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

Percentage Accurate: 77.1% → 98.8%

Time: 11.9s

Alternatives: 11

Speedup: 7.7×

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: 77.1% 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: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e+22)
   (/ (exp (- y)) x)
   (if (<= x 5e-7) (/ 1.0 x) (/ 1.0 (* x (exp y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e+22) {
		tmp = exp(-y) / x;
	} else if (x <= 5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * exp(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 <= (-3.5d+22)) then
        tmp = exp(-y) / x
    else if (x <= 5d-7) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * exp(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.5e+22) {
		tmp = Math.exp(-y) / x;
	} else if (x <= 5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * Math.exp(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.5e+22:
		tmp = math.exp(-y) / x
	elif x <= 5e-7:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * math.exp(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.5e+22)
		tmp = Float64(exp(Float64(-y)) / x);
	elseif (x <= 5e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * exp(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e+22)
		tmp = exp(-y) / x;
	elseif (x <= 5e-7)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * exp(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e+22], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5e-7], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e22

   1. Initial program 69.8%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   if -3.5e22 < x < 4.99999999999999977e-7

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.9%

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

   if 4.99999999999999977e-7 < x

   1. Initial program 65.9%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y\right)} \cdot \frac{1}{x}} \]

      2. exp-neg N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -3.5e+22) t_0 (if (<= x 2.1e-6) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -3.5e+22) {
		tmp = t_0;
	} else if (x <= 2.1e-6) {
		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 <= (-3.5d+22)) then
        tmp = t_0
    else if (x <= 2.1d-6) 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 <= -3.5e+22) {
		tmp = t_0;
	} else if (x <= 2.1e-6) {
		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 <= -3.5e+22:
		tmp = t_0
	elif x <= 2.1e-6:
		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 <= -3.5e+22)
		tmp = t_0;
	elseif (x <= 2.1e-6)
		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 <= -3.5e+22)
		tmp = t_0;
	elseif (x <= 2.1e-6)
		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, -3.5e+22], t$95$0, If[LessEqual[x, 2.1e-6], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e22 or 2.0999999999999998e-6 < x

   1. Initial program 67.8%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   if -3.5e22 < x < 2.0999999999999998e-6

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.9%

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

Alternative 3: 86.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, 0.5, 0.5\right) - x, x\right)}{x}}{x}\\ t_1 := \frac{1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+228}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+236}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ (fma y (- (* y (fma x 0.5 0.5)) x) x) x) x))
        (t_1
         (/
          1.0
          (* x (fma y (fma y (fma y 0.16666666666666666 0.5) 1.0) 1.0)))))
   (if (<= x -1.2e+228)
     t_1
     (if (<= x -3.5e+22)
       t_0
       (if (<= x 2.1e-6) (/ 1.0 x) (if (<= x 1.55e+236) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = (fma(y, ((y * fma(x, 0.5, 0.5)) - x), x) / x) / x;
	double t_1 = 1.0 / (x * fma(y, fma(y, fma(y, 0.16666666666666666, 0.5), 1.0), 1.0));
	double tmp;
	if (x <= -1.2e+228) {
		tmp = t_1;
	} else if (x <= -3.5e+22) {
		tmp = t_0;
	} else if (x <= 2.1e-6) {
		tmp = 1.0 / x;
	} else if (x <= 1.55e+236) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(y, Float64(Float64(y * fma(x, 0.5, 0.5)) - x), x) / x) / x)
	t_1 = Float64(1.0 / Float64(x * fma(y, fma(y, fma(y, 0.16666666666666666, 0.5), 1.0), 1.0)))
	tmp = 0.0
	if (x <= -1.2e+228)
		tmp = t_1;
	elseif (x <= -3.5e+22)
		tmp = t_0;
	elseif (x <= 2.1e-6)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.55e+236)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * N[(N[(y * N[(x * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(x * N[(y * N[(y * N[(y * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e+228], t$95$1, If[LessEqual[x, -3.5e+22], t$95$0, If[LessEqual[x, 2.1e-6], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.55e+236], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999994e228 or 2.0999999999999998e-6 < x < 1.55e236

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y\right)} \cdot \frac{1}{x}} \]

      2. exp-neg N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{1}{\color{blue}{\left(1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)} \cdot x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right) + 1\right)} \cdot x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(y, 1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right), 1\right)} \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right) + 1}, 1\right) \cdot x} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} + \frac{1}{6} \cdot y, 1\right)}, 1\right) \cdot x} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot y + \frac{1}{2}}, 1\right), 1\right) \cdot x} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot x} \]

