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

Percentage Accurate: 99.7% → 99.7%

Time: 10.8s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+23} \lor \neg \left(y \leq 4.2 \cdot 10^{+60}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.45e+23) (not (<= y 4.2e+60)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e+23) || !(y <= 4.2e+60)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.45d+23)) .or. (.not. (y <= 4.2d+60))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e+23) || !(y <= 4.2e+60)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.45e+23) or not (y <= 4.2e+60):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.45e+23) || !(y <= 4.2e+60))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.45e+23) || ~((y <= 4.2e+60)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.45e+23], N[Not[LessEqual[y, 4.2e+60]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45000000000000006e23 or 4.2000000000000002e60 < 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. *-commutative 99.6%

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

      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 99.5%

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

      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 95.3%

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

   if -1.45000000000000006e23 < y < 4.2000000000000002e60

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.0%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
   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. div-inv 99.0%

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

      5. metadata-eval 99.0%

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

      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 99.0%

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

      9. sqrt-unprod 80.6%

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

      10. frac-times 80.6%

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

      11. metadata-eval 80.6%

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

      12. div-inv 80.6%

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

      13. metadata-eval 80.6%

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

      14. div-inv 80.5%

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

      15. frac-times 80.5%

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

      16. clear-num 80.6%

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

      17. clear-num 80.6%

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

      18. frac-times 80.7%

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

      19. metadata-eval 80.7%

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

      20. metadata-eval 80.7%

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

      21. frac-times 80.6%

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

      22. sqrt-unprod 0.0%

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

      23. add-sqr-sqrt 56.6%

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

   7. Applied egg-rr 56.6%

      \[\leadsto \color{blue}{1 - \frac{-0.1111111111111111}{x}} \]
   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 80.6%

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

      3. frac-times 80.7%

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

      4. metadata-eval 80.7%

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

      5. metadata-eval 80.7%

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

      6. frac-times 80.6%

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

      7. clear-num 80.6%

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

      8. clear-num 80.5%

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

      9. frac-times 80.5%

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

      10. div-inv 80.6%

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

      11. metadata-eval 80.6%

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

      12. div-inv 80.6%

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

      13. metadata-eval 80.6%

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

      14. frac-times 80.6%

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

      15. sqrt-unprod 99.0%

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

      16. add-sqr-sqrt 99.2%

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

      17. inv-pow 99.2%

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

   9. Applied egg-rr 99.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+23} \lor \neg \left(y \leq 4.2 \cdot 10^{+60}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+23}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+60}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+23)
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (if (<= y 3.5e+60)
     (- 1.0 (pow (* x 9.0) -1.0))
     (- 1.0 (/ y (* 3.0 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+23) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else if (y <= 3.5e+60) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - (y / (3.0 * 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 <= (-1.45d+23)) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else if (y <= 3.5d+60) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+23) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else if (y <= 3.5e+60) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+23:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	elif y <= 3.5e+60:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+23)
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	elseif (y <= 3.5e+60)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+23)
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	elseif (y <= 3.5e+60)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+23], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+60], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45000000000000006e23

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in x around inf 95.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   6. Applied egg-rr 95.6%

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

      1. unpow1/2 99.6%

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

   8. Simplified 95.6%

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

   if -1.45000000000000006e23 < y < 3.5000000000000002e60

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.0%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
   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. div-inv 99.0%

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

      5. metadata-eval 99.0%

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

      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 99.0%

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

      9. sqrt-unprod 80.6%

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

      10. frac-times 80.6%

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

      11. metadata-eval 80.6%

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

      12. div-inv 80.6%

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

      13. metadata-eval 80.6%

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

      14. div-inv 80.5%

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

      15. frac-times 80.5%

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

      16. clear-num 80.6%

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

      17. clear-num 80.6%

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

      18. frac-times 80.7%

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

      19. metadata-eval 80.7%

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

      20. metadata-eval 80.7%

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

      21. frac-times 80.6%

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

      22. sqrt-unprod 0.0%

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

      23. add-sqr-sqrt 56.6%

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

   7. Applied egg-rr 56.6%

      \[\leadsto \color{blue}{1 - \frac{-0.1111111111111111}{x}} \]
   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 80.6%

