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

Percentage Accurate: 99.7% → 99.7%

Time: 9.2s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+40} \lor \neg \left(y \leq 5.2 \cdot 10^{+44}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{{x}^{-1}}{-9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9e+40) (not (<= y 5.2e+44)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ (pow x -1.0) -9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9e+40) || !(y <= 5.2e+44)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + (pow(x, -1.0) / -9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-9d+40)) .or. (.not. (y <= 5.2d+44))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + ((x ** (-1.0d0)) / (-9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9e+40) || !(y <= 5.2e+44)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (Math.pow(x, -1.0) / -9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9e+40) or not (y <= 5.2e+44):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (math.pow(x, -1.0) / -9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9e+40) || !(y <= 5.2e+44))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64((x ^ -1.0) / -9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9e+40) || ~((y <= 5.2e+44)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + ((x ^ -1.0) / -9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9e+40], N[Not[LessEqual[y, 5.2e+44]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Power[x, -1.0], $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.00000000000000064e40 or 5.1999999999999998e44 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 95.1%

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

   if -9.00000000000000064e40 < y < 5.1999999999999998e44

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.6%

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

      1. cancel-sign-sub-inv 97.6%

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

      2. metadata-eval 97.6%

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

      3. associate-*r/ 97.7%

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

      4. metadata-eval 97.7%

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

   7. Simplified 97.7%

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

      1. expm1-log1p-u 47.2%

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

      2. expm1-undefine 47.2%

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

      3. metadata-eval 47.2%

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

      4. distribute-neg-frac 47.2%

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

      5. log1p-define 47.2%

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

      6. add-exp-log 97.7%

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

      7. div-inv 97.6%

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

      8. distribute-lft-neg-in 97.6%

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

      9. metadata-eval 97.6%

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

      10. div-inv 97.7%

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

   9. Applied egg-rr 97.7%

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

       1. associate--l+ 97.7%

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

   11. Simplified 97.7%

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

       1. associate-+r- 97.7%

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

       2. add-exp-log 47.2%

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

       3. log1p-undefine 47.2%

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

       4. expm1-undefine 47.2%

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

       5. expm1-log1p-u 97.7%

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

       6. clear-num 97.6%

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

       7. div-inv 97.7%

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

       8. metadata-eval 97.7%

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

   13. Applied egg-rr 97.7%

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

       1. associate-/r* 97.7%

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

       2. add-sqr-sqrt 97.5%

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

       3. associate-/l* 97.4%

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

       4. inv-pow 97.4%

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

       5. sqrt-pow1 97.5%

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

       6. metadata-eval 97.5%

          \[\leadsto 1 + {x}^{\color{blue}{-0.5}} \cdot \frac{\sqrt{\frac{1}{x}}}{-9} \]

       7. inv-pow 97.5%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{-9} \]

       8. sqrt-pow1 97.4%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{-9} \]

       9. metadata-eval 97.4%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{{x}^{\color{blue}{-0.5}}}{-9} \]

   15. Applied egg-rr 97.4%

       \[\leadsto 1 + \color{blue}{{x}^{-0.5} \cdot \frac{{x}^{-0.5}}{-9}} \]
   16. Step-by-step derivation

       1. associate-*r/ 97.5%

          \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-9}} \]

       2. pow-sqr 97.7%

          \[\leadsto 1 + \frac{\color{blue}{{x}^{\left(2 \cdot -0.5\right)}}}{-9} \]

       3. metadata-eval 97.7%

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

   17. Simplified 97.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+40} \lor \neg \left(y \leq 5.2 \cdot 10^{+44}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{{x}^{-1}}{-9}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+40}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+43}:\\ \;\;\;\;1 + \frac{{x}^{-1}}{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.25e+40)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 3.7e+43)
     (+ 1.0 (/ (pow x -1.0) -9.0))
     (- 1.0 (/ y (* 3.0 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.25e+40) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 3.7e+43) {
		tmp = 1.0 + (pow(x, -1.0) / -9.0);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.25d+40)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 3.7d+43) then
        tmp = 1.0d0 + ((x ** (-1.0d0)) / (-9.0d0))
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.25e+40) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 3.7e+43) {
		tmp = 1.0 + (Math.pow(x, -1.0) / -9.0);
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.25e+40:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 3.7e+43:
		tmp = 1.0 + (math.pow(x, -1.0) / -9.0)
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.25e+40)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 3.7e+43)
		tmp = Float64(1.0 + Float64((x ^ -1.0) / -9.0));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.25e+40)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 3.7e+43)
		tmp = 1.0 + ((x ^ -1.0) / -9.0);
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.25e+40], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+43], N[(1.0 + N[(N[Power[x, -1.0], $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.25000000000000001e40

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 95.9%

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

   if -1.25000000000000001e40 < y < 3.7000000000000001e43

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.6%

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

      1. cancel-sign-sub-inv 97.6%

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

      2. metadata-eval 97.6%

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

      3. associate-*r/ 97.7%

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

      4. metadata-eval 97.7%

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

   7. Simplified 97.7%

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

      1. expm1-log1p-u 47.2%

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

      2. expm1-undefine 47.2%

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

      3. metadata-eval 47.2%

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

      4. distribute-neg-frac 47.2%

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

      5. log1p-define 47.2%

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

      6. add-exp-log 97.7%

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

      7. div-inv 97.6%

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

      8. distribute-lft-neg-in 97.6%

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

      9. metadata-eval 97.6%

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

      10. div-inv 97.7%

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

   9. Applied egg-rr 97.7%

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

       1. associate--l+ 97.7%

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

   11. Simplified 97.7%

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

       1. associate-+r- 97.7%

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

       2. add-exp-log 47.2%

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

       3. log1p-undefine 47.2%

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

       4. expm1-undefine 47.2%

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

       5. expm1-log1p-u 97.7%

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

       6. clear-num 97.6%

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

       7. div-inv 97.7%

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

       8. metadata-eval 97.7%

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

   13. Applied egg-rr 97.7%

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

       1. associate-/r* 97.7%

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

       2. add-sqr-sqrt 97.5%

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

       3. associate-/l* 97.4%

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

       4. inv-pow 97.4%

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

       5. sqrt-pow1 97.5%

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

       6. metadata-eval 97.5%

          \[\leadsto 1 + {x}^{\color{blue}{-0.5}} \cdot \frac{\sqrt{\frac{1}{x}}}{-9} \]

