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

Percentage Accurate: 77.7% → 98.0%

Time: 13.6s

Alternatives: 10

Speedup: 11.0×

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: 77.7% 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: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -1.7e+26) t_0 (if (<= x 3.2e-36) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -1.7e+26) {
		tmp = t_0;
	} else if (x <= 3.2e-36) {
		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 = exp(-y) / x
    if (x <= (-1.7d+26)) then
        tmp = t_0
    else if (x <= 3.2d-36) 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 = Math.exp(-y) / x;
	double tmp;
	if (x <= -1.7e+26) {
		tmp = t_0;
	} else if (x <= 3.2e-36) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp(-y) / x
	tmp = 0
	if x <= -1.7e+26:
		tmp = t_0
	elif x <= 3.2e-36:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-y)) / x)
	tmp = 0.0
	if (x <= -1.7e+26)
		tmp = t_0;
	elseif (x <= 3.2e-36)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(-y) / x;
	tmp = 0.0;
	if (x <= -1.7e+26)
		tmp = t_0;
	elseif (x <= 3.2e-36)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.7e+26], t$95$0, If[LessEqual[x, 3.2e-36], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7000000000000001e26 or 3.20000000000000021e-36 < x

   1. Initial program 74.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if -1.7000000000000001e26 < x < 3.20000000000000021e-36

   1. Initial program 86.4%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + y \cdot \left(-1 + \frac{\left(x \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot y}{x}\right)}{x}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+147}:\\ \;\;\;\;\left(y + 1\right) \cdot \left(x \cdot \left(1 \cdot \frac{1 - y}{x \cdot \left(x \cdot \left(y + 1\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+ 1.0 (* y (+ -1.0 (/ (* (* x (* y -0.16666666666666666)) y) x))))
          x)))
   (if (<= x -2.15e+52)
     t_0
     (if (<= x 4.3e+65)
       (/ 1.0 x)
       (if (<= x 2.8e+147)
         (* (+ y 1.0) (* x (* 1.0 (/ (- 1.0 y) (* x (* x (+ y 1.0)))))))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 + (y * (-1.0 + (((x * (y * -0.16666666666666666)) * y) / x)))) / x;
	double tmp;
	if (x <= -2.15e+52) {
		tmp = t_0;
	} else if (x <= 4.3e+65) {
		tmp = 1.0 / x;
	} else if (x <= 2.8e+147) {
		tmp = (y + 1.0) * (x * (1.0 * ((1.0 - y) / (x * (x * (y + 1.0))))));
	} 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 + (y * ((-1.0d0) + (((x * (y * (-0.16666666666666666d0))) * y) / x)))) / x
    if (x <= (-2.15d+52)) then
        tmp = t_0
    else if (x <= 4.3d+65) then
        tmp = 1.0d0 / x
    else if (x <= 2.8d+147) then
        tmp = (y + 1.0d0) * (x * (1.0d0 * ((1.0d0 - y) / (x * (x * (y + 1.0d0))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 + (y * (-1.0 + (((x * (y * -0.16666666666666666)) * y) / x)))) / x;
	double tmp;
	if (x <= -2.15e+52) {
		tmp = t_0;
	} else if (x <= 4.3e+65) {
		tmp = 1.0 / x;
	} else if (x <= 2.8e+147) {
		tmp = (y + 1.0) * (x * (1.0 * ((1.0 - y) / (x * (x * (y + 1.0))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 + (y * (-1.0 + (((x * (y * -0.16666666666666666)) * y) / x)))) / x
	tmp = 0
	if x <= -2.15e+52:
		tmp = t_0
	elif x <= 4.3e+65:
		tmp = 1.0 / x
	elif x <= 2.8e+147:
		tmp = (y + 1.0) * (x * (1.0 * ((1.0 - y) / (x * (x * (y + 1.0))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(Float64(Float64(x * Float64(y * -0.16666666666666666)) * y) / x)))) / x)
	tmp = 0.0
	if (x <= -2.15e+52)
		tmp = t_0;
	elseif (x <= 4.3e+65)
		tmp = Float64(1.0 / x);
	elseif (x <= 2.8e+147)
		tmp = Float64(Float64(y + 1.0) * Float64(x * Float64(1.0 * Float64(Float64(1.0 - y) / Float64(x * Float64(x * Float64(y + 1.0)))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 + (y * (-1.0 + (((x * (y * -0.16666666666666666)) * y) / x)))) / x;
	tmp = 0.0;
	if (x <= -2.15e+52)
		tmp = t_0;
	elseif (x <= 4.3e+65)
		tmp = 1.0 / x;
	elseif (x <= 2.8e+147)
		tmp = (y + 1.0) * (x * (1.0 * ((1.0 - y) / (x * (x * (y + 1.0))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + N[(y * N[(-1.0 + N[(N[(N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -2.15e+52], t$95$0, If[LessEqual[x, 4.3e+65], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 2.8e+147], N[(N[(y + 1.0), $MachinePrecision] * N[(x * N[(1.0 * N[(N[(1.0 - y), $MachinePrecision] / N[(x * N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.15e52 or 2.8000000000000001e147 < x

