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

      4. frac-times 75.1%

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

      5. metadata-eval 75.1%

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

      6. metadata-eval 75.1%

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

      7. frac-times 75.0%

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

      8. pow-prod-down 75.0%

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

      9. pow-prod-up 75.0%

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

      10. metadata-eval 75.0%

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

      11. associate-/r* 75.0%

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

      12. *-commutative 75.0%

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

      13. pow-plus 75.0%

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

      14. pow1 75.0%

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

      15. sqrt-unprod 98.9%

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

      16. add-sqr-sqrt 99.2%

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

      17. metadata-eval 99.2%

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

      18. div-inv 99.1%

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

      19. associate-/r/ 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-/r/ 99.1%

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

       2. div-inv 99.2%

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

       3. metadata-eval 99.2%

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

   11. Applied egg-rr 99.2%

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

   if 7.5e8 < y

   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 99.7%

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

      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 inf 90.0%

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

      1. clear-num 89.9%

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

      2. associate-/r/ 90.0%

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

      3. metadata-eval 90.0%

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

      4. sqrt-div 90.1%

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

      5. *-commutative 90.1%

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

      6. associate-/r* 90.1%

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

      7. metadata-eval 90.1%

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

      8. sqrt-div 89.9%

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

      9. metadata-eval 89.9%

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

   9. Applied egg-rr 89.9%

      \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{\sqrt{x}} \cdot y} \]
   10. Step-by-step derivation

       1. metadata-eval 89.9%

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

       2. sqrt-div 90.1%

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

       3. metadata-eval 90.1%

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

       4. associate-*l/ 90.0%

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

       5. pow1/2 90.0%

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

       6. associate-*l/ 90.1%

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

       7. metadata-eval 90.1%

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

   11. Applied egg-rr 90.1%

       \[\leadsto 1 - \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \cdot y \]
   12. Step-by-step derivation

       1. unpow1/2 90.1%

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

   13. Simplified 90.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+36}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{elif}\;y \leq 750000000:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{-0.5} \cdot \frac{y}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112)
   (- (/ -0.1111111111111111 x) (/ y (sqrt (* x 9.0))))
   (- 1.0 (* (pow x -0.5) (/ y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = (-0.1111111111111111 / x) - (y / sqrt((x * 9.0)));
	} else {
		tmp = 1.0 - (pow(x, -0.5) * (y / 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = ((-0.1111111111111111d0) / x) - (y / sqrt((x * 9.0d0)))
    else
        tmp = 1.0d0 - ((x ** (-0.5d0)) * (y / 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = (-0.1111111111111111 / x) - (y / Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 - (Math.pow(x, -0.5) * (y / 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = (-0.1111111111111111 / x) - (y / math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 - (math.pow(x, -0.5) * (y / 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(Float64(-0.1111111111111111 / x) - Float64(y / sqrt(Float64(x * 9.0))));
	else
		tmp = Float64(1.0 - Float64((x ^ -0.5) * Float64(y / 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = (-0.1111111111111111 / x) - (y / sqrt((x * 9.0)));
	else
		tmp = 1.0 - ((x ^ -0.5) * (y / 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Power[x, -0.5], $MachinePrecision] * N[(y / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   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 98.1%

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

   if 0.112000000000000002 < x

   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. Step-by-step derivation

      1. *-un-lft-identity 99.8%

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

      2. sqrt-prod 99.7%

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

      3. metadata-eval 99.7%

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

      4. times-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. sqrt-div 99.8%

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

      7. inv-pow 99.8%

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

      8. sqrt-pow1 99.9%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot \frac{y}{3} \]

      9. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf 97.5%

      \[\leadsto \color{blue}{1} - {x}^{-0.5} \cdot \frac{y}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{-0.5} \cdot \frac{y}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.3%

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

   8. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.3%

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

   8. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 11: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 12: 61.1% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 200:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 200.0) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 200.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 200.0d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 200.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 200.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 200.0)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 200.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 200.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 200

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r- 99.7%

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

      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-*r/ 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* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 62.3%

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

      1. div-inv 62.3%

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

      2. metadata-eval 62.3%

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

      3. cancel-sign-sub-inv 62.3%

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

      4. un-div-inv 62.3%

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

      5. metadata-eval 62.3%

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

      6. associate-/r* 62.4%

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

      7. *-commutative 62.4%

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

      8. add-sqr-sqrt 62.1%

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

      9. sqrt-unprod 38.2%

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

      10. *-commutative 38.2%

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

      11. associate-/r* 38.2%

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

      12. metadata-eval 38.2%

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

      13. *-commutative 38.2%

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

      14. associate-/r* 38.1%

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

      15. metadata-eval 38.1%

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

      16. frac-times 38.2%

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

      17. metadata-eval 38.2%

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

      18. metadata-eval 38.2%

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

      19. frac-times 38.1%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 1.5%

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

   7. Applied egg-rr 1.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 38.1%

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

      3. pow1 38.1%

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

      4. frac-times 38.2%

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

      5. metadata-eval 38.2%

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

      6. metadata-eval 38.2%

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

      7. frac-times 38.1%

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

      8. pow-prod-down 38.1%

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

      9. pow-prod-up 38.1%

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

      10. metadata-eval 38.1%

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

      11. associate-/r* 38.2%

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

      12. *-commutative 38.2%

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

      13. pow-plus 38.2%

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

      14. pow1 38.2%

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

      15. sqrt-unprod 62.1%

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

      16. add-sqr-sqrt 62.4%

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

      17. metadata-eval 62.4%

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

      18. div-inv 62.4%

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

      19. associate-/r/ 62.3%

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

   9. Applied egg-rr 62.3%

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

   10. Taylor expanded in x around 0 60.9%

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

   if 200 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r- 99.7%

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

      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-*r/ 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      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 59.1%

