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

Percentage Accurate: 99.7% → 99.7%

Time: 9.5s

Alternatives: 14

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. Final simplification 99.7%

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

Alternative 2: 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 3: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+65}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+39}:\\ \;\;\;\;1 + {\left(x \cdot -9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \frac{y}{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e+65)
   (+ 1.0 (/ y (* (sqrt x) -3.0)))
   (if (<= y 4.6e+39)
     (+ 1.0 (pow (* x -9.0) -1.0))
     (+ 1.0 (* (pow x -0.5) (/ y -3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+65) {
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	} else if (y <= 4.6e+39) {
		tmp = 1.0 + pow((x * -9.0), -1.0);
	} else {
		tmp = 1.0 + (pow(x, -0.5) * (y / -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.2d+65)) then
        tmp = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    else if (y <= 4.6d+39) then
        tmp = 1.0d0 + ((x * (-9.0d0)) ** (-1.0d0))
    else
        tmp = 1.0d0 + ((x ** (-0.5d0)) * (y / (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+65) {
		tmp = 1.0 + (y / (Math.sqrt(x) * -3.0));
	} else if (y <= 4.6e+39) {
		tmp = 1.0 + Math.pow((x * -9.0), -1.0);
	} else {
		tmp = 1.0 + (Math.pow(x, -0.5) * (y / -3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e+65:
		tmp = 1.0 + (y / (math.sqrt(x) * -3.0))
	elif y <= 4.6e+39:
		tmp = 1.0 + math.pow((x * -9.0), -1.0)
	else:
		tmp = 1.0 + (math.pow(x, -0.5) * (y / -3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e+65)
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	elseif (y <= 4.6e+39)
		tmp = Float64(1.0 + (Float64(x * -9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64((x ^ -0.5) * Float64(y / -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.2e+65)
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	elseif (y <= 4.6e+39)
		tmp = 1.0 + ((x * -9.0) ^ -1.0);
	else
		tmp = 1.0 + ((x ^ -0.5) * (y / -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e+65], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+39], N[(1.0 + N[Power[N[(x * -9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Power[x, -0.5], $MachinePrecision] * N[(y / -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.20000000000000007e65

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. *-commutative 99.6%

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

      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 97.1%

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

      1. associate-*r* 97.0%

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

   6. Simplified 97.0%

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

      1. expm1-log1p-u 90.3%

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

      2. expm1-udef 90.3%

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

      3. associate-*l* 90.3%

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

      4. sqrt-div 90.3%

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

      5. metadata-eval 90.3%

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

      6. un-div-inv 90.3%

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

   8. Applied egg-rr 90.3%

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

      1. expm1-def 90.3%

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

      2. expm1-log1p 97.2%

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

      3. metadata-eval 97.2%

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

      4. times-frac 97.3%

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

      5. neg-mul-1 97.3%

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

      6. *-commutative 97.3%

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

      7. distribute-frac-neg 97.3%

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

      8. *-commutative 97.3%

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

      9. remove-double-neg 97.3%

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

      10. neg-mul-1 97.3%

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

      11. times-frac 97.2%

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

      12. metadata-eval 97.2%

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

      13. metadata-eval 97.2%

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

      14. times-frac 97.3%

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

      15. *-commutative 97.3%

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

      16. associate-*l/ 97.2%

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

      17. *-commutative 97.2%

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

      18. distribute-lft-neg-out 97.2%

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

      19. remove-double-neg 97.2%

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

      20. *-commutative 97.2%

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

      21. associate-/r* 97.2%

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

      22. metadata-eval 97.2%

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

   10. Simplified 97.2%

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

       1. associate-*r/ 97.1%

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

       2. associate-*l/ 97.2%

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

       3. expm1-log1p-u 90.3%

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

       4. expm1-udef 90.3%

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

       5. associate-*l/ 90.3%

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

       6. associate-*r/ 90.3%

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

       7. clear-num 90.3%

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

       8. un-div-inv 90.3%

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

       9. div-inv 90.3%

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

       10. metadata-eval 90.3%

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

   12. Applied egg-rr 90.3%

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

       1. expm1-def 90.3%

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

       2. expm1-log1p 97.3%

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

   14. Simplified 97.3%

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

   if -3.20000000000000007e65 < y < 4.60000000000000024e39

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.6%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. 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/ 97.6%