       7. inv-pow 97.5%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{-9} \]

       8. sqrt-pow1 97.4%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{-9} \]

       9. metadata-eval 97.4%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{{x}^{\color{blue}{-0.5}}}{-9} \]

   15. Applied egg-rr 97.4%

       \[\leadsto 1 + \color{blue}{{x}^{-0.5} \cdot \frac{{x}^{-0.5}}{-9}} \]
   16. Step-by-step derivation

       1. associate-*r/ 97.5%

          \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-9}} \]

       2. pow-sqr 97.7%

          \[\leadsto 1 + \frac{\color{blue}{{x}^{\left(2 \cdot -0.5\right)}}}{-9} \]

       3. metadata-eval 97.7%

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

   17. Simplified 97.7%

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

   if 3.7000000000000001e43 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in x around inf 94.5%

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

Alternative 4: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+40}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+44}:\\ \;\;\;\;1 + \frac{{x}^{-1}}{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.6e+40)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 5.1e+44)
     (+ 1.0 (/ (pow x -1.0) -9.0))
     (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+40) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 5.1e+44) {
		tmp = 1.0 + (pow(x, -1.0) / -9.0);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.6d+40)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 5.1d+44) then
        tmp = 1.0d0 + ((x ** (-1.0d0)) / (-9.0d0))
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+40) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 5.1e+44) {
		tmp = 1.0 + (Math.pow(x, -1.0) / -9.0);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.6e+40:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 5.1e+44:
		tmp = 1.0 + (math.pow(x, -1.0) / -9.0)
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.6e+40)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 5.1e+44)
		tmp = Float64(1.0 + Float64((x ^ -1.0) / -9.0));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.6e+40)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 5.1e+44)
		tmp = 1.0 + ((x ^ -1.0) / -9.0);
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.6e+40], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.1e+44], N[(1.0 + N[(N[Power[x, -1.0], $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.6000000000000005e40

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 95.9%

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

   if -8.6000000000000005e40 < y < 5.1e44

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.6%

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

      1. cancel-sign-sub-inv 97.6%

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

      2. metadata-eval 97.6%

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

      3. associate-*r/ 97.7%

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

      4. metadata-eval 97.7%

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

   7. Simplified 97.7%

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

      1. expm1-log1p-u 47.2%

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

      2. expm1-undefine 47.2%

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

      3. metadata-eval 47.2%

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

      4. distribute-neg-frac 47.2%

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

      5. log1p-define 47.2%

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

      6. add-exp-log 97.7%

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

      7. div-inv 97.6%

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

      8. distribute-lft-neg-in 97.6%

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

      9. metadata-eval 97.6%

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

      10. div-inv 97.7%

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

   9. Applied egg-rr 97.7%

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

       1. associate--l+ 97.7%

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

   11. Simplified 97.7%

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

       1. associate-+r- 97.7%

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

       2. add-exp-log 47.2%

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

       3. log1p-undefine 47.2%

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

       4. expm1-undefine 47.2%

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

       5. expm1-log1p-u 97.7%

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

       6. clear-num 97.6%

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

       7. div-inv 97.7%

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

       8. metadata-eval 97.7%

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

   13. Applied egg-rr 97.7%

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

       1. associate-/r* 97.7%

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

       2. add-sqr-sqrt 97.5%

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

       3. associate-/l* 97.4%

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

       4. inv-pow 97.4%

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

       5. sqrt-pow1 97.5%

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

       6. metadata-eval 97.5%

          \[\leadsto 1 + {x}^{\color{blue}{-0.5}} \cdot \frac{\sqrt{\frac{1}{x}}}{-9} \]

       7. inv-pow 97.5%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{-9} \]

       8. sqrt-pow1 97.4%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{-9} \]

       9. metadata-eval 97.4%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{{x}^{\color{blue}{-0.5}}}{-9} \]

   15. Applied egg-rr 97.4%

       \[\leadsto 1 + \color{blue}{{x}^{-0.5} \cdot \frac{{x}^{-0.5}}{-9}} \]
   16. Step-by-step derivation

       1. associate-*r/ 97.5%

          \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-9}} \]

       2. pow-sqr 97.7%

          \[\leadsto 1 + \frac{\color{blue}{{x}^{\left(2 \cdot -0.5\right)}}}{-9} \]

