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

Percentage Accurate: 99.7% → 99.6%

Time: 8.5s

Alternatives: 13

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/r* 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}} \]
4. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

5. Applied egg-rr 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
6. Step-by-step derivation

   1. unpow1/2 99.7%

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

7. Simplified 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 72.4%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\sqrt{x \cdot 9}}\right)\right)} \]

   2. expm1-udef 72.4%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y}{\sqrt{x \cdot 9}}\right)} - 1\right)} \]

   3. sqrt-prod 72.4%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \left(e^{\mathsf{log1p}\left(\frac{y}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}}\right)} - 1\right) \]

   4. metadata-eval 72.4%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \left(e^{\mathsf{log1p}\left(\frac{y}{\sqrt{x} \cdot \color{blue}{3}}\right)} - 1\right) \]

9. Applied egg-rr 72.4%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y}{\sqrt{x} \cdot 3}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 72.4%

       \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\sqrt{x} \cdot 3}\right)\right)} \]

    2. expm1-log1p 99.7%

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

    3. associate-/r* 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+52} \lor \neg \left(y \leq 2.05 \cdot 10^{+65}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.6e+52) (not (<= y 2.05e+65)))
   (+ 1.0 (* (/ y (sqrt x)) -0.3333333333333333))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.6e+52) || !(y <= 2.05e+65)) {
		tmp = 1.0 + ((y / sqrt(x)) * -0.3333333333333333);
	} 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 <= (-4.6d+52)) .or. (.not. (y <= 2.05d+65))) then
        tmp = 1.0d0 + ((y / sqrt(x)) * (-0.3333333333333333d0))
    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 <= -4.6e+52) || !(y <= 2.05e+65)) {
		tmp = 1.0 + ((y / Math.sqrt(x)) * -0.3333333333333333);
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.6e+52) or not (y <= 2.05e+65):
		tmp = 1.0 + ((y / math.sqrt(x)) * -0.3333333333333333)
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.6e+52) || !(y <= 2.05e+65))
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) * -0.3333333333333333));
	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 <= -4.6e+52) || ~((y <= 2.05e+65)))
		tmp = 1.0 + ((y / sqrt(x)) * -0.3333333333333333);
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.6e+52], N[Not[LessEqual[y, 2.05e+65]], $MachinePrecision]], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $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 < -4.6e52 or 2.0500000000000001e65 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-def 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. distribute-neg-frac 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 95.0%

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

      1. *-commutative 95.0%

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

   6. Simplified 95.0%

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

      1. pow1/2 85.5%

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

      2. inv-pow 85.5%

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

      3. pow-pow 85.5%

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

      4. metadata-eval 85.5%

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

      5. metadata-eval 85.5%

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

      6. pow-prod-up 85.4%

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

      7. pow1/2 85.4%

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

      8. inv-pow 85.4%

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

   8. Applied egg-rr 94.9%

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

      1. associate-*r/ 85.4%

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

      2. *-rgt-identity 85.4%

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

   10. Simplified 94.9%

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

       1. associate-*r/ 71.8%

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

       2. associate-/l* 85.4%

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

       3. pow1 85.4%

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

       4. pow1/2 85.4%

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

       5. pow-div 85.6%

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

       6. metadata-eval 85.6%

          \[\leadsto \frac{y}{{x}^{\color{blue}{0.5}}} \cdot -0.3333333333333333 \]

       7. pow1/2 85.6%

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

       8. expm1-log1p-u 37.5%

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

       9. expm1-udef 37.4%

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

   12. Applied egg-rr 46.2%

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

       1. expm1-def 37.5%

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

       2. expm1-log1p 85.6%

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

   14. Simplified 95.2%

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

   if -4.6e52 < y < 2.0500000000000001e65

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. *-commutative 99.7%

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

      9. fma-def 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-/r* 99.7%

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

      13. distribute-neg-frac 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 95.0%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

   6. Simplified 95.1%

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

      1. div-inv 95.0%

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

      2. metadata-eval 95.0%

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

      3. inv-pow 95.0%

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

      4. unpow-prod-down 95.1%

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

      5. *-commutative 95.1%

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

   8. Applied egg-rr 95.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+52} \lor \neg \left(y \leq 2.05 \cdot 10^{+65}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]