   1. Initial program 68.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   11. Taylor expanded in y around inf 0

       \[\leadsto expr\]

   13. Simplified 0

       \[\leadsto expr\]

   15. Applied egg-rr 0

       \[\leadsto expr\]

   if -2.15e52 < x < 4.30000000000000046e65

   1. Initial program 87.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 4.30000000000000046e65 < x < 2.8000000000000001e147

   1. Initial program 79.1%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + y \cdot \left(-1 + \frac{\left(x \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot y}{x}\right)}{x}\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{x \cdot x}{\left(x \cdot x\right) \cdot \left(\left(y + 1\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+ 1.0 (* y (+ -1.0 (/ (* (* x (* y -0.16666666666666666)) y) x))))
          x)))
   (if (<= x -2.05e+52)
     t_0
     (if (<= x 2.8e+69)
       (/ 1.0 x)
       (if (<= x 1.2e+103) (/ (* x x) (* (* x x) (* (+ y 1.0) x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 + (y * (-1.0 + (((x * (y * -0.16666666666666666)) * y) / x)))) / x;
	double tmp;
	if (x <= -2.05e+52) {
		tmp = t_0;
	} else if (x <= 2.8e+69) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+103) {
		tmp = (x * x) / ((x * x) * ((y + 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 + (y * ((-1.0d0) + (((x * (y * (-0.16666666666666666d0))) * y) / x)))) / x
    if (x <= (-2.05d+52)) then
        tmp = t_0
    else if (x <= 2.8d+69) then
        tmp = 1.0d0 / x
    else if (x <= 1.2d+103) then
        tmp = (x * x) / ((x * x) * ((y + 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 + (y * (-1.0 + (((x * (y * -0.16666666666666666)) * y) / x)))) / x;
	double tmp;
	if (x <= -2.05e+52) {
		tmp = t_0;
	} else if (x <= 2.8e+69) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+103) {
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 + (y * (-1.0 + (((x * (y * -0.16666666666666666)) * y) / x)))) / x
	tmp = 0
	if x <= -2.05e+52:
		tmp = t_0
	elif x <= 2.8e+69:
		tmp = 1.0 / x
	elif x <= 1.2e+103:
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(Float64(Float64(x * Float64(y * -0.16666666666666666)) * y) / x)))) / x)
	tmp = 0.0
	if (x <= -2.05e+52)
		tmp = t_0;
	elseif (x <= 2.8e+69)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.2e+103)
		tmp = Float64(Float64(x * x) / Float64(Float64(x * x) * Float64(Float64(y + 1.0) * x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 + (y * (-1.0 + (((x * (y * -0.16666666666666666)) * y) / x)))) / x;
	tmp = 0.0;
	if (x <= -2.05e+52)
		tmp = t_0;
	elseif (x <= 2.8e+69)
		tmp = 1.0 / x;
	elseif (x <= 1.2e+103)
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + N[(y * N[(-1.0 + N[(N[(N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -2.05e+52], t$95$0, If[LessEqual[x, 2.8e+69], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.2e+103], N[(N[(x * x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.05e52 or 1.1999999999999999e103 < x

   1. Initial program 69.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   11. Taylor expanded in y around inf 0

       \[\leadsto expr\]

   13. Simplified 0

       \[\leadsto expr\]

   15. Applied egg-rr 0

       \[\leadsto expr\]

   if -2.05e52 < x < 2.79999999999999982e69

   1. Initial program 87.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 2.79999999999999982e69 < x < 1.1999999999999999e103

   1. Initial program 83.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in y around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + y \cdot -0.16666666666666666\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+52}:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot t\_0\right)}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{x \cdot x}{\left(x \cdot x\right) \cdot \left(\left(y + 1\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - y\right) + t\_0 \cdot \left(y \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* y -0.16666666666666666))))
   (if (<= x -2.05e+52)
     (/ (+ 1.0 (* y (+ -1.0 (* y t_0)))) x)
     (if (<= x 2.8e+69)
       (/ 1.0 x)
       (if (<= x 1.2e+103)
         (/ (* x x) (* (* x x) (* (+ y 1.0) x)))
         (/ (+ (- 1.0 y) (* t_0 (* y y))) x))))))