       3. metadata-eval 97.7%

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

   17. Simplified 97.7%

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

   if 5.1e44 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l/ 94.3%

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

      3. associate-*r/ 94.4%

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

      4. frac-2neg 94.4%

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

      5. associate-*r/ 94.3%

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

      6. metadata-eval 94.3%

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

      7. metadata-eval 94.3%

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

      8. div-inv 94.5%

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

      9. distribute-neg-frac2 94.5%

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

      10. associate-/r* 94.5%

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

      11. clear-num 94.4%

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

      12. distribute-neg-frac2 94.4%

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

      13. *-commutative 94.4%

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

      14. associate-/l* 94.4%

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

   7. Applied egg-rr 94.4%

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

      1. distribute-frac-neg2 94.4%

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

      2. associate-*r/ 94.4%

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

      3. associate-/r/ 94.4%

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

      4. *-commutative 94.4%

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

      5. associate-/r* 94.4%

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

      6. metadata-eval 94.4%

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

      7. /-rgt-identity 94.4%

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

      8. times-frac 94.3%

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

      9. *-commutative 94.3%

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

      10. *-rgt-identity 94.3%

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

      11. distribute-frac-neg 94.3%

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

      12. distribute-rgt-neg-in 94.3%

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

      13. metadata-eval 94.3%

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

      14. associate-/l* 94.4%

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

   9. Simplified 94.4%

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

Alternative 5: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 3.45 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{{x}^{-1}}{-9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.4e+70)
   (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))
   (if (<= y 3.45e+53)
     (+ 1.0 (/ (pow x -1.0) -9.0))
     (* y (/ -0.3333333333333333 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.4e+70) {
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} else if (y <= 3.45e+53) {
		tmp = 1.0 + (pow(x, -1.0) / -9.0);
	} else {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.4d+70)) then
        tmp = sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0))
    else if (y <= 3.45d+53) then
        tmp = 1.0d0 + ((x ** (-1.0d0)) / (-9.0d0))
    else
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.4e+70) {
		tmp = Math.sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} else if (y <= 3.45e+53) {
		tmp = 1.0 + (Math.pow(x, -1.0) / -9.0);
	} else {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.4e+70:
		tmp = math.sqrt((1.0 / x)) * (y * -0.3333333333333333)
	elif y <= 3.45e+53:
		tmp = 1.0 + (math.pow(x, -1.0) / -9.0)
	else:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.4e+70)
		tmp = Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333));
	elseif (y <= 3.45e+53)
		tmp = Float64(1.0 + Float64((x ^ -1.0) / -9.0));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.4e+70)
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	elseif (y <= 3.45e+53)
		tmp = 1.0 + ((x ^ -1.0) / -9.0);
	else
		tmp = y * (-0.3333333333333333 / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.4e+70], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.45e+53], N[(1.0 + N[(N[Power[x, -1.0], $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.4000000000000001e70

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 95.7%

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

   6. Taylor expanded in y around inf 91.0%

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

      1. *-commutative 91.0%

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

      2. associate-*l* 91.1%

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

      3. *-commutative 91.1%

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

   8. Simplified 91.1%

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

   if -3.4000000000000001e70 < y < 3.4500000000000001e53

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 95.9%

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

      1. cancel-sign-sub-inv 95.9%

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

      2. metadata-eval 95.9%

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

      3. associate-*r/ 95.9%

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

      4. metadata-eval 95.9%

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

   7. Simplified 95.9%

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

      1. expm1-log1p-u 46.7%

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

      2. expm1-undefine 46.7%

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

      3. metadata-eval 46.7%

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

      4. distribute-neg-frac 46.7%

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

      5. log1p-define 46.7%

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

      6. add-exp-log 95.9%

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

      7. div-inv 95.9%

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

      8. distribute-lft-neg-in 95.9%

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

      9. metadata-eval 95.9%

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

      10. div-inv 95.9%

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

   9. Applied egg-rr 95.9%

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

       1. associate--l+ 95.9%

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

   11. Simplified 95.9%

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

       1. associate-+r- 95.9%

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

       2. add-exp-log 46.7%

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

       3. log1p-undefine 46.7%

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

       4. expm1-undefine 46.7%

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

       5. expm1-log1p-u 95.9%

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

       6. clear-num 95.9%

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

       7. div-inv 95.9%

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

       8. metadata-eval 95.9%

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

   13. Applied egg-rr 95.9%

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

       1. associate-/r* 96.0%

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

       2. add-sqr-sqrt 95.8%

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

       3. associate-/l* 95.7%

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

       4. inv-pow 95.7%

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

       5. sqrt-pow1 95.8%

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

       6. metadata-eval 95.8%

          \[\leadsto 1 + {x}^{\color{blue}{-0.5}} \cdot \frac{\sqrt{\frac{1}{x}}}{-9} \]

       7. inv-pow 95.8%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{-9} \]

       8. sqrt-pow1 95.7%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{-9} \]

       9. metadata-eval 95.7%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{{x}^{\color{blue}{-0.5}}}{-9} \]

   15. Applied egg-rr 95.7%

       \[\leadsto 1 + \color{blue}{{x}^{-0.5} \cdot \frac{{x}^{-0.5}}{-9}} \]
   16. Step-by-step derivation

       1. associate-*r/ 95.8%

          \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-9}} \]

       2. pow-sqr 96.0%

          \[\leadsto 1 + \frac{\color{blue}{{x}^{\left(2 \cdot -0.5\right)}}}{-9} \]

       3. metadata-eval 96.0%

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

   17. Simplified 96.0%

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

   if 3.4500000000000001e53 < 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-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 87.7%

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

      1. associate-*r* 87.8%

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

   7. Simplified 87.8%

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

      1. sqrt-div 87.7%

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

      2. metadata-eval 87.7%

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

      3. div-inv 87.8%

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

      4. frac-2neg 87.8%

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

      5. metadata-eval 87.8%

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

   9. Applied egg-rr 87.8%

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

       1. distribute-frac-neg2 87.8%

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

       2. distribute-neg-frac 87.8%

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

       3. metadata-eval 87.8%

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

   11. Simplified 87.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 3.45 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{{x}^{-1}}{-9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+70}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{{x}^{-1}}{-9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.1e+70)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 3.5e+53)
     (+ 1.0 (/ (pow x -1.0) -9.0))
     (* y (/ -0.3333333333333333 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.1e+70) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 3.5e+53) {
		tmp = 1.0 + (pow(x, -1.0) / -9.0);
	} else {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.1d+70)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 3.5d+53) then
        tmp = 1.0d0 + ((x ** (-1.0d0)) / (-9.0d0))
    else
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.1e+70) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 3.5e+53) {
		tmp = 1.0 + (Math.pow(x, -1.0) / -9.0);
	} else {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.1e+70:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 3.5e+53:
		tmp = 1.0 + (math.pow(x, -1.0) / -9.0)
	else:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.1e+70)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 3.5e+53)
		tmp = Float64(1.0 + Float64((x ^ -1.0) / -9.0));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.1e+70)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 3.5e+53)
		tmp = 1.0 + ((x ^ -1.0) / -9.0);
	else
		tmp = y * (-0.3333333333333333 / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.1e+70], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+53], N[(1.0 + N[(N[Power[x, -1.0], $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.10000000000000014e70

