Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Percentage Accurate: 77.3% → 99.4%

Time: 8.3s

Alternatives: 6

Speedup: 18.8×

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Initial Program: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+82} \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.2e+82) (not (<= x 3.0)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.2e+82) || !(x <= 3.0)) {
		tmp = exp(-y) / x;
	} else {
		tmp = pow(exp(x), log((x / (x + y)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.2d+82)) .or. (.not. (x <= 3.0d0))) then
        tmp = exp(-y) / x
    else
        tmp = (exp(x) ** log((x / (x + y)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.2e+82) || !(x <= 3.0)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.2e+82) or not (x <= 3.0):
		tmp = math.exp(-y) / x
	else:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.2e+82) || !(x <= 3.0))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.2e+82) || ~((x <= 3.0)))
		tmp = exp(-y) / x;
	else
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.2e+82], N[Not[LessEqual[x, 3.0]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000001e82 or 3 < x

   1. Initial program 70.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. *-commutative 70.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{x}{x + y}\right) \cdot x}}}{x} \]

      2. exp-to-pow 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x} \]

   if -2.2000000000000001e82 < x < 3

   1. Initial program 86.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+82} \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.062 \lor \neg \left(x \leq 0.38\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.062) (not (<= x 0.38))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.062) || !(x <= 0.38)) {
		tmp = exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.062d0)) .or. (.not. (x <= 0.38d0))) then
        tmp = exp(-y) / x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.062) || !(x <= 0.38)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.062) or not (x <= 0.38):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.062) || !(x <= 0.38))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.062) || ~((x <= 0.38)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.062], N[Not[LessEqual[x, 0.38]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.062 or 0.38 < x

   1. Initial program 74.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. *-commutative 74.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{x}{x + y}\right) \cdot x}}}{x} \]

      2. exp-to-pow 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x} \]

   if -0.062 < x < 0.38

   1. Initial program 83.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.062 \lor \neg \left(x \leq 0.38\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 82.7% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + x \cdot y}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -0.85:\\ \;\;\;\;\left(\frac{1}{x} + 0.5 \cdot \left(y \cdot \frac{y}{x}\right)\right) - \frac{y}{x}\\ \mathbf{elif}\;x \leq 0.156:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x (* x y)))))
   (if (<= x -4.8e+173)
     t_0
     (if (<= x -0.85)
       (- (+ (/ 1.0 x) (* 0.5 (* y (/ y x)))) (/ y x))
       (if (<= x 0.156) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x + (x * y));
	double tmp;
	if (x <= -4.8e+173) {
		tmp = t_0;
	} else if (x <= -0.85) {
		tmp = ((1.0 / x) + (0.5 * (y * (y / x)))) - (y / x);
	} else if (x <= 0.156) {
		tmp = 1.0 / x;
	} else {
		tmp = 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 = 1.0d0 / (x + (x * y))
    if (x <= (-4.8d+173)) then
        tmp = t_0
    else if (x <= (-0.85d0)) then
        tmp = ((1.0d0 / x) + (0.5d0 * (y * (y / x)))) - (y / x)
    else if (x <= 0.156d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (x + (x * y));
	double tmp;
	if (x <= -4.8e+173) {
		tmp = t_0;
	} else if (x <= -0.85) {
		tmp = ((1.0 / x) + (0.5 * (y * (y / x)))) - (y / x);
	} else if (x <= 0.156) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x + (x * y))
	tmp = 0
	if x <= -4.8e+173:
		tmp = t_0
	elif x <= -0.85:
		tmp = ((1.0 / x) + (0.5 * (y * (y / x)))) - (y / x)
	elif x <= 0.156:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x + Float64(x * y)))
	tmp = 0.0
	if (x <= -4.8e+173)
		tmp = t_0;
	elseif (x <= -0.85)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(0.5 * Float64(y * Float64(y / x)))) - Float64(y / x));
	elseif (x <= 0.156)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (x + (x * y));
	tmp = 0.0;
	if (x <= -4.8e+173)
		tmp = t_0;
	elseif (x <= -0.85)
		tmp = ((1.0 / x) + (0.5 * (y * (y / x)))) - (y / x);
	elseif (x <= 0.156)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e+173], t$95$0, If[LessEqual[x, -0.85], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 * N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.156], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7999999999999998e173 or 0.156 < x

   1. Initial program 65.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. exp-prod 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in x around inf 55.4%

