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

Percentage Accurate: 99.7% → 99.7%

Time: 8.4s

Alternatives: 9

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

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

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(1.0 + Float64(Float64(-1.0 / Float64(x * 9.0)) - Float64(Float64(y / 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[(1.0 + N[(N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-/r* 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+25}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+60}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8e+25)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 9.8e+60)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* y (* -0.3333333333333333 (pow x -0.5)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8e+25) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 9.8e+60) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 * pow(x, -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8d+25)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 9.8d+60) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) * (x ** (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8e+25) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 9.8e+60) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 * Math.pow(x, -0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8e+25:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 9.8e+60:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 * math.pow(x, -0.5)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8e+25)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 9.8e+60)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8e+25)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 9.8e+60)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + (y * (-0.3333333333333333 * (x ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8e+25], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.8e+60], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000072e25

   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-neg-frac 99.5%

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

      6. neg-mul-1 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*r/ 99.4%

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

      9. fma-def 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.1%

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

      1. associate-*r* 93.1%

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

      2. *-commutative 93.1%

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

      3. *-commutative 93.1%

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

      4. *-commutative 93.1%

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

   6. Simplified 93.1%

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

      1. sqrt-div 93.1%

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

      2. metadata-eval 93.1%

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

      3. frac-2neg 93.1%

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

      4. metadata-eval 93.1%

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

   8. Applied egg-rr 93.1%

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

      1. neg-mul-1 93.1%

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

      2. associate-/r* 93.1%

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

      3. metadata-eval 93.1%

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

   10. Simplified 93.1%

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

       1. expm1-log1p-u 87.4%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}} \cdot \left(-0.3333333333333333 \cdot y\right)\right)\right)} \]

       2. expm1-udef 87.4%

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

       3. associate-*l/ 87.4%

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

       4. *-un-lft-identity 87.4%

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

       5. *-commutative 87.4%

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

   12. Applied egg-rr 87.4%

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

       1. expm1-def 87.4%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\right)\right)} \]

       2. expm1-log1p 93.2%

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

       3. associate-/l* 93.0%

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

       4. associate-/r/ 93.2%

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

       5. metadata-eval 93.2%

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

       6. times-frac 93.1%

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

       7. *-commutative 93.1%

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

       8. *-commutative 93.1%

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

       9. times-frac 93.2%

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

       10. metadata-eval 93.2%

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

   14. Simplified 93.2%

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

   if -8.00000000000000072e25 < y < 9.8000000000000005e60

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r* 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 98.5%

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

      1. cancel-sign-sub-inv 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   if 9.8000000000000005e60 < y

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. neg-mul-1 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r/ 99.5%

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

      9. fma-def 99.5%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.5%

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

      14. distribute-neg-frac 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 96.0%

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

      1. associate-*r* 96.1%

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

      2. *-commutative 96.1%

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

      3. *-commutative 96.1%

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

      4. *-commutative 96.1%

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

   6. Simplified 96.1%

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

      1. expm1-log1p-u 2.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot \left(-0.3333333333333333 \cdot y\right)\right)\right)} \]

      2. expm1-udef 2.1%

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

      3. *-commutative 2.1%

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

      4. *-commutative 2.1%

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

      5. inv-pow 2.1%

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

      6. sqrt-pow1 2.1%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(y \cdot -0.3333333333333333\right) \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      7. metadata-eval 2.1%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(y \cdot -0.3333333333333333\right) \cdot {x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   8. Applied egg-rr 2.1%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 2.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\right)\right)} \]

      2. expm1-log1p 96.1%

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

      3. associate-*l* 96.2%

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

   10. Simplified 96.2%

       \[\leadsto 1 + \color{blue}{y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+25}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+60}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+25} \lor \neg \left(y \leq 5.1 \cdot 10^{+60}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.8e+25) (not (<= y 5.1e+60)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e+25) || !(y <= 5.1e+60)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.8d+25)) .or. (.not. (y <= 5.1d+60))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e+25) || !(y <= 5.1e+60)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.8e+25) or not (y <= 5.1e+60):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.8e+25) || !(y <= 5.1e+60))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.8e+25) || ~((y <= 5.1e+60)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.8e+25], N[Not[LessEqual[y, 5.1e+60]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8000000000000004e25 or 5.09999999999999996e60 < 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-neg-frac 99.5%

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

      6. neg-mul-1 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*r/ 99.4%

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

      9. fma-def 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.3%

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

      1. associate-*r* 94.4%

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

      2. *-commutative 94.4%

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

      3. *-commutative 94.4%

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

      4. *-commutative 94.4%

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

   6. Simplified 94.4%

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

      1. sqrt-div 94.3%

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

      2. metadata-eval 94.3%

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

      3. frac-2neg 94.3%

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

      4. metadata-eval 94.3%

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

   8. Applied egg-rr 94.3%

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

      1. neg-mul-1 94.3%

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

      2. associate-/r* 94.3%

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

      3. metadata-eval 94.3%

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

   10. Simplified 94.3%

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

       1. expm1-log1p-u 51.2%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}} \cdot \left(-0.3333333333333333 \cdot y\right)\right)\right)} \]