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

      4. metadata-eval 97.6%

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

      5. +-commutative 97.6%

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

   6. Simplified 97.6%

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

      1. clear-num 97.6%

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

      2. inv-pow 97.6%

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

      3. div-inv 97.8%

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

      4. metadata-eval 97.8%

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

   8. Applied egg-rr 97.8%

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

   if 4.60000000000000024e39 < 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/ 99.6%

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

      9. fma-def 99.6%

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

      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 89.7%

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

      1. associate-*r* 89.7%

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

   6. Simplified 89.7%

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

      1. expm1-log1p-u 1.8%

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

      2. expm1-udef 1.8%

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

      3. associate-*l* 1.8%

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

      4. sqrt-div 1.8%

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

      5. metadata-eval 1.8%

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

      6. un-div-inv 1.8%

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

   8. Applied egg-rr 1.8%

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

      1. expm1-def 1.8%

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

      2. expm1-log1p 89.7%

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

      3. metadata-eval 89.7%

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

      4. times-frac 89.8%

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

      5. neg-mul-1 89.8%

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

      6. *-commutative 89.8%

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

      7. distribute-frac-neg 89.8%

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

      8. *-commutative 89.8%

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

      9. remove-double-neg 89.8%

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

      10. neg-mul-1 89.8%

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

      11. times-frac 89.7%

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

      12. metadata-eval 89.7%

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

      13. metadata-eval 89.7%

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

      14. times-frac 89.8%

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

      15. *-commutative 89.8%

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

      16. associate-*l/ 90.0%

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

      17. *-commutative 90.0%

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

      18. distribute-lft-neg-out 90.0%

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

      19. remove-double-neg 90.0%

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

      20. *-commutative 90.0%

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

      21. associate-/r* 89.9%

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

      22. metadata-eval 89.9%

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

   10. Simplified 89.9%

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

       1. associate-*r/ 89.9%

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

       2. associate-*l/ 89.7%

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

       3. expm1-log1p-u 1.8%

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

       4. expm1-udef 1.8%

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

       5. associate-*l/ 1.8%

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

       6. associate-*r/ 1.8%

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

       7. clear-num 1.8%

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

       8. un-div-inv 1.8%

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

       9. div-inv 1.8%

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

       10. metadata-eval 1.8%

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

   12. Applied egg-rr 1.8%

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

       1. expm1-def 1.8%

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

       2. expm1-log1p 89.8%

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

   14. Simplified 89.8%

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

       1. *-un-lft-identity 89.8%

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

       2. times-frac 89.9%

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

       3. metadata-eval 89.9%

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

       4. sqrt-div 89.9%

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

       5. inv-pow 89.9%

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

       6. sqrt-pow1 90.0%

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

       7. metadata-eval 90.0%

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

   16. Applied egg-rr 90.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+65}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+39}:\\ \;\;\;\;1 + {\left(x \cdot -9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \frac{y}{-3}\\ \end{array} \]

Alternative 4: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+65} \lor \neg \left(y \leq 2.9 \cdot 10^{+32}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + {\left(x \cdot -9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.1e+65) (not (<= y 2.9e+32)))
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))
   (+ 1.0 (pow (* x -9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.1e+65) || !(y <= 2.9e+32)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	} else {
		tmp = 1.0 + pow((x * -9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.1d+65)) .or. (.not. (y <= 2.9d+32))) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    else
        tmp = 1.0d0 + ((x * (-9.0d0)) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.1e+65) || !(y <= 2.9e+32)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	} else {
		tmp = 1.0 + Math.pow((x * -9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.1e+65) or not (y <= 2.9e+32):
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	else:
		tmp = 1.0 + math.pow((x * -9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.1e+65) || !(y <= 2.9e+32))
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	else
		tmp = Float64(1.0 + (Float64(x * -9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.1e+65) || ~((y <= 2.9e+32)))
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	else
		tmp = 1.0 + ((x * -9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.1e+65], N[Not[LessEqual[y, 2.9e+32]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Power[N[(x * -9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1000000000000001e65 or 2.90000000000000003e32 < 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/ 99.5%