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

      3. frac-times 80.7%

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

      4. metadata-eval 80.7%

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

      5. metadata-eval 80.7%

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

      6. frac-times 80.6%

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

      7. clear-num 80.6%

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

      8. clear-num 80.5%

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

      9. frac-times 80.5%

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

      10. div-inv 80.6%

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

      11. metadata-eval 80.6%

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

      12. div-inv 80.6%

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

      13. metadata-eval 80.6%

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

      14. frac-times 80.6%

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

      15. sqrt-unprod 99.0%

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

      16. add-sqr-sqrt 99.2%

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

      17. inv-pow 99.2%

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

   9. Applied egg-rr 99.2%

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

   if 3.5000000000000002e60 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   4. Taylor expanded in x around inf 95.0%

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

Alternative 4: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+23}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+60}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.02e+23)
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (if (<= y 1.7e+60)
     (- 1.0 (pow (* x 9.0) -1.0))
     (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.02e+23) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else if (y <= 1.7e+60) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (-0.3333333333333333 * (y / 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 <= (-1.02d+23)) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else if (y <= 1.7d+60) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.02e+23) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else if (y <= 1.7e+60) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.02e+23:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	elif y <= 1.7e+60:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.02e+23)
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	elseif (y <= 1.7e+60)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.02e+23)
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	elseif (y <= 1.7e+60)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.02e+23], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+60], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.02e23

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in x around inf 95.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   6. Applied egg-rr 95.6%

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

      1. unpow1/2 99.6%

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

   8. Simplified 95.6%

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

   if -1.02e23 < y < 1.7e60

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.0%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
   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. div-inv 99.0%

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

      5. metadata-eval 99.0%

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

      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 99.0%

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

      9. sqrt-unprod 80.6%

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

      10. frac-times 80.6%

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

      11. metadata-eval 80.6%

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

      12. div-inv 80.6%

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

      13. metadata-eval 80.6%

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

      14. div-inv 80.5%

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

      15. frac-times 80.5%

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

      16. clear-num 80.6%

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

      17. clear-num 80.6%

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

      18. frac-times 80.7%

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

      19. metadata-eval 80.7%

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

      20. metadata-eval 80.7%

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

      21. frac-times 80.6%

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

      22. sqrt-unprod 0.0%

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

      23. add-sqr-sqrt 56.6%

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

   7. Applied egg-rr 56.6%

      \[\leadsto \color{blue}{1 - \frac{-0.1111111111111111}{x}} \]
   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 80.6%

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

      3. frac-times 80.7%

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

      4. metadata-eval 80.7%

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

      5. metadata-eval 80.7%

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

      6. frac-times 80.6%

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

      7. clear-num 80.6%

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

      8. clear-num 80.5%

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

      9. frac-times 80.5%

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

      10. div-inv 80.6%

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

      11. metadata-eval 80.6%

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

      12. div-inv 80.6%

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

      13. metadata-eval 80.6%

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

      14. frac-times 80.6%

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

      15. sqrt-unprod 99.0%

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

      16. add-sqr-sqrt 99.2%

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

      17. inv-pow 99.2%

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

   9. Applied egg-rr 99.2%

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

   if 1.7e60 < y

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.5%

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

      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 94.9%

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

Alternative 5: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+49} \lor \neg \left(y \leq 5.8 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7e+49) (not (<= y 5.8e+95)))
   (/ y (* (sqrt x) -3.0))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7e+49) || !(y <= 5.8e+95)) {
		tmp = y / (sqrt(x) * -3.0);
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7d+49)) .or. (.not. (y <= 5.8d+95))) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7e+49) || !(y <= 5.8e+95)) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7e+49) or not (y <= 5.8e+95):
		tmp = y / (math.sqrt(x) * -3.0)
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7e+49) || !(y <= 5.8e+95))
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7e+49) || ~((y <= 5.8e+95)))
		tmp = y / (sqrt(x) * -3.0);
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7e+49], N[Not[LessEqual[y, 5.8e+95]], $MachinePrecision]], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9999999999999995e49 or 5.80000000000000027e95 < 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. *-commutative 99.6%