       2. expm1-udef 51.2%

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

       3. associate-*l/ 51.2%

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

       4. *-un-lft-identity 51.2%

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

       5. *-commutative 51.2%

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

   12. Applied egg-rr 51.2%

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

       1. expm1-def 51.2%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\right)\right)} \]

       2. expm1-log1p 94.4%

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

       3. associate-/l* 94.3%

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

       4. associate-/r/ 94.4%

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

       5. metadata-eval 94.4%

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

       6. times-frac 94.3%

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

       7. *-commutative 94.3%

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

       8. *-commutative 94.3%

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

       9. times-frac 94.4%

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

       10. metadata-eval 94.4%

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

   14. Simplified 94.4%

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

   if -7.8000000000000004e25 < y < 5.09999999999999996e60

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r* 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 98.5%

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

      1. cancel-sign-sub-inv 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+25} \lor \neg \left(y \leq 5.1 \cdot 10^{+60}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 4: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+31}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+60}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.8e+31)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 2.05e+60)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.8e+31) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 2.05e+60) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} 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 <= (-5.8d+31)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 2.05d+60) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    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 <= -5.8e+31) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 2.05e+60) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.8e+31:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 2.05e+60:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.8e+31)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 2.05e+60)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	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 <= -5.8e+31)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 2.05e+60)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.8e+31], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+60], N[(1.0 + N[(-0.1111111111111111 / x), $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 < -5.8000000000000001e31

   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-neg-frac 99.5%

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

      6. neg-mul-1 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*r/ 99.4%

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

      9. fma-def 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.1%

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

      1. associate-*r* 93.1%

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

      2. *-commutative 93.1%

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

      3. *-commutative 93.1%

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

      4. *-commutative 93.1%

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

   6. Simplified 93.1%

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

      1. sqrt-div 93.1%

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

      2. metadata-eval 93.1%

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

      3. frac-2neg 93.1%

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

      4. metadata-eval 93.1%

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

   8. Applied egg-rr 93.1%

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

      1. neg-mul-1 93.1%

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

      2. associate-/r* 93.1%

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

      3. metadata-eval 93.1%

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

   10. Simplified 93.1%

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

       1. expm1-log1p-u 87.4%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}} \cdot \left(-0.3333333333333333 \cdot y\right)\right)\right)} \]

       2. expm1-udef 87.4%

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

       3. associate-*l/ 87.4%

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

       4. *-un-lft-identity 87.4%

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

       5. *-commutative 87.4%

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

   12. Applied egg-rr 87.4%

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

       1. expm1-def 87.4%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\right)\right)} \]

       2. expm1-log1p 93.2%

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

       3. associate-/l* 93.0%

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

       4. associate-/r/ 93.2%

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

       5. metadata-eval 93.2%

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

       6. times-frac 93.1%

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

       7. *-commutative 93.1%

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

       8. *-commutative 93.1%

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

       9. times-frac 93.2%

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

       10. metadata-eval 93.2%

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

   14. Simplified 93.2%

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

   if -5.8000000000000001e31 < y < 2.05e60

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r* 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 98.5%

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

      1. cancel-sign-sub-inv 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   if 2.05e60 < y

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. neg-mul-1 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r/ 99.5%

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

      9. fma-def 99.5%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.5%

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

      14. distribute-neg-frac 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 96.0%

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

      1. associate-*r* 96.1%

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

      2. *-commutative 96.1%

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

      3. *-commutative 96.1%

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

      4. *-commutative 96.1%

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

   6. Simplified 96.1%

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

      1. sqrt-div 96.0%

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

      2. metadata-eval 96.0%

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

      3. frac-2neg 96.0%

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

      4. metadata-eval 96.0%

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

   8. Applied egg-rr 96.0%

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

      1. neg-mul-1 96.0%

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

      2. associate-/r* 96.0%

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

      3. metadata-eval 96.0%

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

   10. Simplified 96.0%

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

       1. *-commutative 96.0%

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

       2. metadata-eval 96.0%

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

       3. div-inv 95.9%

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

       4. frac-times 95.9%

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

       5. *-un-lft-identity 95.9%

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

   12. Applied egg-rr 95.9%

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

       1. clear-num 95.9%

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

       2. *-commutative 95.9%

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

       3. associate-*l/ 95.9%

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

       4. frac-2neg 95.9%

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

       5. metadata-eval 95.9%

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

       6. div-inv 95.9%

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

       7. associate-*l/ 95.9%

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

       8. *-commutative 95.9%

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

       9. distribute-neg-frac 95.9%

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

       10. distribute-rgt-neg-in 95.9%

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

       11. metadata-eval 95.9%

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

       12. clear-num 95.9%

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

       13. associate-/r* 96.0%

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

       14. div-inv 95.9%

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

       15. pow1/2 95.9%

           \[\leadsto 1 + -1 \cdot \frac{y \cdot \frac{1}{\color{blue}{{x}^{0.5}}}}{3} \]