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.3%

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

      10. fma-neg 99.3%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 91.0%

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

      1. associate-*r* 91.0%

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

   7. Simplified 91.0%

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

      1. pow1 91.0%

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

      2. associate-*l* 91.0%

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

      3. sqrt-div 90.9%

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

      4. metadata-eval 90.9%

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

      5. associate-*l/ 91.1%

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

      6. *-un-lft-identity 91.1%

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

   9. Applied egg-rr 91.1%

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

       1. unpow1 91.1%

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

   11. Simplified 91.1%

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

   if -5.10000000000000014e70 < y < 3.50000000000000019e53

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 95.9%

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

      1. cancel-sign-sub-inv 95.9%

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

      2. metadata-eval 95.9%

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

      3. associate-*r/ 95.9%

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

      4. metadata-eval 95.9%

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

   7. Simplified 95.9%

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

      1. expm1-log1p-u 46.7%

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

      2. expm1-undefine 46.7%

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

      3. metadata-eval 46.7%

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

      4. distribute-neg-frac 46.7%

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

      5. log1p-define 46.7%

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

      6. add-exp-log 95.9%

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

      7. div-inv 95.9%

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

      8. distribute-lft-neg-in 95.9%

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

      9. metadata-eval 95.9%

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

      10. div-inv 95.9%

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

   9. Applied egg-rr 95.9%

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

       1. associate--l+ 95.9%

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

   11. Simplified 95.9%

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

       1. associate-+r- 95.9%

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

       2. add-exp-log 46.7%

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

       3. log1p-undefine 46.7%

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

       4. expm1-undefine 46.7%

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

       5. expm1-log1p-u 95.9%

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

       6. clear-num 95.9%

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

       7. div-inv 95.9%

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

       8. metadata-eval 95.9%

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

   13. Applied egg-rr 95.9%

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

       1. associate-/r* 96.0%

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

       2. add-sqr-sqrt 95.8%

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

       3. associate-/l* 95.7%

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

       4. inv-pow 95.7%

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

       5. sqrt-pow1 95.8%

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

       6. metadata-eval 95.8%

          \[\leadsto 1 + {x}^{\color{blue}{-0.5}} \cdot \frac{\sqrt{\frac{1}{x}}}{-9} \]

       7. inv-pow 95.8%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{-9} \]

       8. sqrt-pow1 95.7%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{-9} \]

       9. metadata-eval 95.7%

          \[\leadsto 1 + {x}^{-0.5} \cdot \frac{{x}^{\color{blue}{-0.5}}}{-9} \]

   15. Applied egg-rr 95.7%

       \[\leadsto 1 + \color{blue}{{x}^{-0.5} \cdot \frac{{x}^{-0.5}}{-9}} \]
   16. Step-by-step derivation

       1. associate-*r/ 95.8%

          \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-9}} \]

       2. pow-sqr 96.0%

          \[\leadsto 1 + \frac{\color{blue}{{x}^{\left(2 \cdot -0.5\right)}}}{-9} \]

       3. metadata-eval 96.0%

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

   17. Simplified 96.0%

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

   if 3.50000000000000019e53 < 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-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 87.7%

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

      1. associate-*r* 87.8%

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

   7. Simplified 87.8%

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

      1. sqrt-div 87.7%

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

      2. metadata-eval 87.7%

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

      3. div-inv 87.8%

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

      4. frac-2neg 87.8%

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

      5. metadata-eval 87.8%

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

   9. Applied egg-rr 87.8%

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

       1. distribute-frac-neg2 87.8%

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

       2. distribute-neg-frac 87.8%

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

       3. metadata-eval 87.8%

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

   11. Simplified 87.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+70}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{{x}^{-1}}{-9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+68} \lor \neg \left(y \leq 3.45 \cdot 10^{+53}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.02e+68) (not (<= y 3.45e+53)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ 1.0 (* x -9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.02e+68) || !(y <= 3.45e+53)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 + (1.0 / (x * -9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.02d+68)) .or. (.not. (y <= 3.45d+53))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.02e+68) || !(y <= 3.45e+53)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 + (1.0 / (x * -9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.02e+68) or not (y <= 3.45e+53):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 + (1.0 / (x * -9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.02e+68) || !(y <= 3.45e+53))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.02e+68) || ~((y <= 3.45e+53)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 + (1.0 / (x * -9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.02e+68], N[Not[LessEqual[y, 3.45e+53]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02e68 or 3.4500000000000001e53 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 89.2%

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

      1. associate-*r* 89.3%

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

   7. Simplified 89.3%

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

      1. pow1 89.3%

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

      2. associate-*l* 89.2%

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

      3. sqrt-div 89.2%

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

      4. metadata-eval 89.2%

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

      5. associate-*l/ 89.3%

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

      6. *-un-lft-identity 89.3%

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

   9. Applied egg-rr 89.3%

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

       1. unpow1 89.3%

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

   11. Simplified 89.3%

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

   if -1.02e68 < y < 3.4500000000000001e53

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 95.9%

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

      1. cancel-sign-sub-inv 95.9%

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

      2. metadata-eval 95.9%

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

      3. associate-*r/ 95.9%

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

      4. metadata-eval 95.9%

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

   7. Simplified 95.9%

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

      1. expm1-log1p-u 46.7%

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

      2. expm1-undefine 46.7%

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

      3. metadata-eval 46.7%

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

      4. distribute-neg-frac 46.7%

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

      5. log1p-define 46.7%

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

      6. add-exp-log 95.9%

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

      7. div-inv 95.9%

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

      8. distribute-lft-neg-in 95.9%

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

      9. metadata-eval 95.9%

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

      10. div-inv 95.9%

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

   9. Applied egg-rr 95.9%

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

       1. associate--l+ 95.9%

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

   11. Simplified 95.9%

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

       1. associate-+r- 95.9%

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

       2. add-exp-log 46.7%

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

       3. log1p-undefine 46.7%

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

       4. expm1-undefine 46.7%

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

       5. expm1-log1p-u 95.9%

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

       6. clear-num 95.9%

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

       7. div-inv 95.9%

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

       8. metadata-eval 95.9%

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

   13. Applied egg-rr 95.9%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+68} \lor \neg \left(y \leq 3.45 \cdot 10^{+53}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+73}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.72e+73)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 3.5e+53)
     (+ 1.0 (/ 1.0 (* x -9.0)))
     (* y (/ -0.3333333333333333 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.72e+73) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 3.5e+53) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.72d+73)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 3.5d+53) then
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    else
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.72e+73) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 3.5e+53) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.72e+73:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 3.5e+53:
		tmp = 1.0 + (1.0 / (x * -9.0))
	else:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.72e+73)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 3.5e+53)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.72e+73)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 3.5e+53)
		tmp = 1.0 + (1.0 / (x * -9.0));
	else
		tmp = y * (-0.3333333333333333 / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.72e+73], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+53], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7200000000000001e73