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

      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 93.0%

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

      1. associate-*r* 92.9%

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

   6. Simplified 92.9%

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

      1. expm1-log1p-u 40.6%

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

      2. expm1-udef 40.6%

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

      3. associate-*l* 40.6%

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

      4. sqrt-div 40.6%

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

      5. metadata-eval 40.6%

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

      6. un-div-inv 40.6%

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

   8. Applied egg-rr 40.6%

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

      1. expm1-def 40.6%

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

      2. expm1-log1p 93.0%

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

      3. metadata-eval 93.0%

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

      4. times-frac 93.1%

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

      5. neg-mul-1 93.1%

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

      6. *-commutative 93.1%

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

      7. distribute-frac-neg 93.1%

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

      8. *-commutative 93.1%

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

      9. remove-double-neg 93.1%

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

      10. neg-mul-1 93.1%

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

      11. times-frac 93.0%

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

      12. metadata-eval 93.0%

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

      13. metadata-eval 93.0%

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

      14. times-frac 93.1%

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

      15. *-commutative 93.1%

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

      16. associate-*l/ 93.1%

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

      17. *-commutative 93.1%

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

      18. distribute-lft-neg-out 93.1%

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

      19. remove-double-neg 93.1%

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

      20. *-commutative 93.1%

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

      21. associate-/r* 93.1%

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

      22. metadata-eval 93.1%

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

   10. Simplified 93.1%

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

   if -4.1000000000000001e65 < y < 2.90000000000000003e32

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.6%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. 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/ 97.6%

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

      4. metadata-eval 97.6%

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

      5. +-commutative 97.6%

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

   6. Simplified 97.6%

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

      1. clear-num 97.6%

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

      2. inv-pow 97.6%

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

      3. div-inv 97.8%

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

      4. metadata-eval 97.8%

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

   8. Applied egg-rr 97.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+65} \lor \neg \left(y \leq 2.9 \cdot 10^{+32}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + {\left(x \cdot -9\right)}^{-1}\\ \end{array} \]

Alternative 5: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+65}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+37}:\\ \;\;\;\;1 + {\left(x \cdot -9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.9e+65)
   (+ 1.0 (/ y (* (sqrt x) -3.0)))
   (if (<= y 6e+37)
     (+ 1.0 (pow (* x -9.0) -1.0))
     (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.9e+65) {
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	} else if (y <= 6e+37) {
		tmp = 1.0 + pow((x * -9.0), -1.0);
	} 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 <= (-3.9d+65)) then
        tmp = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    else if (y <= 6d+37) then
        tmp = 1.0d0 + ((x * (-9.0d0)) ** (-1.0d0))
    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 <= -3.9e+65) {
		tmp = 1.0 + (y / (Math.sqrt(x) * -3.0));
	} else if (y <= 6e+37) {
		tmp = 1.0 + Math.pow((x * -9.0), -1.0);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.9e+65:
		tmp = 1.0 + (y / (math.sqrt(x) * -3.0))
	elif y <= 6e+37:
		tmp = 1.0 + math.pow((x * -9.0), -1.0)
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.9e+65)
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	elseif (y <= 6e+37)
		tmp = Float64(1.0 + (Float64(x * -9.0) ^ -1.0));
	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 <= -3.9e+65)
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	elseif (y <= 6e+37)
		tmp = 1.0 + ((x * -9.0) ^ -1.0);
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.9e+65], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+37], N[(1.0 + N[Power[N[(x * -9.0), $MachinePrecision], -1.0], $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 < -3.8999999999999998e65