      7. accelerator-lowering-fma.f64 81.6

         \[\leadsto \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot x} \]

   10. Simplified 81.6%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot x} \]

   if -1.19999999999999994e228 < x < -3.5e22 or 1.55e236 < x

   1. Initial program 66.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{x} \]

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \frac{1}{2} + \frac{\color{blue}{\frac{1}{2}}}{x}, -1\right), 1\right)}{x} \]

      9. /-lowering-/.f64 66.5

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

   5. Simplified 66.5%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

   8. Simplified 78.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, 0.5, 0.5\right) - x, x\right)}{x}}}{x} \]

   if -3.5e22 < x < 2.0999999999999998e-6

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+228}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, 0.5, 0.5\right) - x, x\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+236}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, 0.5, 0.5\right) - x, x\right)}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+228}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          1.0
          (* x (fma y (fma y (fma y 0.16666666666666666 0.5) 1.0) 1.0)))))
   (if (<= x -1.7e+228)
     t_0
     (if (<= x -3.5e+22)
       (/ (fma y (fma y (fma y -0.16666666666666666 0.5) -1.0) 1.0) x)
       (if (<= x 2.1e-6) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x * fma(y, fma(y, fma(y, 0.16666666666666666, 0.5), 1.0), 1.0));
	double tmp;
	if (x <= -1.7e+228) {
		tmp = t_0;
	} else if (x <= -3.5e+22) {
		tmp = fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x;
	} else if (x <= 2.1e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x * fma(y, fma(y, fma(y, 0.16666666666666666, 0.5), 1.0), 1.0)))
	tmp = 0.0
	if (x <= -1.7e+228)
		tmp = t_0;
	elseif (x <= -3.5e+22)
		tmp = Float64(fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x);
	elseif (x <= 2.1e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(y * N[(y * N[(y * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e+228], t$95$0, If[LessEqual[x, -3.5e+22], N[(N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.1e-6], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6999999999999999e228 or 2.0999999999999998e-6 < x

   1. Initial program 62.4%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y\right)} \cdot \frac{1}{x}} \]

      2. exp-neg N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{1}{\color{blue}{\left(1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)} \cdot x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right) + 1\right)} \cdot x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(y, 1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right), 1\right)} \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right) + 1}, 1\right) \cdot x} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} + \frac{1}{6} \cdot y, 1\right)}, 1\right) \cdot x} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot y + \frac{1}{2}}, 1\right), 1\right) \cdot x} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot x} \]

      7. accelerator-lowering-fma.f64 75.7

         \[\leadsto \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot x} \]

   10. Simplified 75.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot x} \]

   if -1.6999999999999999e228 < x < -3.5e22

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right) + 1}}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1, 1\right)}}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \color{blue}{-1}, 1\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} + \frac{-1}{6} \cdot y, -1\right)}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{-1}{6} \cdot y + \frac{1}{2}}, -1\right), 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{x} \]

      8. accelerator-lowering-fma.f64 78.2

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{x} \]

   8. Simplified 78.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{x} \]

   if -3.5e22 < x < 2.0999999999999998e-6

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+228}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot 0.5, y\right), x\right)}\\ \mathbf{if}\;x \leq -4 \cdot 10^{+226}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma x (fma y (* y 0.5) y) x))))
   (if (<= x -4e+226)
     t_0
     (if (<= x -3.5e+22)
       (/ (fma y (fma y (fma y -0.16666666666666666 0.5) -1.0) 1.0) x)
       (if (<= x 2.1e-6) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / fma(x, fma(y, (y * 0.5), y), x);
	double tmp;
	if (x <= -4e+226) {
		tmp = t_0;
	} else if (x <= -3.5e+22) {
		tmp = fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x;
	} else if (x <= 2.1e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / fma(x, fma(y, Float64(y * 0.5), y), x))
	tmp = 0.0
	if (x <= -4e+226)
		tmp = t_0;
	elseif (x <= -3.5e+22)
		tmp = Float64(fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x);
	elseif (x <= 2.1e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(y * N[(y * 0.5), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+226], t$95$0, If[LessEqual[x, -3.5e+22], N[(N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.1e-6], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999985e226 or 2.0999999999999998e-6 < x

   1. Initial program 62.4%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y\right)} \cdot \frac{1}{x}} \]

      2. exp-neg N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-out N/A

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

      9. *-rgt-identity N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, y \cdot \left(1 + \frac{1}{2} \cdot y\right), x\right)}} \]