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

      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 99.5%

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

      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 96.9%

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

   6. Taylor expanded in y around inf 92.3%

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

      1. associate-*r* 92.2%

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

      2. *-commutative 92.2%

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

      3. *-commutative 92.2%

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

      4. unpow1/2 92.2%

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

      5. rem-exp-log 87.4%

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

      6. exp-neg 87.4%

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

      7. exp-prod 87.4%

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

      8. distribute-lft-neg-out 87.4%

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

      9. exp-neg 87.4%

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

      10. exp-to-pow 92.3%

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

      11. unpow1/2 92.3%

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

      12. unpow-1 92.3%

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

      13. metadata-eval 92.3%

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

      14. associate-/l* 92.3%

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

      15. *-rgt-identity 92.3%

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

      16. unpow-1 92.3%

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

      17. associate-/r* 92.3%

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

      18. associate-*r/ 92.5%

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

      19. *-commutative 92.5%

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

      20. *-rgt-identity 92.5%

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

      21. associate-/r* 92.4%

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

   8. Simplified 92.4%

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

      1. div-inv 92.3%

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

      2. associate-/l* 92.2%

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

      3. metadata-eval 92.2%

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

   10. Applied egg-rr 92.2%

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

       1. clear-num 92.2%

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

       2. un-div-inv 92.4%

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

       3. div-inv 92.5%

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

       4. metadata-eval 92.5%

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

   12. Applied egg-rr 92.5%

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

   if -6.9999999999999995e49 < y < 5.80000000000000027e95

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 96.4%

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

      1. div-inv 96.4%

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

      2. metadata-eval 96.4%

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

      3. cancel-sign-sub-inv 96.4%

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

      4. div-inv 96.4%

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

      5. metadata-eval 96.4%

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

      6. associate-/r* 96.5%

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

      7. *-commutative 96.5%

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

      8. add-sqr-sqrt 96.3%

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

      9. sqrt-unprod 78.0%

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

      10. frac-times 78.0%

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

      11. metadata-eval 78.0%

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

      12. div-inv 78.0%

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

      13. metadata-eval 78.0%

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

      14. div-inv 78.0%

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

      15. frac-times 78.0%

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

      16. clear-num 78.0%

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

      17. clear-num 78.0%

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

      18. frac-times 78.1%

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

      19. metadata-eval 78.1%

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

      20. metadata-eval 78.1%

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

      21. frac-times 78.0%

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

      22. sqrt-unprod 0.0%

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

      23. add-sqr-sqrt 55.6%

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

   7. Applied egg-rr 55.6%

      \[\leadsto \color{blue}{1 - \frac{-0.1111111111111111}{x}} \]
   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 78.0%

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

      3. frac-times 78.1%

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

      4. metadata-eval 78.1%

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

      5. metadata-eval 78.1%

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

      6. frac-times 78.0%

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

      7. clear-num 78.0%

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

      8. clear-num 78.0%

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

      9. frac-times 78.0%

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

      10. div-inv 78.0%

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

      11. metadata-eval 78.0%

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

      12. div-inv 78.0%

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

      13. metadata-eval 78.0%

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

      14. frac-times 78.0%

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

      15. sqrt-unprod 96.3%

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

      16. add-sqr-sqrt 96.5%

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

      17. inv-pow 96.5%

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

   9. Applied egg-rr 96.5%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+49} \lor \neg \left(y \leq 5.8 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+49} \lor \neg \left(y \leq 2.3 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7e+49) (not (<= y 2.3e+95)))
   (/ y (* (sqrt x) -3.0))
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7e+49) || !(y <= 2.3e+95)) {
		tmp = y / (sqrt(x) * -3.0);
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7d+49)) .or. (.not. (y <= 2.3d+95))) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7e+49) || !(y <= 2.3e+95)) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7e+49) or not (y <= 2.3e+95):
		tmp = y / (math.sqrt(x) * -3.0)
	else:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7e+49) || !(y <= 2.3e+95))
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	else
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7e+49) || ~((y <= 2.3e+95)))
		tmp = y / (sqrt(x) * -3.0);
	else
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7e+49], N[Not[LessEqual[y, 2.3e+95]], $MachinePrecision]], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9999999999999995e49 or 2.29999999999999997e95 < 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. *-commutative 99.6%