Alternative 3: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+54} \lor \neg \left(y \leq 4.2 \cdot 10^{+66}\right):\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.1e+54) (not (<= y 4.2e+66)))
   (+ 1.0 (/ (/ y (sqrt x)) -3.0))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+54) || !(y <= 4.2e+66)) {
		tmp = 1.0 + ((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 <= (-2.1d+54)) .or. (.not. (y <= 4.2d+66))) then
        tmp = 1.0d0 + ((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 <= -2.1e+54) || !(y <= 4.2e+66)) {
		tmp = 1.0 + ((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 <= -2.1e+54) or not (y <= 4.2e+66):
		tmp = 1.0 + ((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 <= -2.1e+54) || !(y <= 4.2e+66))
		tmp = Float64(1.0 + Float64(Float64(y / 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 <= -2.1e+54) || ~((y <= 4.2e+66)))
		tmp = 1.0 + ((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, -2.1e+54], N[Not[LessEqual[y, 4.2e+66]], $MachinePrecision]], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $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 < -2.09999999999999986e54 or 4.20000000000000011e66 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-def 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. distribute-neg-frac 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 95.0%

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

      1. *-commutative 95.0%

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

   6. Simplified 95.0%

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

      1. pow1/2 85.5%

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

      2. inv-pow 85.5%

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

      3. pow-pow 85.5%

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

      4. metadata-eval 85.5%

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

      5. metadata-eval 85.5%

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

      6. pow-prod-up 85.4%

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

      7. pow1/2 85.4%

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

      8. inv-pow 85.4%

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

   8. Applied egg-rr 94.9%

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

      1. associate-*r/ 85.4%

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

      2. *-rgt-identity 85.4%

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

   10. Simplified 94.9%

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

       1. associate-*l* 85.4%

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

       2. pow1/2 85.4%

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

       3. pow1 85.4%

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

       4. pow-div 85.4%

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

       5. metadata-eval 85.4%

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

       6. metadata-eval 85.4%

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

       7. sqrt-pow1 85.4%

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

       8. inv-pow 85.4%

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

       9. associate-*r* 85.5%

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

       10. add-sqr-sqrt 85.5%

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

       11. sqr-neg 85.5%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 1.0%

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

       14. distribute-rgt-neg-in 1.0%

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

       15. sqrt-div 1.0%

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

       16. metadata-eval 1.0%

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

       17. div-inv 1.0%

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

       18. metadata-eval 1.0%

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

       19. metadata-eval 1.0%

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

       20. div-inv 1.0%

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

   12. Applied egg-rr 95.4%

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

   if -2.09999999999999986e54 < y < 4.20000000000000011e66

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. *-commutative 99.7%

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

      9. fma-def 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-/r* 99.7%

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

      13. distribute-neg-frac 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 95.0%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

   6. Simplified 95.1%

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

      1. div-inv 95.0%

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

      2. metadata-eval 95.0%

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

      3. inv-pow 95.0%

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

      4. unpow-prod-down 95.1%

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

      5. *-commutative 95.1%

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

   8. Applied egg-rr 95.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+54} \lor \neg \left(y \leq 4.2 \cdot 10^{+66}\right):\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.7%

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

   8. *-commutative 99.7%

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

   9. fma-def 99.7%

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

   10. metadata-eval 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-/r* 99.6%

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

   13. distribute-neg-frac 99.6%

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

   14. metadata-eval 99.6%

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

   15. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-/r* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/r* 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}} \]
4. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

5. Applied egg-rr 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
6. Step-by-step derivation

   1. unpow1/2 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/r* 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 8: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+55} \lor \neg \left(y \leq 1.35 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.3e+55) (not (<= y 1.35e+100)))
   (/ (/ y (sqrt x)) -3.0)
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.3e+55) || !(y <= 1.35e+100)) {
		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 <= (-3.3d+55)) .or. (.not. (y <= 1.35d+100))) 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 <= -3.3e+55) || !(y <= 1.35e+100)) {
		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 <= -3.3e+55) or not (y <= 1.35e+100):
		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 <= -3.3e+55) || !(y <= 1.35e+100))
		tmp = Float64(Float64(y / 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 <= -3.3e+55) || ~((y <= 1.35e+100)))
		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, -3.3e+55], N[Not[LessEqual[y, 1.35e+100]], $MachinePrecision]], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $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 < -3.3e55 or 1.34999999999999999e100 < 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. *-commutative 99.6%