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.3%

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

      10. fma-neg 99.3%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 91.0%

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

      1. associate-*r* 91.0%

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

   7. Simplified 91.0%

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

      1. pow1 91.0%

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

      2. associate-*l* 91.0%

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

      3. sqrt-div 90.9%

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

      4. metadata-eval 90.9%

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

      5. associate-*l/ 91.1%

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

      6. *-un-lft-identity 91.1%

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

   9. Applied egg-rr 91.1%

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

       1. unpow1 91.1%

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

   11. Simplified 91.1%

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

   if -1.7200000000000001e73 < y < 3.50000000000000019e53

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 95.9%

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

      1. cancel-sign-sub-inv 95.9%

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

      2. metadata-eval 95.9%

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

      3. associate-*r/ 95.9%

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

      4. metadata-eval 95.9%

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

   7. Simplified 95.9%

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

      1. expm1-log1p-u 46.7%

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

      2. expm1-undefine 46.7%

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

      3. metadata-eval 46.7%

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

      4. distribute-neg-frac 46.7%

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

      5. log1p-define 46.7%

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

      6. add-exp-log 95.9%

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

      7. div-inv 95.9%

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

      8. distribute-lft-neg-in 95.9%

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

      9. metadata-eval 95.9%

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

      10. div-inv 95.9%

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

   9. Applied egg-rr 95.9%

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

       1. associate--l+ 95.9%

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

   11. Simplified 95.9%

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

       1. associate-+r- 95.9%

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

       2. add-exp-log 46.7%

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

       3. log1p-undefine 46.7%

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

       4. expm1-undefine 46.7%

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

       5. expm1-log1p-u 95.9%

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

       6. clear-num 95.9%

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

       7. div-inv 95.9%

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

       8. metadata-eval 95.9%

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

   13. Applied egg-rr 95.9%

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

   if 3.50000000000000019e53 < 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-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 87.7%

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

      1. associate-*r* 87.8%

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

   7. Simplified 87.8%

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

      1. sqrt-div 87.7%

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

      2. metadata-eval 87.7%

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

      3. div-inv 87.8%

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

      4. frac-2neg 87.8%

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

      5. metadata-eval 87.8%

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

   9. Applied egg-rr 87.8%

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

       1. distribute-frac-neg2 87.8%

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

       2. distribute-neg-frac 87.8%

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

       3. metadata-eval 87.8%

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

   11. Simplified 87.8%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+73}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1400:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1400.0)
   (/ (- (* -0.3333333333333333 (* y (sqrt x))) 0.1111111111111111) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1400.0) {
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1400.0d0) then
        tmp = (((-0.3333333333333333d0) * (y * sqrt(x))) - 0.1111111111111111d0) / x
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1400.0) {
		tmp = ((-0.3333333333333333 * (y * Math.sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1400.0:
		tmp = ((-0.3333333333333333 * (y * math.sqrt(x))) - 0.1111111111111111) / x
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1400.0)
		tmp = Float64(Float64(Float64(-0.3333333333333333 * Float64(y * sqrt(x))) - 0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1400.0)
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1400.0], N[(N[(N[(-0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1400

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 99.2%

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

   if 1400 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.8%

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

   4. Taylor expanded in x around inf 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1400:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0 99.7%

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

Alternative 11: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 12: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 13: 65.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{-0.012345679012345678}{x \cdot x}}{-1 + \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e+154)
   (+ 1.0 (/ 1.0 (* x -9.0)))
   (/
    (- -1.0 (/ -0.012345679012345678 (* x x)))
    (+ -1.0 (/ -0.1111111111111111 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.4e+154) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = (-1.0 - (-0.012345679012345678 / (x * x))) / (-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 <= 1.4d+154) then
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    else
        tmp = ((-1.0d0) - ((-0.012345679012345678d0) / (x * x))) / ((-1.0d0) + ((-0.1111111111111111d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.4e+154) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = (-1.0 - (-0.012345679012345678 / (x * x))) / (-1.0 + (-0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.4e+154:
		tmp = 1.0 + (1.0 / (x * -9.0))
	else:
		tmp = (-1.0 - (-0.012345679012345678 / (x * x))) / (-1.0 + (-0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.4e+154)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = Float64(Float64(-1.0 - Float64(-0.012345679012345678 / Float64(x * x))) / Float64(-1.0 + Float64(-0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e+154)
		tmp = 1.0 + (1.0 / (x * -9.0));
	else
		tmp = (-1.0 - (-0.012345679012345678 / (x * x))) / (-1.0 + (-0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.4e+154], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 - N[(-0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4e154

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.6%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 68.5%

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

      1. cancel-sign-sub-inv 68.5%

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

      2. metadata-eval 68.5%

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

      3. associate-*r/ 68.5%

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

      4. metadata-eval 68.5%

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

   7. Simplified 68.5%

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

      1. expm1-log1p-u 34.3%

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

      2. expm1-undefine 34.3%

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

      3. metadata-eval 34.3%

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

      4. distribute-neg-frac 34.3%

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

      5. log1p-define 34.3%

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

      6. add-exp-log 68.5%

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

      7. div-inv 68.5%

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

      8. distribute-lft-neg-in 68.5%

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

      9. metadata-eval 68.5%

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

      10. div-inv 68.5%

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

   9. Applied egg-rr 68.5%

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

       1. associate--l+ 68.5%

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

   11. Simplified 68.5%

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

       1. associate-+r- 68.5%

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

       2. add-exp-log 34.3%

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

       3. log1p-undefine 34.3%

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

       4. expm1-undefine 34.3%

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

       5. expm1-log1p-u 68.5%

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

       6. clear-num 68.5%

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

       7. div-inv 68.5%

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

       8. metadata-eval 68.5%

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

   13. Applied egg-rr 68.5%

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

   if 1.4e154 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 3.8%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 3.8%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 3.8%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 3.8%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 3.8%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