C

double code(double x, double y) {
	double t_0 = 0.5 + (y * -0.16666666666666666);
	double tmp;
	if (x <= -2.05e+52) {
		tmp = (1.0 + (y * (-1.0 + (y * t_0)))) / x;
	} else if (x <= 2.8e+69) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+103) {
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	} else {
		tmp = ((1.0 - y) + (t_0 * (y * y))) / x;
	}
	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 = 0.5d0 + (y * (-0.16666666666666666d0))
    if (x <= (-2.05d+52)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * t_0)))) / x
    else if (x <= 2.8d+69) then
        tmp = 1.0d0 / x
    else if (x <= 1.2d+103) then
        tmp = (x * x) / ((x * x) * ((y + 1.0d0) * x))
    else
        tmp = ((1.0d0 - y) + (t_0 * (y * y))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 + (y * -0.16666666666666666);
	double tmp;
	if (x <= -2.05e+52) {
		tmp = (1.0 + (y * (-1.0 + (y * t_0)))) / x;
	} else if (x <= 2.8e+69) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+103) {
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	} else {
		tmp = ((1.0 - y) + (t_0 * (y * y))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 + (y * -0.16666666666666666)
	tmp = 0
	if x <= -2.05e+52:
		tmp = (1.0 + (y * (-1.0 + (y * t_0)))) / x
	elif x <= 2.8e+69:
		tmp = 1.0 / x
	elif x <= 1.2e+103:
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x))
	else:
		tmp = ((1.0 - y) + (t_0 * (y * y))) / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 + Float64(y * -0.16666666666666666))
	tmp = 0.0
	if (x <= -2.05e+52)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * t_0)))) / x);
	elseif (x <= 2.8e+69)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.2e+103)
		tmp = Float64(Float64(x * x) / Float64(Float64(x * x) * Float64(Float64(y + 1.0) * x)));
	else
		tmp = Float64(Float64(Float64(1.0 - y) + Float64(t_0 * Float64(y * y))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 + (y * -0.16666666666666666);
	tmp = 0.0;
	if (x <= -2.05e+52)
		tmp = (1.0 + (y * (-1.0 + (y * t_0)))) / x;
	elseif (x <= 2.8e+69)
		tmp = 1.0 / x;
	elseif (x <= 1.2e+103)
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	else
		tmp = ((1.0 - y) + (t_0 * (y * y))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.05e+52], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.8e+69], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.2e+103], N[(N[(x * x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] + N[(t$95$0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.05e52

   1. Initial program 72.1%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   if -2.05e52 < x < 2.79999999999999982e69

   1. Initial program 87.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 2.79999999999999982e69 < x < 1.1999999999999999e103

   1. Initial program 83.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in y around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if 1.1999999999999999e103 < x

   1. Initial program 66.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   12. Applied egg-rr 0

       \[\leadsto expr\]

   14. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + y \cdot \left(-1 + y \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{x \cdot x}{\left(x \cdot x\right) \cdot \left(\left(y + 1\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+ 1.0 (* y (+ -1.0 (* y (+ 0.5 (* y -0.16666666666666666))))))
          x)))
   (if (<= x -2.05e+52)
     t_0
     (if (<= x 2.8e+69)
       (/ 1.0 x)
       (if (<= x 1.2e+103) (/ (* x x) (* (* x x) (* (+ y 1.0) x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 + (y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x;
	double tmp;
	if (x <= -2.05e+52) {
		tmp = t_0;
	} else if (x <= 2.8e+69) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+103) {
		tmp = (x * x) / ((x * x) * ((y + 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 + (y * ((-1.0d0) + (y * (0.5d0 + (y * (-0.16666666666666666d0))))))) / x
    if (x <= (-2.05d+52)) then
        tmp = t_0
    else if (x <= 2.8d+69) then
        tmp = 1.0d0 / x
    else if (x <= 1.2d+103) then
        tmp = (x * x) / ((x * x) * ((y + 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 + (y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x;
	double tmp;
	if (x <= -2.05e+52) {
		tmp = t_0;
	} else if (x <= 2.8e+69) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+103) {
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 + (y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x
	tmp = 0
	if x <= -2.05e+52:
		tmp = t_0
	elif x <= 2.8e+69:
		tmp = 1.0 / x
	elif x <= 1.2e+103:
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666)))))) / x)
	tmp = 0.0
	if (x <= -2.05e+52)
		tmp = t_0;
	elseif (x <= 2.8e+69)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.2e+103)
		tmp = Float64(Float64(x * x) / Float64(Float64(x * x) * Float64(Float64(y + 1.0) * x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 + (y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x;
	tmp = 0.0;
	if (x <= -2.05e+52)
		tmp = t_0;
	elseif (x <= 2.8e+69)
		tmp = 1.0 / x;
	elseif (x <= 1.2e+103)
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -2.05e+52], t$95$0, If[LessEqual[x, 2.8e+69], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.2e+103], N[(N[(x * x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.05e52 or 1.1999999999999999e103 < x