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

      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 99.5%

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

      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 96.9%

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

   6. Taylor expanded in y around inf 92.3%

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

      1. associate-*r* 92.2%

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

      2. *-commutative 92.2%

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

      3. *-commutative 92.2%

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

      4. unpow1/2 92.2%

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

      5. rem-exp-log 87.4%

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

      6. exp-neg 87.4%

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

      7. exp-prod 87.4%

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

      8. distribute-lft-neg-out 87.4%

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

      9. exp-neg 87.4%

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

      10. exp-to-pow 92.3%

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

      11. unpow1/2 92.3%

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

      12. unpow-1 92.3%

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

      13. metadata-eval 92.3%

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

      14. associate-/l* 92.3%

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

      15. *-rgt-identity 92.3%

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

      16. unpow-1 92.3%

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

      17. associate-/r* 92.3%

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

      18. associate-*r/ 92.5%

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

      19. *-commutative 92.5%

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

      20. *-rgt-identity 92.5%

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

      21. associate-/r* 92.4%

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

   8. Simplified 92.4%

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

      1. div-inv 92.3%

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

      2. associate-/l* 92.2%

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

      3. metadata-eval 92.2%

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

   10. Applied egg-rr 92.2%

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

       1. clear-num 92.2%

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

       2. un-div-inv 92.4%

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

       3. div-inv 92.5%

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

       4. metadata-eval 92.5%

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

   12. Applied egg-rr 92.5%

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

   if -6.9999999999999995e49 < y < 2.29999999999999997e95

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 96.4%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+49} \lor \neg \left(y \leq 2.3 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+49} \lor \neg \left(y \leq 2.6 \cdot 10^{+95}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7e+49) (not (<= y 2.6e+95)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7e+49) || !(y <= 2.6e+95)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7d+49)) .or. (.not. (y <= 2.6d+95))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7e+49) || !(y <= 2.6e+95)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7e+49) or not (y <= 2.6e+95):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7e+49) || !(y <= 2.6e+95))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7e+49) || ~((y <= 2.6e+95)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7e+49], N[Not[LessEqual[y, 2.6e+95]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9999999999999995e49 or 2.5999999999999999e95 < 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. *-commutative 99.6%

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

      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 99.5%

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

      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 96.9%

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

   6. Taylor expanded in y around inf 92.3%

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

      1. associate-*r* 92.2%

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

      2. *-commutative 92.2%

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

      3. *-commutative 92.2%

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

      4. unpow1/2 92.2%

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

      5. rem-exp-log 87.4%

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

      6. exp-neg 87.4%

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

      7. exp-prod 87.4%

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

      8. distribute-lft-neg-out 87.4%

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

      9. exp-neg 87.4%

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

      10. exp-to-pow 92.3%

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

      11. unpow1/2 92.3%

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

      12. unpow-1 92.3%

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

      13. metadata-eval 92.3%

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

      14. associate-/l* 92.3%

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

      15. *-rgt-identity 92.3%

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

      16. unpow-1 92.3%

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

      17. associate-/r* 92.3%

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

      18. associate-*r/ 92.5%

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

      19. *-commutative 92.5%

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

      20. *-rgt-identity 92.5%

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

      21. associate-/r* 92.4%

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

   8. Simplified 92.4%

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

      1. expm1-log1p-u 46.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{y}{-3}}{\sqrt{x}}\right)\right)} \]

      2. expm1-undefine 46.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{y}{-3}}{\sqrt{x}}\right)} - 1} \]

   10. Applied egg-rr 92.1%

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

       1. +-commutative 92.1%

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

       2. associate--l+ 92.2%

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

       3. metadata-eval 92.2%

          \[\leadsto y \cdot \frac{-0.3333333333333333}{\sqrt{x}} + \color{blue}{0} \]

       4. +-commutative 92.2%

          \[\leadsto \color{blue}{0 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

       5. +-lft-identity 92.2%

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

       6. *-commutative 92.2%

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

       7. associate-*l/ 92.3%

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

       8. associate-/l* 92.4%

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

   12. Simplified 92.4%

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

   if -6.9999999999999995e49 < y < 2.5999999999999999e95

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 96.4%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+49} \lor \neg \left(y \leq 2.6 \cdot 10^{+95}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (- (/ -0.1111111111111111 x) (/ y (sqrt (* x 9.0))))
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (-0.1111111111111111 / x) - (y / sqrt((x * 9.0)));
	} else {
		tmp = 1.0 - (y / (3.0 * 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 (x <= 0.11d0) then
        tmp = ((-0.1111111111111111d0) / x) - (y / sqrt((x * 9.0d0)))
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (-0.1111111111111111 / x) - (y / Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = (-0.1111111111111111 / x) - (y / math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(Float64(-0.1111111111111111 / x) - Float64(y / sqrt(Float64(x * 9.0))));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = (-0.1111111111111111 / x) - (y / sqrt((x * 9.0)));
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 97.4%

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

   if 0.110000000000000001 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.9%

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

   4. Taylor expanded in x around inf 99.3%

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

Alternative 9: 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 99.8%

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

3. Taylor expanded in x around 0 99.7%

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

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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. *-commutative 99.8%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 11: 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 99.8%

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

   1. associate--l- 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. distribute-neg-in 99.8%

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

   5. distribute-frac-neg 99.8%

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

   6. sub-neg 99.8%

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

   7. neg-mul-1 99.8%

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

   8. *-commutative 99.8%

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

   9. associate-/l* 99.7%

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

   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* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 65.7%

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

Alternative 12: 31.5% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.8%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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. *-commutative 99.8%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around inf 72.6%

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

6. Taylor expanded in y around 0 38.4%

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

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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