   7. Simplified 3.8%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 0.5%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 0.5%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)} - 1\right)} \]

      3. metadata-eval 0.5%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{-0.1111111111111111}}{x}\right)} - 1\right) \]

      4. distribute-neg-frac 0.5%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{-\frac{0.1111111111111111}{x}}\right)} - 1\right) \]

      5. log1p-define 0.5%

         \[\leadsto 1 + \left(e^{\color{blue}{\log \left(1 + \left(-\frac{0.1111111111111111}{x}\right)\right)}} - 1\right) \]

      6. add-exp-log 3.8%

         \[\leadsto 1 + \left(\color{blue}{\left(1 + \left(-\frac{0.1111111111111111}{x}\right)\right)} - 1\right) \]

      7. div-inv 3.8%

         \[\leadsto 1 + \left(\left(1 + \left(-\color{blue}{0.1111111111111111 \cdot \frac{1}{x}}\right)\right) - 1\right) \]

      8. distribute-lft-neg-in 3.8%

         \[\leadsto 1 + \left(\left(1 + \color{blue}{\left(-0.1111111111111111\right) \cdot \frac{1}{x}}\right) - 1\right) \]

      9. metadata-eval 3.8%

         \[\leadsto 1 + \left(\left(1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x}\right) - 1\right) \]

      10. div-inv 3.8%

          \[\leadsto 1 + \left(\left(1 + \color{blue}{\frac{-0.1111111111111111}{x}}\right) - 1\right) \]

   9. Applied egg-rr 3.8%

      \[\leadsto 1 + \color{blue}{\left(\left(1 + \frac{-0.1111111111111111}{x}\right) - 1\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 3.8%

          \[\leadsto 1 + \color{blue}{\left(1 + \left(\frac{-0.1111111111111111}{x} - 1\right)\right)} \]

   11. Simplified 3.8%

       \[\leadsto 1 + \color{blue}{\left(1 + \left(\frac{-0.1111111111111111}{x} - 1\right)\right)} \]
   12. Step-by-step derivation

       1. +-commutative 3.8%

          \[\leadsto \color{blue}{\left(1 + \left(\frac{-0.1111111111111111}{x} - 1\right)\right) + 1} \]

       2. associate-+r- 3.8%

          \[\leadsto \color{blue}{\left(\left(1 + \frac{-0.1111111111111111}{x}\right) - 1\right)} + 1 \]

       3. add-exp-log 0.5%

          \[\leadsto \left(\color{blue}{e^{\log \left(1 + \frac{-0.1111111111111111}{x}\right)}} - 1\right) + 1 \]

       4. log1p-undefine 0.5%

          \[\leadsto \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)}} - 1\right) + 1 \]

       5. expm1-undefine 0.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} + 1 \]

       6. expm1-log1p-u 3.8%

          \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} + 1 \]

       7. flip-+ 25.4%

          \[\leadsto \color{blue}{\frac{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x} - 1 \cdot 1}{\frac{-0.1111111111111111}{x} - 1}} \]

       8. frac-times 25.4%

          \[\leadsto \frac{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}} - 1 \cdot 1}{\frac{-0.1111111111111111}{x} - 1} \]

       9. metadata-eval 25.4%

          \[\leadsto \frac{\frac{\color{blue}{0.012345679012345678}}{x \cdot x} - 1 \cdot 1}{\frac{-0.1111111111111111}{x} - 1} \]

       10. metadata-eval 25.4%

           \[\leadsto \frac{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x} - 1 \cdot 1}{\frac{-0.1111111111111111}{x} - 1} \]

       11. frac-times 25.4%

           \[\leadsto \frac{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}} - 1 \cdot 1}{\frac{-0.1111111111111111}{x} - 1} \]

       12. metadata-eval 25.4%

           \[\leadsto \frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - \color{blue}{1}}{\frac{-0.1111111111111111}{x} - 1} \]

       13. sub-neg 25.4%

           \[\leadsto \frac{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} + \left(-1\right)}}{\frac{-0.1111111111111111}{x} - 1} \]

       14. div-inv 25.4%

           \[\leadsto \frac{\color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x}\right)} \cdot \frac{0.1111111111111111}{x} + \left(-1\right)}{\frac{-0.1111111111111111}{x} - 1} \]

       15. div-inv 25.4%

           \[\leadsto \frac{\left(0.1111111111111111 \cdot \frac{1}{x}\right) \cdot \color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x}\right)} + \left(-1\right)}{\frac{-0.1111111111111111}{x} - 1} \]

       16. swap-sqr 25.4%

           \[\leadsto \frac{\color{blue}{\left(0.1111111111111111 \cdot 0.1111111111111111\right) \cdot \left(\frac{1}{x} \cdot \frac{1}{x}\right)} + \left(-1\right)}{\frac{-0.1111111111111111}{x} - 1} \]

       17. metadata-eval 25.4%

           \[\leadsto \frac{\color{blue}{0.012345679012345678} \cdot \left(\frac{1}{x} \cdot \frac{1}{x}\right) + \left(-1\right)}{\frac{-0.1111111111111111}{x} - 1} \]

       18. inv-pow 25.4%

           \[\leadsto \frac{0.012345679012345678 \cdot \left(\color{blue}{{x}^{-1}} \cdot \frac{1}{x}\right) + \left(-1\right)}{\frac{-0.1111111111111111}{x} - 1} \]