   1. Initial program 69.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   if -2.05e52 < x < 2.79999999999999982e69

   1. Initial program 87.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 2.79999999999999982e69 < x < 1.1999999999999999e103

   1. Initial program 83.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in y around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+52}:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{x \cdot x}{\left(x \cdot x\right) \cdot \left(\left(y + 1\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - y \cdot x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.05e+52)
   (/ (+ 1.0 (* y (+ -1.0 (* y (* y -0.16666666666666666))))) x)
   (if (<= x 2.8e+69)
     (/ 1.0 x)
     (if (<= x 1.2e+103)
       (/ (* x x) (* (* x x) (* (+ y 1.0) x)))
       (/ (/ (- x (* y x)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.05e+52) {
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x;
	} else if (x <= 2.8e+69) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+103) {
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	} else {
		tmp = ((x - (y * x)) / x) / 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 <= (-2.05d+52)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * (y * (-0.16666666666666666d0)))))) / x
    else if (x <= 2.8d+69) then
        tmp = 1.0d0 / x
    else if (x <= 1.2d+103) then
        tmp = (x * x) / ((x * x) * ((y + 1.0d0) * x))
    else
        tmp = ((x - (y * x)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.05e+52) {
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x;
	} else if (x <= 2.8e+69) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+103) {
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	} else {
		tmp = ((x - (y * x)) / x) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.05e+52:
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x
	elif x <= 2.8e+69:
		tmp = 1.0 / x
	elif x <= 1.2e+103:
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x))
	else:
		tmp = ((x - (y * x)) / x) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.05e+52)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * Float64(y * -0.16666666666666666))))) / x);
	elseif (x <= 2.8e+69)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.2e+103)
		tmp = Float64(Float64(x * x) / Float64(Float64(x * x) * Float64(Float64(y + 1.0) * x)));
	else
		tmp = Float64(Float64(Float64(x - Float64(y * x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.05e+52)
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x;
	elseif (x <= 2.8e+69)
		tmp = 1.0 / x;
	elseif (x <= 1.2e+103)
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	else
		tmp = ((x - (y * x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.05e+52], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.8e+69], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.2e+103], N[(N[(x * x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.05e52

   1. Initial program 72.1%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   11. Taylor expanded in y around inf 0

       \[\leadsto expr\]

   13. Simplified 0

       \[\leadsto expr\]

   if -2.05e52 < x < 2.79999999999999982e69

   1. Initial program 87.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 2.79999999999999982e69 < x < 1.1999999999999999e103

   1. Initial program 83.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in y around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if 1.1999999999999999e103 < x

   1. Initial program 66.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x - y \cdot x}{x}}{x}\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{x \cdot x}{\left(x \cdot x\right) \cdot \left(\left(y + 1\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ (- x (* y x)) x) x)))
   (if (<= x -2.05e+52)
     t_0
     (if (<= x 2.8e+69)
       (/ 1.0 x)
       (if (<= x 1.2e+103) (/ (* x x) (* (* x x) (* (+ y 1.0) x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = ((x - (y * x)) / x) / x;
	double tmp;
	if (x <= -2.05e+52) {
		tmp = t_0;
	} else if (x <= 2.8e+69) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+103) {
		tmp = (x * x) / ((x * x) * ((y + 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 = ((x - (y * x)) / x) / x
    if (x <= (-2.05d+52)) then
        tmp = t_0
    else if (x <= 2.8d+69) then
        tmp = 1.0d0 / x
    else if (x <= 1.2d+103) then
        tmp = (x * x) / ((x * x) * ((y + 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 = ((x - (y * x)) / x) / x;
	double tmp;
	if (x <= -2.05e+52) {
		tmp = t_0;
	} else if (x <= 2.8e+69) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+103) {
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - (y * x)) / x) / x
	tmp = 0
	if x <= -2.05e+52:
		tmp = t_0
	elif x <= 2.8e+69:
		tmp = 1.0 / x
	elif x <= 1.2e+103:
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - Float64(y * x)) / x) / x)
	tmp = 0.0
	if (x <= -2.05e+52)
		tmp = t_0;
	elseif (x <= 2.8e+69)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.2e+103)
		tmp = Float64(Float64(x * x) / Float64(Float64(x * x) * Float64(Float64(y + 1.0) * x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - (y * x)) / x) / x;
	tmp = 0.0;
	if (x <= -2.05e+52)
		tmp = t_0;
	elseif (x <= 2.8e+69)
		tmp = 1.0 / x;
	elseif (x <= 1.2e+103)
		tmp = (x * x) / ((x * x) * ((y + 1.0) * x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -2.05e+52], t$95$0, If[LessEqual[x, 2.8e+69], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.2e+103], N[(N[(x * x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.05e52 or 1.1999999999999999e103 < x