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

      1. mul-1-neg 55.4%

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

      2. unsub-neg 55.4%

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

   6. Simplified 55.4%

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

      1. frac-2neg 55.4%

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

      2. div-inv 55.4%

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

      3. sub-neg 55.4%

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

      4. distribute-neg-in 55.4%

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

      5. metadata-eval 55.4%

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

      6. add-sqr-sqrt 25.6%

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

      7. sqrt-unprod 57.8%

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

      8. sqr-neg 57.8%

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

      9. sqrt-unprod 29.5%

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

      10. add-sqr-sqrt 54.5%

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

      11. add-sqr-sqrt 25.1%

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

      12. sqrt-unprod 54.7%

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

      13. sqr-neg 54.7%

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

      14. sqrt-unprod 29.8%

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

      15. add-sqr-sqrt 55.4%

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

   8. Applied egg-rr 55.4%

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

      1. *-commutative 55.4%

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

      2. distribute-rgt-in 55.4%

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

      3. div-inv 55.4%

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

      4. metadata-eval 55.4%

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

      5. frac-2neg 55.4%

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

      6. add-sqr-sqrt 15.2%

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

      7. sqrt-unprod 55.4%

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

      8. frac-times 55.4%

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

      9. sqr-neg 55.4%

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

      10. frac-times 55.4%

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

      11. sqrt-unprod 39.4%

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

      12. add-sqr-sqrt 54.5%

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

      13. div-inv 54.5%

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

      14. frac-add 25.1%

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

   10. Applied egg-rr 55.4%

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

   11. Taylor expanded in y around 0 76.4%

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

       1. *-commutative 76.4%

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

   13. Simplified 76.4%

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

   if -4.7999999999999998e173 < x < -0.849999999999999978

   1. Initial program 91.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. *-commutative 91.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{x}{x + y}\right) \cdot x}}}{x} \]

      2. exp-to-pow 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x} \]

   7. Taylor expanded in y around 0 78.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{x} + \left(0.5 \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. div-inv 78.9%

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

      2. unpow2 78.9%

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

      3. associate-*l* 77.0%

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

      4. div-inv 77.0%

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

   9. Applied egg-rr 77.0%

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

   if -0.849999999999999978 < x < 0.156

   1. Initial program 83.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+173}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{elif}\;x \leq -0.85:\\ \;\;\;\;\left(\frac{1}{x} + 0.5 \cdot \left(y \cdot \frac{y}{x}\right)\right) - \frac{y}{x}\\ \mathbf{elif}\;x \leq 0.156:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]

Alternative 4: 83.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;\left(\frac{1}{x} + \frac{y \cdot \left(y \cdot 0.5\right)}{x}\right) - \frac{y}{x}\\ \mathbf{elif}\;x \leq 0.49:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.7)
   (- (+ (/ 1.0 x) (/ (* y (* y 0.5)) x)) (/ y x))
   (if (<= x 0.49) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.7) {
		tmp = ((1.0 / x) + ((y * (y * 0.5)) / x)) - (y / x);
	} else if (x <= 0.49) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.7d0)) then
        tmp = ((1.0d0 / x) + ((y * (y * 0.5d0)) / x)) - (y / x)
    else if (x <= 0.49d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.7) {
		tmp = ((1.0 / x) + ((y * (y * 0.5)) / x)) - (y / x);
	} else if (x <= 0.49) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.7:
		tmp = ((1.0 / x) + ((y * (y * 0.5)) / x)) - (y / x)
	elif x <= 0.49:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.7)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(Float64(y * Float64(y * 0.5)) / x)) - Float64(y / x));
	elseif (x <= 0.49)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.7)
		tmp = ((1.0 / x) + ((y * (y * 0.5)) / x)) - (y / x);
	elseif (x <= 0.49)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.7], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.49], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.69999999999999996

   1. Initial program 78.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. *-commutative 78.7%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{x}{x + y}\right) \cdot x}}}{x} \]

      2. exp-to-pow 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x} \]

   7. Taylor expanded in y around 0 71.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{x} + \left(0.5 \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. div-inv 71.6%

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

      2. unpow2 71.6%

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

      3. associate-*l* 69.2%

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

      4. div-inv 69.2%

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

   9. Applied egg-rr 69.2%

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

       1. associate-*r* 69.2%

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

       2. associate-*r/ 71.6%

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

       3. *-commutative 71.6%

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

   11. Applied egg-rr 71.6%

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

   if -0.69999999999999996 < x < 0.48999999999999999

   1. Initial program 83.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.6%

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

   if 0.48999999999999999 < x

   1. Initial program 69.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. exp-prod 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in x around inf 55.4%

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

      1. mul-1-neg 55.4%

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

      2. unsub-neg 55.4%

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

   6. Simplified 55.4%

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

      1. frac-2neg 55.4%

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

      2. div-inv 55.4%

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

      3. sub-neg 55.4%

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

      4. distribute-neg-in 55.4%

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

      5. metadata-eval 55.4%

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

      6. add-sqr-sqrt 27.1%

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

      7. sqrt-unprod 57.3%

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

      8. sqr-neg 57.3%

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

      9. sqrt-unprod 27.8%

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

      10. add-sqr-sqrt 54.3%

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

      11. add-sqr-sqrt 26.4%

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

      12. sqrt-unprod 54.6%

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

      13. sqr-neg 54.6%

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

      14. sqrt-unprod 28.4%

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

      15. add-sqr-sqrt 55.4%

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

   8. Applied egg-rr 55.4%

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

      1. *-commutative 55.4%

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

      2. distribute-rgt-in 55.4%

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

      3. div-inv 55.4%

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

      4. metadata-eval 55.4%

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

      5. frac-2neg 55.4%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 55.1%

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

      8. frac-times 55.1%

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

      9. sqr-neg 55.1%

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

      10. frac-times 55.1%

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

      11. sqrt-unprod 54.3%

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

      12. add-sqr-sqrt 54.3%

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

      13. div-inv 54.3%

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

      14. frac-add 27.7%

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

   10. Applied egg-rr 55.4%

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

   11. Taylor expanded in y around 0 77.8%

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

       1. *-commutative 77.8%

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

   13. Simplified 77.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;\left(\frac{1}{x} + \frac{y \cdot \left(y \cdot 0.5\right)}{x}\right) - \frac{y}{x}\\ \mathbf{elif}\;x \leq 0.49:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]