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. *-commutative 99.6%

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

      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 97.1%

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

      1. associate-*r* 97.0%

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

   6. Simplified 97.0%

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

      1. expm1-log1p-u 90.3%

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

      2. expm1-udef 90.3%

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

      3. associate-*l* 90.3%

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

      4. sqrt-div 90.3%

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

      5. metadata-eval 90.3%

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

      6. un-div-inv 90.3%

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

   8. Applied egg-rr 90.3%

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

      1. expm1-def 90.3%

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

      2. expm1-log1p 97.2%

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

      3. metadata-eval 97.2%

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

      4. times-frac 97.3%

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

      5. neg-mul-1 97.3%

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

      6. *-commutative 97.3%

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

      7. distribute-frac-neg 97.3%

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

      8. *-commutative 97.3%

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

      9. remove-double-neg 97.3%

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

      10. neg-mul-1 97.3%

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

      11. times-frac 97.2%

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

      12. metadata-eval 97.2%

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

      13. metadata-eval 97.2%

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

      14. times-frac 97.3%

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

      15. *-commutative 97.3%

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

      16. associate-*l/ 97.2%

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

      17. *-commutative 97.2%

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

      18. distribute-lft-neg-out 97.2%

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

      19. remove-double-neg 97.2%

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

      20. *-commutative 97.2%

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

      21. associate-/r* 97.2%

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

      22. metadata-eval 97.2%

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

   10. Simplified 97.2%

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

       1. associate-*r/ 97.1%

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

       2. associate-*l/ 97.2%

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

       3. expm1-log1p-u 90.3%

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

       4. expm1-udef 90.3%

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

       5. associate-*l/ 90.3%

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

       6. associate-*r/ 90.3%

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

       7. clear-num 90.3%

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

       8. un-div-inv 90.3%

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

       9. div-inv 90.3%

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

       10. metadata-eval 90.3%

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

   12. Applied egg-rr 90.3%

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

       1. expm1-def 90.3%

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

       2. expm1-log1p 97.3%

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

   14. Simplified 97.3%

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

   if -3.8999999999999998e65 < y < 6.00000000000000043e37

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.6%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. 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/ 97.6%

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

      4. metadata-eval 97.6%

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

      5. +-commutative 97.6%

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

   6. Simplified 97.6%

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

      1. clear-num 97.6%

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

      2. inv-pow 97.6%

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

      3. div-inv 97.8%

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

      4. metadata-eval 97.8%

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

   8. Applied egg-rr 97.8%

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

   if 6.00000000000000043e37 < 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/ 99.6%