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

      2. associate-/r* 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 89.9%

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

      1. *-commutative 89.9%

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

   10. Simplified 89.9%

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

       1. pow1/2 89.9%

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

       2. inv-pow 89.9%

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

       3. pow-pow 89.9%

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

       4. metadata-eval 89.9%

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

       5. metadata-eval 89.9%

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

       6. pow-prod-up 89.8%

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

       7. pow1/2 89.8%

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

       8. inv-pow 89.8%

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

   12. Applied egg-rr 89.8%

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

       1. associate-*r/ 89.8%

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

       2. *-rgt-identity 89.8%

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

   14. Simplified 89.8%

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

       1. associate-*l* 89.7%

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

       2. pow1/2 89.7%

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

       3. pow1 89.7%

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

       4. pow-div 89.8%

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

       5. metadata-eval 89.8%

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

       6. metadata-eval 89.8%

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

       7. sqrt-pow1 89.8%

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

       8. inv-pow 89.8%

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

       9. associate-*r* 89.9%

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

       10. add-sqr-sqrt 89.9%

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

       11. sqr-neg 89.9%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 0.8%

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

       14. distribute-rgt-neg-in 0.8%

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

       15. sqrt-div 0.8%

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

       16. metadata-eval 0.8%

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

       17. div-inv 0.8%

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

       18. metadata-eval 0.8%

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

       19. metadata-eval 0.8%

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

       20. div-inv 0.8%

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

   16. Applied egg-rr 90.2%

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

   if -3.3e55 < y < 1.34999999999999999e100

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. *-commutative 99.7%

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

      9. fma-def 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-/r* 99.7%

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

      13. distribute-neg-frac 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 92.7%

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

      1. associate-*r/ 92.8%

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

      2. metadata-eval 92.8%

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

   6. Simplified 92.8%

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

      1. div-inv 92.7%

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

      2. metadata-eval 92.7%

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

      3. inv-pow 92.7%

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

      4. unpow-prod-down 92.8%

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

      5. *-commutative 92.8%

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

   8. Applied egg-rr 92.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+55} \lor \neg \left(y \leq 1.35 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]

Alternative 9: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+55} \lor \neg \left(y \leq 2.6 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.3e+55) (not (<= y 2.6e+100)))
   (* (/ y (sqrt x)) -0.3333333333333333)
   (- 1.0 (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.3e+55) || !(y <= 2.6e+100)) {
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3.3d+55)) .or. (.not. (y <= 2.6d+100))) then
        tmp = (y / sqrt(x)) * (-0.3333333333333333d0)
    else
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.3e+55) || !(y <= 2.6e+100)) {
		tmp = (y / Math.sqrt(x)) * -0.3333333333333333;
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.3e+55) or not (y <= 2.6e+100):
		tmp = (y / math.sqrt(x)) * -0.3333333333333333
	else:
		tmp = 1.0 - (0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.3e+55) || !(y <= 2.6e+100))
		tmp = Float64(Float64(y / sqrt(x)) * -0.3333333333333333);
	else
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.3e+55) || ~((y <= 2.6e+100)))
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	else
		tmp = 1.0 - (0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.3e+55], N[Not[LessEqual[y, 2.6e+100]], $MachinePrecision]], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e55 or 2.6000000000000002e100 < 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. *-commutative 99.6%

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

      2. associate-/r* 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 89.9%

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

      1. *-commutative 89.9%

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

   10. Simplified 89.9%

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

       1. pow1/2 89.9%

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

       2. inv-pow 89.9%

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

       3. pow-pow 89.9%

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

       4. metadata-eval 89.9%

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

       5. metadata-eval 89.9%

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

       6. pow-prod-up 89.8%

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

       7. pow1/2 89.8%

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

       8. inv-pow 89.8%

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

   12. Applied egg-rr 89.8%

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

       1. associate-*r/ 89.8%

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

       2. *-rgt-identity 89.8%

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

   14. Simplified 89.8%

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

       1. associate-*r/ 74.9%

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

       2. associate-/l* 89.8%

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

       3. pow1 89.8%

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

       4. pow1/2 89.8%

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

       5. pow-div 90.0%

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

       6. metadata-eval 90.0%

          \[\leadsto \frac{y}{{x}^{\color{blue}{0.5}}} \cdot -0.3333333333333333 \]

       7. pow1/2 90.0%

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

       8. expm1-log1p-u 37.4%

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

       9. expm1-udef 37.2%

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

   16. Applied egg-rr 37.2%

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

       1. expm1-def 37.4%

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

       2. expm1-log1p 90.0%

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

   18. Simplified 90.0%

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

   if -3.3e55 < y < 2.6000000000000002e100

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. *-commutative 99.7%

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

      9. fma-def 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-/r* 99.7%

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

      13. distribute-neg-frac 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 92.7%

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

      1. associate-*r/ 92.8%

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

      2. metadata-eval 92.8%

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

   6. Simplified 92.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+55} \lor \neg \left(y \leq 2.6 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]