       19. inv-pow 25.4%

           \[\leadsto \frac{0.012345679012345678 \cdot \left({x}^{-1} \cdot \color{blue}{{x}^{-1}}\right) + \left(-1\right)}{\frac{-0.1111111111111111}{x} - 1} \]

       20. pow-prod-up 25.4%

           \[\leadsto \frac{0.012345679012345678 \cdot \color{blue}{{x}^{\left(-1 + -1\right)}} + \left(-1\right)}{\frac{-0.1111111111111111}{x} - 1} \]

       21. metadata-eval 25.4%

           \[\leadsto \frac{0.012345679012345678 \cdot {x}^{\color{blue}{-2}} + \left(-1\right)}{\frac{-0.1111111111111111}{x} - 1} \]

       22. metadata-eval 25.4%

           \[\leadsto \frac{0.012345679012345678 \cdot {x}^{-2} + \color{blue}{-1}}{\frac{-0.1111111111111111}{x} - 1} \]

   13. Applied egg-rr 25.4%

       \[\leadsto \color{blue}{\frac{0.012345679012345678 \cdot {x}^{-2} + -1}{\frac{-0.1111111111111111}{x} + -1}} \]
   14. Step-by-step derivation

       1. metadata-eval 25.4%

          \[\leadsto \frac{\color{blue}{\left(-0.1111111111111111 \cdot -0.1111111111111111\right)} \cdot {x}^{-2} + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       2. metadata-eval 25.4%

          \[\leadsto \frac{\left(-0.1111111111111111 \cdot -0.1111111111111111\right) \cdot {x}^{\color{blue}{\left(2 \cdot -1\right)}} + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       3. pow-sqr 25.4%

          \[\leadsto \frac{\left(-0.1111111111111111 \cdot -0.1111111111111111\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       4. inv-pow 25.4%

          \[\leadsto \frac{\left(-0.1111111111111111 \cdot -0.1111111111111111\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       5. inv-pow 25.4%

          \[\leadsto \frac{\left(-0.1111111111111111 \cdot -0.1111111111111111\right) \cdot \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       6. swap-sqr 25.4%

          \[\leadsto \frac{\color{blue}{\left(-0.1111111111111111 \cdot \frac{1}{x}\right) \cdot \left(-0.1111111111111111 \cdot \frac{1}{x}\right)} + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       7. div-inv 25.4%

          \[\leadsto \frac{\color{blue}{\frac{-0.1111111111111111}{x}} \cdot \left(-0.1111111111111111 \cdot \frac{1}{x}\right) + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       8. frac-2neg 25.4%

          \[\leadsto \frac{\color{blue}{\frac{--0.1111111111111111}{-x}} \cdot \left(-0.1111111111111111 \cdot \frac{1}{x}\right) + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       9. metadata-eval 25.4%

          \[\leadsto \frac{\frac{\color{blue}{0.1111111111111111}}{-x} \cdot \left(-0.1111111111111111 \cdot \frac{1}{x}\right) + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       10. div-inv 25.4%

           \[\leadsto \frac{\frac{0.1111111111111111}{-x} \cdot \color{blue}{\frac{-0.1111111111111111}{x}} + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       11. frac-times 25.4%

           \[\leadsto \frac{\color{blue}{\frac{0.1111111111111111 \cdot -0.1111111111111111}{\left(-x\right) \cdot x}} + -1}{\frac{-0.1111111111111111}{x} + -1} \]

       12. metadata-eval 25.4%

           \[\leadsto \frac{\frac{\color{blue}{-0.012345679012345678}}{\left(-x\right) \cdot x} + -1}{\frac{-0.1111111111111111}{x} + -1} \]

   15. Applied egg-rr 25.4%

       \[\leadsto \frac{\color{blue}{\frac{-0.012345679012345678}{\left(-x\right) \cdot x}} + -1}{\frac{-0.1111111111111111}{x} + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{-0.012345679012345678}{x \cdot x}}{-1 + \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.1% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.032) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.032) {
		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 <= 0.032d0) 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 <= 0.032) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.032:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.032)
		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 <= 0.032)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.032], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.032000000000000001

   1. Initial program 99.5%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.5%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.5%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.5%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.5%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.5%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.5%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.5%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. *-commutative 99.5%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      9. associate-/l* 99.5%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fma-neg 99.5%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      11. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      17. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 59.7%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 59.7%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 59.7%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 59.7%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 59.7%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 0.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 0.0%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)} - 1\right)} \]

      3. metadata-eval 0.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{-0.1111111111111111}}{x}\right)} - 1\right) \]

      4. distribute-neg-frac 0.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{-\frac{0.1111111111111111}{x}}\right)} - 1\right) \]

      5. log1p-define 0.0%

         \[\leadsto 1 + \left(e^{\color{blue}{\log \left(1 + \left(-\frac{0.1111111111111111}{x}\right)\right)}} - 1\right) \]

      6. add-exp-log 59.7%

         \[\leadsto 1 + \left(\color{blue}{\left(1 + \left(-\frac{0.1111111111111111}{x}\right)\right)} - 1\right) \]

      7. div-inv 59.7%

         \[\leadsto 1 + \left(\left(1 + \left(-\color{blue}{0.1111111111111111 \cdot \frac{1}{x}}\right)\right) - 1\right) \]

      8. distribute-lft-neg-in 59.7%

         \[\leadsto 1 + \left(\left(1 + \color{blue}{\left(-0.1111111111111111\right) \cdot \frac{1}{x}}\right) - 1\right) \]

      9. metadata-eval 59.7%

         \[\leadsto 1 + \left(\left(1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x}\right) - 1\right) \]

      10. div-inv 59.7%

          \[\leadsto 1 + \left(\left(1 + \color{blue}{\frac{-0.1111111111111111}{x}}\right) - 1\right) \]

   9. Applied egg-rr 59.7%

      \[\leadsto 1 + \color{blue}{\left(\left(1 + \frac{-0.1111111111111111}{x}\right) - 1\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 59.7%

          \[\leadsto 1 + \color{blue}{\left(1 + \left(\frac{-0.1111111111111111}{x} - 1\right)\right)} \]