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

      9. fma-def 99.6%

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

      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 89.7%

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

      1. associate-*r* 89.7%

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

   6. Simplified 89.7%

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

      1. expm1-log1p-u 1.8%

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

      2. expm1-udef 1.8%

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

      3. associate-*l* 1.8%

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

      4. sqrt-div 1.8%

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

      5. metadata-eval 1.8%

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

      6. un-div-inv 1.8%

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

   8. Applied egg-rr 1.8%

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

      1. expm1-def 1.8%

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

      2. expm1-log1p 89.7%

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

      3. metadata-eval 89.7%

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

      4. times-frac 89.8%

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

      5. neg-mul-1 89.8%

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

      6. *-commutative 89.8%

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

      7. distribute-frac-neg 89.8%

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

      8. *-commutative 89.8%

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

      9. remove-double-neg 89.8%

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

      10. neg-mul-1 89.8%

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

      11. times-frac 89.7%

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

      12. metadata-eval 89.7%

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

      13. metadata-eval 89.7%

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

      14. times-frac 89.8%

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

      15. *-commutative 89.8%

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

      16. associate-*l/ 90.0%

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

      17. *-commutative 90.0%

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

      18. distribute-lft-neg-out 90.0%

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

      19. remove-double-neg 90.0%

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

      20. *-commutative 90.0%

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

      21. associate-/r* 89.9%

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

      22. metadata-eval 89.9%

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

   10. Simplified 89.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+65}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+37}:\\ \;\;\;\;1 + {\left(x \cdot -9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 6: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+65}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+38}:\\ \;\;\;\;1 + {\left(x \cdot -9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e+65)
   (+ 1.0 (/ y (* (sqrt x) -3.0)))
   (if (<= y 2.3e+38)
     (+ 1.0 (pow (* x -9.0) -1.0))
     (+ 1.0 (/ (/ y -3.0) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+65) {
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	} else if (y <= 2.3e+38) {
		tmp = 1.0 + pow((x * -9.0), -1.0);
	} else {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.2d+65)) then
        tmp = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    else if (y <= 2.3d+38) then
        tmp = 1.0d0 + ((x * (-9.0d0)) ** (-1.0d0))
    else
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+65) {
		tmp = 1.0 + (y / (Math.sqrt(x) * -3.0));
	} else if (y <= 2.3e+38) {
		tmp = 1.0 + Math.pow((x * -9.0), -1.0);
	} else {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e+65:
		tmp = 1.0 + (y / (math.sqrt(x) * -3.0))
	elif y <= 2.3e+38:
		tmp = 1.0 + math.pow((x * -9.0), -1.0)
	else:
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e+65)
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	elseif (y <= 2.3e+38)
		tmp = Float64(1.0 + (Float64(x * -9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.2e+65)
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	elseif (y <= 2.3e+38)
		tmp = 1.0 + ((x * -9.0) ^ -1.0);
	else
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e+65], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e+38], N[(1.0 + N[Power[N[(x * -9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.20000000000000007e65

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. *-commutative 99.6%

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

      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 97.1%

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

      1. associate-*r* 97.0%

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

   6. Simplified 97.0%

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

      1. expm1-log1p-u 90.3%

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

      2. expm1-udef 90.3%

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

      3. associate-*l* 90.3%

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

      4. sqrt-div 90.3%

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

      5. metadata-eval 90.3%

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

      6. un-div-inv 90.3%

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

   8. Applied egg-rr 90.3%

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

      1. expm1-def 90.3%

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

      2. expm1-log1p 97.2%

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

      3. metadata-eval 97.2%

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

      4. times-frac 97.3%

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

      5. neg-mul-1 97.3%

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

      6. *-commutative 97.3%

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

      7. distribute-frac-neg 97.3%

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

      8. *-commutative 97.3%

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

      9. remove-double-neg 97.3%

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

      10. neg-mul-1 97.3%

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

      11. times-frac 97.2%

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

      12. metadata-eval 97.2%

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

      13. metadata-eval 97.2%

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

      14. times-frac 97.3%

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

      15. *-commutative 97.3%

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

      16. associate-*l/ 97.2%

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

      17. *-commutative 97.2%

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

      18. distribute-lft-neg-out 97.2%

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

      19. remove-double-neg 97.2%

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

      20. *-commutative 97.2%

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

      21. associate-/r* 97.2%

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

      22. metadata-eval 97.2%

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

   10. Simplified 97.2%

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

       1. associate-*r/ 97.1%

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

       2. associate-*l/ 97.2%

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

       3. expm1-log1p-u 90.3%

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

       4. expm1-udef 90.3%

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

       5. associate-*l/ 90.3%

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

       6. associate-*r/ 90.3%

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

       7. clear-num 90.3%

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

       8. un-div-inv 90.3%

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

       9. div-inv 90.3%

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

       10. metadata-eval 90.3%

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

   12. Applied egg-rr 90.3%

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

       1. expm1-def 90.3%

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

       2. expm1-log1p 97.3%

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

   14. Simplified 97.3%

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

   if -3.20000000000000007e65 < y < 2.3000000000000001e38

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.6%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. 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/ 97.6%

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

      4. metadata-eval 97.6%

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

      5. +-commutative 97.6%

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

   6. Simplified 97.6%

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

      1. clear-num 97.6%

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

      2. inv-pow 97.6%

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

      3. div-inv 97.8%

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

      4. metadata-eval 97.8%

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

   8. Applied egg-rr 97.8%

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

   if 2.3000000000000001e38 < 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/ 99.6%