   1. Initial program 69.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if -2.05e52 < x < 2.79999999999999982e69

   1. Initial program 87.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 2.79999999999999982e69 < x < 1.1999999999999999e103

   1. Initial program 83.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in y around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.7% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x - y \cdot x}{x}}{x}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ (- x (* y x)) x) x)))
   (if (<= x -3e+52) t_0 (if (<= x 3.8e+103) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - (y * x)) / x) / x;
	double tmp;
	if (x <= -3e+52) {
		tmp = t_0;
	} else if (x <= 3.8e+103) {
		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 = ((x - (y * x)) / x) / x
    if (x <= (-3d+52)) then
        tmp = t_0
    else if (x <= 3.8d+103) 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 = ((x - (y * x)) / x) / x;
	double tmp;
	if (x <= -3e+52) {
		tmp = t_0;
	} else if (x <= 3.8e+103) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - (y * x)) / x) / x
	tmp = 0
	if x <= -3e+52:
		tmp = t_0
	elif x <= 3.8e+103:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - Float64(y * x)) / x) / x)
	tmp = 0.0
	if (x <= -3e+52)
		tmp = t_0;
	elseif (x <= 3.8e+103)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - (y * x)) / x) / x;
	tmp = 0.0;
	if (x <= -3e+52)
		tmp = t_0;
	elseif (x <= 3.8e+103)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -3e+52], t$95$0, If[LessEqual[x, 3.8e+103], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3e52 or 3.7999999999999997e103 < x

   1. Initial program 69.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if -3e52 < x < 3.7999999999999997e103

   1. Initial program 87.0%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.5% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+192}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+16}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8e+192)
   (/ 1.0 x)
   (if (<= y -1.65e+16)
     (/ (* -0.16666666666666666 (* y (* y y))) x)
     (/ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8e+192) {
		tmp = 1.0 / x;
	} else if (y <= -1.65e+16) {
		tmp = (-0.16666666666666666 * (y * (y * 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 (y <= (-8d+192)) then
        tmp = 1.0d0 / x
    else if (y <= (-1.65d+16)) then
        tmp = ((-0.16666666666666666d0) * (y * (y * 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 (y <= -8e+192) {
		tmp = 1.0 / x;
	} else if (y <= -1.65e+16) {
		tmp = (-0.16666666666666666 * (y * (y * y))) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8e+192:
		tmp = 1.0 / x
	elif y <= -1.65e+16:
		tmp = (-0.16666666666666666 * (y * (y * y))) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8e+192)
		tmp = Float64(1.0 / x);
	elseif (y <= -1.65e+16)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(y * Float64(y * y))) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8e+192)
		tmp = 1.0 / x;
	elseif (y <= -1.65e+16)
		tmp = (-0.16666666666666666 * (y * (y * y))) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8e+192], N[(1.0 / x), $MachinePrecision], If[LessEqual[y, -1.65e+16], N[(N[(-0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000033e192 or -1.65e16 < y

   1. Initial program 85.5%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if -8.00000000000000033e192 < y < -1.65e16

   1. Initial program 32.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   11. Taylor expanded in y around inf 0

       \[\leadsto expr\]

   13. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 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 79.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in x around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. Add Preprocessing

Developer Target 1: 77.5% accurate, 0.6× 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 2024111 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -37311844206647956000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (exp (/ -1 y)) x) (if (< y 28179592427282880000000000000000000000) (/ (pow (/ x (+ y x)) x) x) (if (< y 23473874151669980000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x)))))

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