   11. Simplified 59.7%

       \[\leadsto 1 + \color{blue}{\left(1 + \left(\frac{-0.1111111111111111}{x} - 1\right)\right)} \]

   12. Taylor expanded in x around 0 59.4%

       \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} \]

   if 0.032000000000000001 < 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. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. *-commutative 99.8%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right) \]

      3. associate-/r* 99.8%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right) \]

      5. distribute-frac-neg 99.8%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      6. neg-mul-1 99.8%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{1} + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \]

   6. Taylor expanded in y around 0 60.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 63.2% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{1}{x \cdot -9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ 1.0 (* x -9.0))))

C

double code(double x, double y) {
	return 1.0 + (1.0 / (x * -9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (1.0 / (x * -9.0));
}

Python

def code(x, y):
	return 1.0 + (1.0 / (x * -9.0))

Julia

function code(x, y)
	return Float64(1.0 + Float64(1.0 / Float64(x * -9.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (1.0 / (x * -9.0));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.7%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. sub-neg 99.7%

      \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

   3. +-commutative 99.7%

      \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

   4. distribute-neg-in 99.7%

      \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

   5. distribute-frac-neg 99.7%

      \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   6. sub-neg 99.7%

      \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. *-commutative 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   9. associate-/l* 99.6%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fma-neg 99.6%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   11. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   12. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   17. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 59.9%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 59.9%

      \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

   2. metadata-eval 59.9%

      \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

   3. associate-*r/ 59.9%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

   4. metadata-eval 59.9%

      \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

7. Simplified 59.9%

   \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 29.8%

      \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

   2. expm1-undefine 29.8%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)} - 1\right)} \]

   3. metadata-eval 29.8%

      \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{-0.1111111111111111}}{x}\right)} - 1\right) \]

   4. distribute-neg-frac 29.8%

      \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{-\frac{0.1111111111111111}{x}}\right)} - 1\right) \]

   5. log1p-define 29.8%

      \[\leadsto 1 + \left(e^{\color{blue}{\log \left(1 + \left(-\frac{0.1111111111111111}{x}\right)\right)}} - 1\right) \]

   6. add-exp-log 59.9%

      \[\leadsto 1 + \left(\color{blue}{\left(1 + \left(-\frac{0.1111111111111111}{x}\right)\right)} - 1\right) \]

   7. div-inv 59.9%

      \[\leadsto 1 + \left(\left(1 + \left(-\color{blue}{0.1111111111111111 \cdot \frac{1}{x}}\right)\right) - 1\right) \]

   8. distribute-lft-neg-in 59.9%

      \[\leadsto 1 + \left(\left(1 + \color{blue}{\left(-0.1111111111111111\right) \cdot \frac{1}{x}}\right) - 1\right) \]

   9. metadata-eval 59.9%

      \[\leadsto 1 + \left(\left(1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x}\right) - 1\right) \]

   10. div-inv 59.9%

       \[\leadsto 1 + \left(\left(1 + \color{blue}{\frac{-0.1111111111111111}{x}}\right) - 1\right) \]

9. Applied egg-rr 59.9%

   \[\leadsto 1 + \color{blue}{\left(\left(1 + \frac{-0.1111111111111111}{x}\right) - 1\right)} \]
10. Step-by-step derivation

    1. associate--l+ 59.9%

       \[\leadsto 1 + \color{blue}{\left(1 + \left(\frac{-0.1111111111111111}{x} - 1\right)\right)} \]

11. Simplified 59.9%

    \[\leadsto 1 + \color{blue}{\left(1 + \left(\frac{-0.1111111111111111}{x} - 1\right)\right)} \]
12. Step-by-step derivation

    1. associate-+r- 59.9%

       \[\leadsto 1 + \color{blue}{\left(\left(1 + \frac{-0.1111111111111111}{x}\right) - 1\right)} \]

    2. add-exp-log 29.8%

       \[\leadsto 1 + \left(\color{blue}{e^{\log \left(1 + \frac{-0.1111111111111111}{x}\right)}} - 1\right) \]

    3. log1p-undefine 29.8%

       \[\leadsto 1 + \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)}} - 1\right) \]

    4. expm1-undefine 29.8%

       \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.1111111111111111}{x}\right)\right)} \]

    5. expm1-log1p-u 59.9%

       \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]

    6. clear-num 59.9%

       \[\leadsto 1 + \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}} \]

    7. div-inv 59.9%

       \[\leadsto 1 + \frac{1}{\color{blue}{x \cdot \frac{1}{-0.1111111111111111}}} \]

    8. metadata-eval 59.9%

       \[\leadsto 1 + \frac{1}{x \cdot \color{blue}{-9}} \]

13. Applied egg-rr 59.9%

    \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot -9}} \]
14. Add Preprocessing

Alternative 16: 63.2% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ -0.1111111111111111 x)))

C

double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-0.1111111111111111d0) / x)
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

function code(x, y)
	return Float64(1.0 + Float64(-0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-0.1111111111111111 / x);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.7%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. sub-neg 99.7%

      \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

   3. +-commutative 99.7%

      \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

   4. distribute-neg-in 99.7%

      \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

   5. distribute-frac-neg 99.7%

      \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   6. sub-neg 99.7%

      \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. *-commutative 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   9. associate-/l* 99.6%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fma-neg 99.6%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   11. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   12. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   17. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 59.9%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 59.9%

      \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

   2. metadata-eval 59.9%

      \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

   3. associate-*r/ 59.9%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

   4. metadata-eval 59.9%

      \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

7. Simplified 59.9%

   \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
8. Add Preprocessing

Alternative 17: 31.8% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. *-commutative 99.7%

      \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right) \]

   3. associate-/r* 99.7%

      \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right) \]

   4. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right) \]

   5. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   6. neg-mul-1 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

   7. times-frac 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

   8. metadata-eval 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 70.1%

   \[\leadsto \color{blue}{1} + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \]

6. Taylor expanded in y around 0 30.5%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer Target 1: 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 2024132 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x)))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))