Alternative 5: 81.0% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+158} \lor \neg \left(x \leq 0.495\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.15e+158) (not (<= x 0.495)))
   (/ 1.0 (+ x (* x y)))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.15e+158) || !(x <= 0.495)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3.15d+158)) .or. (.not. (x <= 0.495d0))) then
        tmp = 1.0d0 / (x + (x * y))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.15e+158) || !(x <= 0.495)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.15e+158) or not (x <= 0.495):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.15e+158) || !(x <= 0.495))
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.15e+158) || ~((x <= 0.495)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.15e+158], N[Not[LessEqual[x, 0.495]], $MachinePrecision]], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1499999999999998e158 or 0.495 < x

   1. Initial program 67.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. exp-prod 67.4%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in x around inf 55.9%

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

      1. mul-1-neg 55.9%

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

      2. unsub-neg 55.9%

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

   6. Simplified 55.9%

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

      1. frac-2neg 55.9%

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

      2. div-inv 55.9%

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

      3. sub-neg 55.9%

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

      4. distribute-neg-in 55.9%

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

      5. metadata-eval 55.9%

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

      6. add-sqr-sqrt 25.7%

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

      7. sqrt-unprod 60.1%

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

      8. sqr-neg 60.1%

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

      9. sqrt-unprod 29.5%

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

      10. add-sqr-sqrt 54.5%

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

      11. add-sqr-sqrt 25.1%

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

      12. sqrt-unprod 55.1%

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

      13. sqr-neg 55.1%

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

      14. sqrt-unprod 30.2%

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

      15. add-sqr-sqrt 55.9%

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

   8. Applied egg-rr 55.9%

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

      1. *-commutative 55.9%

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

      2. distribute-rgt-in 55.9%

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

      3. div-inv 55.9%

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

      4. metadata-eval 55.9%

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

      5. frac-2neg 55.9%

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

      6. add-sqr-sqrt 19.9%

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

      7. sqrt-unprod 55.7%

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

      8. frac-times 55.4%

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

      9. sqr-neg 55.4%

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

      10. frac-times 55.7%

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

      11. sqrt-unprod 35.2%

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

      12. add-sqr-sqrt 54.5%

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

      13. div-inv 54.5%

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

      14. frac-add 22.8%

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

   10. Applied egg-rr 55.9%

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

   11. Taylor expanded in y around 0 74.6%

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

       1. *-commutative 74.6%

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

   13. Simplified 74.6%

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

   if -3.1499999999999998e158 < x < 0.495

   1. Initial program 86.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. exp-prod 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around 0 89.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+158} \lor \neg \left(x \leq 0.495\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 6: 74.7% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
2. Step-by-step derivation

   1. exp-prod 84.7%

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

3. Simplified 84.7%

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

4. Taylor expanded in x around 0 74.3%

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

5. Final simplification 74.3%

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

Developer target: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{-1}{y}}}{x}\\ t_1 := \frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (/ -1.0 y)) x)) (t_1 (/ (pow (/ x (+ y x)) x) x)))
   (if (< y -3.7311844206647956e+94)
     t_0
     (if (< y 2.817959242728288e+37)
       t_1
       (if (< y 2.347387415166998e+178) (log (exp t_1)) t_0)))))

C

double code(double x, double y) {
	double t_0 = exp((-1.0 / y)) / x;
	double t_1 = pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = log(exp(t_1));
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = exp(((-1.0d0) / y)) / x
    t_1 = ((x / (y + x)) ** x) / x
    if (y < (-3.7311844206647956d+94)) then
        tmp = t_0
    else if (y < 2.817959242728288d+37) then
        tmp = t_1
    else if (y < 2.347387415166998d+178) then
        tmp = log(exp(t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-1.0 / y)) / x;
	double t_1 = Math.pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = Math.log(Math.exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-1.0 / y)) / x
	t_1 = math.pow((x / (y + x)), x) / x
	tmp = 0
	if y < -3.7311844206647956e+94:
		tmp = t_0
	elif y < 2.817959242728288e+37:
		tmp = t_1
	elif y < 2.347387415166998e+178:
		tmp = math.log(math.exp(t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-1.0 / y)) / x)
	t_1 = Float64((Float64(x / Float64(y + x)) ^ x) / x)
	tmp = 0.0
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-1.0 / y)) / x;
	t_1 = ((x / (y + x)) ^ x) / x;
	tmp = 0.0;
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision]}, If[Less[y, -3.7311844206647956e+94], t$95$0, If[Less[y, 2.817959242728288e+37], t$95$1, If[Less[y, 2.347387415166998e+178], N[Log[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1.0 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))