Alternative 10: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+55} \lor \neg \left(y \leq 1.35 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.45e+55) (not (<= y 1.35e+100)))
   (/ (/ y (sqrt x)) -3.0)
   (- 1.0 (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e+55) || !(y <= 1.35e+100)) {
		tmp = (y / sqrt(x)) / -3.0;
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.45d+55)) .or. (.not. (y <= 1.35d+100))) then
        tmp = (y / sqrt(x)) / (-3.0d0)
    else
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e+55) || !(y <= 1.35e+100)) {
		tmp = (y / Math.sqrt(x)) / -3.0;
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.45e+55) or not (y <= 1.35e+100):
		tmp = (y / math.sqrt(x)) / -3.0
	else:
		tmp = 1.0 - (0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.45e+55) || !(y <= 1.35e+100))
		tmp = Float64(Float64(y / sqrt(x)) / -3.0);
	else
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.45e+55) || ~((y <= 1.35e+100)))
		tmp = (y / sqrt(x)) / -3.0;
	else
		tmp = 1.0 - (0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.45e+55], N[Not[LessEqual[y, 1.35e+100]], $MachinePrecision]], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4499999999999999e55 or 1.34999999999999999e100 < 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. *-commutative 99.6%

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

      2. associate-/r* 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 89.9%

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

      1. *-commutative 89.9%

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

   10. Simplified 89.9%

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

       1. pow1/2 89.9%

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

       2. inv-pow 89.9%

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

       3. pow-pow 89.9%

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

       4. metadata-eval 89.9%

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

       5. metadata-eval 89.9%

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

       6. pow-prod-up 89.8%

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

       7. pow1/2 89.8%

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

       8. inv-pow 89.8%

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

   12. Applied egg-rr 89.8%

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

       1. associate-*r/ 89.8%

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

       2. *-rgt-identity 89.8%

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

   14. Simplified 89.8%

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

       1. associate-*l* 89.7%

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

       2. pow1/2 89.7%

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

       3. pow1 89.7%

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

       4. pow-div 89.8%

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

       5. metadata-eval 89.8%

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

       6. metadata-eval 89.8%

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

       7. sqrt-pow1 89.8%

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

       8. inv-pow 89.8%

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

       9. associate-*r* 89.9%

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

       10. add-sqr-sqrt 89.9%

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

       11. sqr-neg 89.9%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 0.8%

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

       14. distribute-rgt-neg-in 0.8%

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

       15. sqrt-div 0.8%

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

       16. metadata-eval 0.8%

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

       17. div-inv 0.8%

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

       18. metadata-eval 0.8%

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

       19. metadata-eval 0.8%

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

       20. div-inv 0.8%

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

   16. Applied egg-rr 90.2%

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

   if -1.4499999999999999e55 < y < 1.34999999999999999e100

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. *-commutative 99.7%

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

      9. fma-def 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-/r* 99.7%

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

      13. distribute-neg-frac 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 92.7%

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

      1. associate-*r/ 92.8%

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

      2. metadata-eval 92.8%

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

   6. Simplified 92.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+55} \lor \neg \left(y \leq 1.35 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]

Alternative 11: 61.7% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.05:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.05) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.05) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.05d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.05) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.05:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.05)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.05)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.05], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.050000000000000003

   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. *-commutative 99.6%

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

      2. associate-/r* 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
   6. Step-by-step derivation

      1. unpow1/2 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 66.0%

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

   if 0.050000000000000003 < x

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. *-commutative 99.8%

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

      9. fma-def 99.8%

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

      10. metadata-eval 99.8%

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

      11. *-commutative 99.8%

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

      12. associate-/r* 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 57.1%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.05:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 62.6% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.7%

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

   8. *-commutative 99.7%

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

   9. fma-def 99.7%

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

   10. metadata-eval 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-/r* 99.6%

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

   13. distribute-neg-frac 99.6%

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

   14. metadata-eval 99.6%

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

   15. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 62.2%

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

   1. associate-*r/ 62.3%

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

   2. metadata-eval 62.3%

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

6. Simplified 62.3%

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

7. Final simplification 62.3%

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

Alternative 13: 31.5% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.7%

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

   8. *-commutative 99.7%

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

   9. fma-def 99.7%

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

   10. metadata-eval 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-/r* 99.6%

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

   13. distribute-neg-frac 99.6%

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

   14. metadata-eval 99.6%

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

   15. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around inf 30.1%

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

5. Final simplification 30.1%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023195 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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