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

      9. fma-def 99.6%

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

      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 89.7%

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

      1. associate-*r* 89.7%

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

   6. Simplified 89.7%

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

      1. expm1-log1p-u 1.8%

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

      2. expm1-udef 1.8%

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

      3. associate-*l* 1.8%

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

      4. sqrt-div 1.8%

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

      5. metadata-eval 1.8%

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

      6. un-div-inv 1.8%

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

   8. Applied egg-rr 1.8%

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

      1. expm1-def 1.8%

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

      2. expm1-log1p 89.7%

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

      3. metadata-eval 89.7%

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

      4. times-frac 89.8%

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

      5. neg-mul-1 89.8%

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

      6. *-commutative 89.8%

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

      7. distribute-frac-neg 89.8%

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

      8. *-commutative 89.8%

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

      9. remove-double-neg 89.8%

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

      10. neg-mul-1 89.8%

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

      11. times-frac 89.7%

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

      12. metadata-eval 89.7%

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

      13. metadata-eval 89.7%

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

      14. times-frac 89.8%

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

      15. *-commutative 89.8%

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

      16. associate-*l/ 90.0%

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

      17. *-commutative 90.0%

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

      18. distribute-lft-neg-out 90.0%

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

      19. remove-double-neg 90.0%

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

      20. *-commutative 90.0%

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

      21. associate-/r* 89.9%

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

      22. metadata-eval 89.9%

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

   10. Simplified 89.9%

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

       1. associate-*r/ 89.9%

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

       2. associate-*l/ 89.7%

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

       3. expm1-log1p-u 1.8%

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

       4. expm1-udef 1.8%

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

       5. associate-*l/ 1.8%

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

       6. associate-*r/ 1.8%

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

       7. clear-num 1.8%

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

       8. un-div-inv 1.8%

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

       9. div-inv 1.8%

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

       10. metadata-eval 1.8%

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

   12. Applied egg-rr 1.8%

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

       1. expm1-def 1.8%

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

       2. expm1-log1p 89.8%

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

   14. Simplified 89.8%

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

       1. add-cube-cbrt 88.7%

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

       2. *-un-lft-identity 88.7%

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

       3. times-frac 88.7%

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

       4. pow2 88.7%

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

   16. Applied egg-rr 88.7%

       \[\leadsto 1 + \color{blue}{\frac{{\left(\sqrt[3]{y}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{y}}{\sqrt{x} \cdot -3}} \]
   17. Step-by-step derivation

       1. /-rgt-identity 88.7%

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

       2. associate-*r/ 88.7%

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

       3. *-commutative 88.7%

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

       4. unpow2 88.7%

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

       5. rem-3cbrt-lft 89.8%

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

       6. associate-/r* 89.9%

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

   18. Simplified 89.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+65}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+38}:\\ \;\;\;\;1 + {\left(x \cdot -9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \]

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. *-commutative 99.6%

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

   9. fma-def 99.6%

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

   10. metadata-eval 99.6%

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

   11. *-commutative 99.6%

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

   12. associate-/r* 99.5%

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

   13. distribute-neg-frac 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 8: 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 99.6%