       16. pow-flip 96.0%

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

       17. metadata-eval 96.0%

           \[\leadsto 1 + -1 \cdot \frac{y \cdot {x}^{\color{blue}{-0.5}}}{3} \]

       18. associate-*r/ 96.2%

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

       19. div-inv 96.2%

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

       20. metadata-eval 96.2%

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

       21. metadata-eval 96.2%

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

   14. Applied egg-rr 96.0%

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

       1. neg-mul-1 96.0%

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

       2. distribute-rgt-neg-in 96.0%

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

       3. distribute-neg-frac 96.0%

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

       4. metadata-eval 96.0%

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

   16. Simplified 96.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+31}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+60}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 5: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+31)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 4.5e+61)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+31) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 4.5e+61) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} 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 <= (-3.8d+31)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 4.5d+61) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    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 <= -3.8e+31) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 4.5e+61) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+31:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 4.5e+61:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+31)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 4.5e+61)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.8e+31)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 4.5e+61)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+31], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+61], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000001e31

   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-neg-frac 99.5%

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

      6. neg-mul-1 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*r/ 99.4%

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

      9. fma-def 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.1%

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

      1. associate-*r* 93.1%

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

      2. *-commutative 93.1%

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

      3. *-commutative 93.1%

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

      4. *-commutative 93.1%

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

   6. Simplified 93.1%

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

      1. sqrt-div 93.1%

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

      2. metadata-eval 93.1%

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

      3. frac-2neg 93.1%

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

      4. metadata-eval 93.1%

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

   8. Applied egg-rr 93.1%

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

      1. neg-mul-1 93.1%

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

      2. associate-/r* 93.1%

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

      3. metadata-eval 93.1%

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

   10. Simplified 93.1%

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

       1. expm1-log1p-u 87.4%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}} \cdot \left(-0.3333333333333333 \cdot y\right)\right)\right)} \]

       2. expm1-udef 87.4%

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

       3. associate-*l/ 87.4%

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

       4. *-un-lft-identity 87.4%

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

       5. *-commutative 87.4%

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

   12. Applied egg-rr 87.4%

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

       1. expm1-def 87.4%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\right)\right)} \]

       2. expm1-log1p 93.2%

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

       3. associate-/l* 93.0%

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

       4. associate-/r/ 93.2%

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

       5. metadata-eval 93.2%

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

       6. times-frac 93.1%

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

       7. *-commutative 93.1%

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

       8. *-commutative 93.1%

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

       9. times-frac 93.2%

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

       10. metadata-eval 93.2%

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

   14. Simplified 93.2%

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

   if -3.8000000000000001e31 < y < 4.5e61

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r* 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 98.5%

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

      1. cancel-sign-sub-inv 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   if 4.5e61 < y

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. neg-mul-1 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r/ 99.5%

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

      9. fma-def 99.5%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.5%

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

      14. distribute-neg-frac 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 96.0%

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

      1. associate-*r* 96.1%

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

      2. *-commutative 96.1%

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

      3. *-commutative 96.1%

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

      4. *-commutative 96.1%

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

   6. Simplified 96.1%

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

      1. sqrt-div 96.0%

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

      2. metadata-eval 96.0%

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

      3. frac-2neg 96.0%

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

      4. metadata-eval 96.0%

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

   8. Applied egg-rr 96.0%

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

      1. neg-mul-1 96.0%

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

      2. associate-/r* 96.0%

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

      3. metadata-eval 96.0%

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

   10. Simplified 96.0%

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

       1. associate-*l/ 96.1%

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

       2. *-un-lft-identity 96.1%

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

       3. *-commutative 96.1%

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

   12. Applied egg-rr 96.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 6: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-/r* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 99.7%

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

5. Final simplification 99.7%

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

Alternative 7: 62.6% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / 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. *-commutative 99.7%

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

   2. associate-/r* 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 59.4%

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

5. Final simplification 59.4%

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

Alternative 8: 62.6% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/r* 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 59.4%

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

   1. cancel-sign-sub-inv 59.4%

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

   2. metadata-eval 59.4%

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

   3. associate-*r/ 59.4%

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

   4. metadata-eval 59.4%

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

   5. +-commutative 59.4%

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

6. Simplified 59.4%

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

7. Final simplification 59.4%

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

Alternative 9: 31.2% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/r* 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around inf 30.5%

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

5. Final simplification 30.5%

   \[\leadsto 1 \]

Developer target: 99.6% 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 2023240 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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