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

      1. div-inv 59.1%

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

      2. metadata-eval 59.1%

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

      3. cancel-sign-sub-inv 59.1%

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

      4. un-div-inv 59.1%

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

      5. metadata-eval 59.1%

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

      6. associate-/r* 59.1%

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

      7. *-commutative 59.1%

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

      8. add-sqr-sqrt 59.1%

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

      9. sqrt-unprod 59.1%

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

      10. *-commutative 59.1%

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

      11. associate-/r* 59.1%

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

      12. metadata-eval 59.1%

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

      13. *-commutative 59.1%

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

      14. associate-/r* 59.1%

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

      15. metadata-eval 59.1%

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

      16. frac-times 59.1%

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

      17. metadata-eval 59.1%

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

      18. metadata-eval 59.1%

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

      19. frac-times 59.1%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 56.7%

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

   7. Applied egg-rr 56.7%

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

   8. Taylor expanded in x around inf 56.8%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 200:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.3% 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. sub-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-+r- 99.7%

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

   5. +-commutative 99.7%

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

   6. associate-+r- 99.7%

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

   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-*r/ 99.6%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.7%

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

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 60.8%

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

   1. div-inv 60.8%

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

   2. metadata-eval 60.8%

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

   3. cancel-sign-sub-inv 60.8%

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

   4. un-div-inv 60.8%

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

   5. metadata-eval 60.8%

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

   6. associate-/r* 60.9%

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

   7. *-commutative 60.9%

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

   8. add-sqr-sqrt 60.7%

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

   9. sqrt-unprod 48.1%

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

   10. *-commutative 48.1%

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

   11. associate-/r* 48.1%

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

   12. metadata-eval 48.1%

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

   13. *-commutative 48.1%

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

   14. associate-/r* 48.0%

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

   15. metadata-eval 48.0%

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

   16. frac-times 48.1%

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

   17. metadata-eval 48.1%

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

   18. metadata-eval 48.1%

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

   19. frac-times 48.0%

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

   20. sqrt-unprod 0.0%

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

   21. add-sqr-sqrt 27.6%

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

7. Applied egg-rr 27.6%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 48.0%

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

   3. pow1 48.0%

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

   4. frac-times 48.1%

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

   5. metadata-eval 48.1%

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

   6. metadata-eval 48.1%

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

   7. frac-times 48.0%

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

   8. pow-prod-down 48.0%

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

   9. pow-prod-up 48.0%

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

   10. metadata-eval 48.0%

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

   11. associate-/r* 48.1%

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

   12. *-commutative 48.1%

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

   13. pow-plus 48.1%

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

   14. pow1 48.1%

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

   15. sqrt-unprod 60.7%

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

   16. add-sqr-sqrt 60.9%

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

   17. metadata-eval 60.9%

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

   18. div-inv 60.9%

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

   19. associate-/r/ 60.8%

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

9. Applied egg-rr 60.8%

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

    1. associate-/r/ 60.9%

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

    2. div-inv 60.9%

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

    3. metadata-eval 60.9%

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

11. Applied egg-rr 60.9%

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

12. Final simplification 60.9%

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

Alternative 14: 62.2% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-+r- 99.7%

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

   5. +-commutative 99.7%

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

   6. associate-+r- 99.7%

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

   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-*r/ 99.6%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.7%

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

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 60.8%

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

6. Final simplification 60.8%

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

Alternative 15: 32.2% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-+r- 99.7%

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

   5. +-commutative 99.7%

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

   6. associate-+r- 99.7%

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

   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-*r/ 99.6%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.7%

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

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 60.8%

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

   1. div-inv 60.8%

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

   2. metadata-eval 60.8%

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

   3. cancel-sign-sub-inv 60.8%

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

   4. un-div-inv 60.8%

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

   5. metadata-eval 60.8%

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

   6. associate-/r* 60.9%

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

   7. *-commutative 60.9%

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

   8. add-sqr-sqrt 60.7%

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

   9. sqrt-unprod 48.1%

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

   10. *-commutative 48.1%

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

   11. associate-/r* 48.1%

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

   12. metadata-eval 48.1%

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

   13. *-commutative 48.1%

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

   14. associate-/r* 48.0%

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

   15. metadata-eval 48.0%

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

   16. frac-times 48.1%

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

   17. metadata-eval 48.1%

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

   18. metadata-eval 48.1%

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

   19. frac-times 48.0%

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

   20. sqrt-unprod 0.0%

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

   21. add-sqr-sqrt 27.6%

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

7. Applied egg-rr 27.6%

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

8. Taylor expanded in x around inf 27.6%

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

9. Final simplification 27.6%

   \[\leadsto 1 \]
10. 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 2024031 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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