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

5. Final simplification 99.6%

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

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

Alternative 10: 64.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.1111111111111111}{x} + -1\\ \mathbf{if}\;y \leq 4.1 \cdot 10^{+104}:\\ \;\;\;\;1 + {\left(x \cdot -9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t_0 \cdot t_0}{1 - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ -0.1111111111111111 x) -1.0)))
   (if (<= y 4.1e+104)
     (+ 1.0 (pow (* x -9.0) -1.0))
     (/ (- 1.0 (* t_0 t_0)) (- 1.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = (-0.1111111111111111 / x) + -1.0;
	double tmp;
	if (y <= 4.1e+104) {
		tmp = 1.0 + pow((x * -9.0), -1.0);
	} else {
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-0.1111111111111111d0) / x) + (-1.0d0)
    if (y <= 4.1d+104) then
        tmp = 1.0d0 + ((x * (-9.0d0)) ** (-1.0d0))
    else
        tmp = (1.0d0 - (t_0 * t_0)) / (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (-0.1111111111111111 / x) + -1.0;
	double tmp;
	if (y <= 4.1e+104) {
		tmp = 1.0 + Math.pow((x * -9.0), -1.0);
	} else {
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (-0.1111111111111111 / x) + -1.0
	tmp = 0
	if y <= 4.1e+104:
		tmp = 1.0 + math.pow((x * -9.0), -1.0)
	else:
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-0.1111111111111111 / x) + -1.0)
	tmp = 0.0
	if (y <= 4.1e+104)
		tmp = Float64(1.0 + (Float64(x * -9.0) ^ -1.0));
	else
		tmp = Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (-0.1111111111111111 / x) + -1.0;
	tmp = 0.0;
	if (y <= 4.1e+104)
		tmp = 1.0 + ((x * -9.0) ^ -1.0);
	else
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, 4.1e+104], N[(1.0 + N[Power[N[(x * -9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.09999999999999985e104

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.6%

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

      14. distribute-neg-frac 99.6%

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

      15. metadata-eval 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 75.4%

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

      1. cancel-sign-sub-inv 75.4%

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

      2. metadata-eval 75.4%

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

      3. associate-*r/ 75.5%

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

      4. metadata-eval 75.5%

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

      5. +-commutative 75.5%

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

   6. Simplified 75.5%

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

      1. clear-num 75.4%

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

      2. inv-pow 75.4%

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

      3. div-inv 75.6%

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

      4. metadata-eval 75.6%

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

   8. Applied egg-rr 75.6%

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

   if 4.09999999999999985e104 < 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/ 99.6%

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

      9. fma-def 99.6%

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

      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 0 3.7%

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

      1. cancel-sign-sub-inv 3.7%

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

      2. metadata-eval 3.7%

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

      3. associate-*r/ 3.7%

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

      4. metadata-eval 3.7%

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

      5. +-commutative 3.7%

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

   6. Simplified 3.7%

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

   7. Taylor expanded in x around 0 4.8%

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

      1. div-inv 4.8%

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

   9. Applied egg-rr 4.8%

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

       1. div-inv 4.8%

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

       2. expm1-log1p-u 1.7%

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

       3. expm1-udef 1.1%

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

       4. log1p-udef 1.1%

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

       5. add-exp-log 4.3%

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

       6. associate--l+ 4.3%

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

       7. flip-+ 18.3%

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

       8. metadata-eval 18.3%

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

       9. sub-neg 18.3%

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

       10. metadata-eval 18.3%

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

       11. sub-neg 18.3%

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

       12. metadata-eval 18.3%

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

       13. sub-neg 18.3%

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

       14. metadata-eval 18.3%

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

   11. Applied egg-rr 18.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{+104}:\\ \;\;\;\;1 + {\left(x \cdot -9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{-0.1111111111111111}{x} + -1\right) \cdot \left(\frac{-0.1111111111111111}{x} + -1\right)}{1 - \left(\frac{-0.1111111111111111}{x} + -1\right)}\\ \end{array} \]

Alternative 11: 64.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.1111111111111111}{x} + -1\\ \mathbf{if}\;y \leq 4.1 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t_0 \cdot t_0}{1 - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ -0.1111111111111111 x) -1.0)))
   (if (<= y 4.1e+104)
     (+ 1.0 (/ -0.1111111111111111 x))
     (/ (- 1.0 (* t_0 t_0)) (- 1.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = (-0.1111111111111111 / x) + -1.0;
	double tmp;
	if (y <= 4.1e+104) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-0.1111111111111111d0) / x) + (-1.0d0)
    if (y <= 4.1d+104) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 - (t_0 * t_0)) / (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (-0.1111111111111111 / x) + -1.0;
	double tmp;
	if (y <= 4.1e+104) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (-0.1111111111111111 / x) + -1.0
	tmp = 0
	if y <= 4.1e+104:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-0.1111111111111111 / x) + -1.0)
	tmp = 0.0
	if (y <= 4.1e+104)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (-0.1111111111111111 / x) + -1.0;
	tmp = 0.0;
	if (y <= 4.1e+104)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, 4.1e+104], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.09999999999999985e104

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.6%

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

      14. distribute-neg-frac 99.6%

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

      15. metadata-eval 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 75.4%

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

      1. cancel-sign-sub-inv 75.4%

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

      2. metadata-eval 75.4%

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

      3. associate-*r/ 75.5%

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

      4. metadata-eval 75.5%

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

      5. +-commutative 75.5%

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

   6. Simplified 75.5%

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

   if 4.09999999999999985e104 < 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/ 99.6%

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

      9. fma-def 99.6%

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

      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 0 3.7%

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

      1. cancel-sign-sub-inv 3.7%

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

      2. metadata-eval 3.7%

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

      3. associate-*r/ 3.7%

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

      4. metadata-eval 3.7%

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

      5. +-commutative 3.7%

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

   6. Simplified 3.7%

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

   7. Taylor expanded in x around 0 4.8%

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

      1. div-inv 4.8%

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

   9. Applied egg-rr 4.8%

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

       1. div-inv 4.8%

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

       2. expm1-log1p-u 1.7%

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

       3. expm1-udef 1.1%

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

       4. log1p-udef 1.1%

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

       5. add-exp-log 4.3%

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

       6. associate--l+ 4.3%

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

       7. flip-+ 18.3%

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

       8. metadata-eval 18.3%

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

       9. sub-neg 18.3%

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

       10. metadata-eval 18.3%

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

       11. sub-neg 18.3%

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

       12. metadata-eval 18.3%

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

       13. sub-neg 18.3%

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

       14. metadata-eval 18.3%

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

   11. Applied egg-rr 18.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{-0.1111111111111111}{x} + -1\right) \cdot \left(\frac{-0.1111111111111111}{x} + -1\right)}{1 - \left(\frac{-0.1111111111111111}{x} + -1\right)}\\ \end{array} \]

Alternative 12: 61.8% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 125:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 125.0) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 125.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 125.0d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 125.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 125.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 125.0)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 125.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 125.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 125

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*r/ 99.6%

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

      9. fma-def 99.6%

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

      10. associate-/r* 99.6%

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

      11. metadata-eval 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r* 99.3%

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

      14. distribute-neg-frac 99.3%

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

      15. metadata-eval 99.3%

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

      16. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 69.8%

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

      1. cancel-sign-sub-inv 69.8%

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

      2. metadata-eval 69.8%

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

      3. associate-*r/ 69.8%

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

      4. metadata-eval 69.8%

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

      5. +-commutative 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in x around 0 69.1%

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

   if 125 < x

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.8%

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

      14. distribute-neg-frac 99.8%

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

      15. metadata-eval 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 58.6%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 125:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 62.9% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. neg-mul-1 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-*r/ 99.7%

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

   9. fma-def 99.7%

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

   10. associate-/r* 99.7%

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

   11. metadata-eval 99.7%

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

   12. *-commutative 99.7%

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

   13. associate-/r* 99.6%

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

   14. distribute-neg-frac 99.6%

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

   15. metadata-eval 99.6%

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

   16. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 64.2%

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

   1. cancel-sign-sub-inv 64.2%

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

   2. metadata-eval 64.2%

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

   3. associate-*r/ 64.2%

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

   4. metadata-eval 64.2%

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

   5. +-commutative 64.2%

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

6. Simplified 64.2%

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

7. Final simplification 64.2%

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

Alternative 14: 31.9% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. neg-mul-1 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-*r/ 99.7%

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

   9. fma-def 99.7%

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

   10. associate-/r* 99.7%

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

   11. metadata-eval 99.7%

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

   12. *-commutative 99.7%

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

   13. associate-/r* 99.6%

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

   14. distribute-neg-frac 99.6%

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

   15. metadata-eval 99.6%

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

   16. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around inf 33.3%

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

5. Final simplification 33.3%

   \[\leadsto 1 \]

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 2023185 
(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)))))