      15. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, y \cdot \color{blue}{\left(\frac{1}{2} \cdot y + 1\right)}, x\right)} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{y \cdot \left(\frac{1}{2} \cdot y\right) + y \cdot 1}, x\right)} \]

      17. *-rgt-identity N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, y \cdot \left(\frac{1}{2} \cdot y\right) + \color{blue}{y}, x\right)} \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} \cdot y, y\right)}, x\right)} \]

      19. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{2}}, y\right), x\right)} \]

      20. *-lowering-*.f64 72.7

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.5}, y\right), x\right)} \]

   10. Simplified 72.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot 0.5, y\right), x\right)}} \]

   if -3.99999999999999985e226 < x < -3.5e22

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right) + 1}}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1, 1\right)}}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \color{blue}{-1}, 1\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} + \frac{-1}{6} \cdot y, -1\right)}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{-1}{6} \cdot y + \frac{1}{2}}, -1\right), 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{x} \]

      8. accelerator-lowering-fma.f64 78.2

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{x} \]

   8. Simplified 78.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{x} \]

   if -3.5e22 < x < 2.0999999999999998e-6

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.9%

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

Alternative 6: 85.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot 0.5, y\right), x\right)}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+228}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot -0.16666666666666666, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma x (fma y (* y 0.5) y) x))))
   (if (<= x -1.1e+228)
     t_0
     (if (<= x -3.5e+22)
       (/ (fma y (fma y (* y -0.16666666666666666) -1.0) 1.0) x)
       (if (<= x 2.1e-6) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / fma(x, fma(y, (y * 0.5), y), x);
	double tmp;
	if (x <= -1.1e+228) {
		tmp = t_0;
	} else if (x <= -3.5e+22) {
		tmp = fma(y, fma(y, (y * -0.16666666666666666), -1.0), 1.0) / x;
	} else if (x <= 2.1e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / fma(x, fma(y, Float64(y * 0.5), y), x))
	tmp = 0.0
	if (x <= -1.1e+228)
		tmp = t_0;
	elseif (x <= -3.5e+22)
		tmp = Float64(fma(y, fma(y, Float64(y * -0.16666666666666666), -1.0), 1.0) / x);
	elseif (x <= 2.1e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(y * N[(y * 0.5), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.1e+228], t$95$0, If[LessEqual[x, -3.5e+22], N[(N[(y * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.1e-6], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e228 or 2.0999999999999998e-6 < x

   1. Initial program 62.4%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y\right)} \cdot \frac{1}{x}} \]

      2. exp-neg N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-out N/A

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

      9. *-rgt-identity N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, y \cdot \left(1 + \frac{1}{2} \cdot y\right), x\right)}} \]

      15. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, y \cdot \color{blue}{\left(\frac{1}{2} \cdot y + 1\right)}, x\right)} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{y \cdot \left(\frac{1}{2} \cdot y\right) + y \cdot 1}, x\right)} \]

      17. *-rgt-identity N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, y \cdot \left(\frac{1}{2} \cdot y\right) + \color{blue}{y}, x\right)} \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} \cdot y, y\right)}, x\right)} \]

      19. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{2}}, y\right), x\right)} \]

      20. *-lowering-*.f64 72.7

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.5}, y\right), x\right)} \]

   10. Simplified 72.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot 0.5, y\right), x\right)}} \]

   if -1.1e228 < x < -3.5e22

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right) + 1}}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1, 1\right)}}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \color{blue}{-1}, 1\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} + \frac{-1}{6} \cdot y, -1\right)}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{-1}{6} \cdot y + \frac{1}{2}}, -1\right), 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{x} \]

      8. accelerator-lowering-fma.f64 78.2

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{x} \]

   8. Simplified 78.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{x} \]

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{6}}, -1\right), 1\right)}{x} \]

       2. *-lowering-*.f64 77.8

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot -0.16666666666666666}, -1\right), 1\right)}{x} \]

   11. Simplified 77.8%

       \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot -0.16666666666666666}, -1\right), 1\right)}{x} \]

   if -3.5e22 < x < 2.0999999999999998e-6

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.9%

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

Alternative 7: 85.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot 0.5, y\right), x\right)}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+227}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot y\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma x (fma y (* y 0.5) y) x))))
   (if (<= x -5e+227)
     t_0
     (if (<= x -3.5e+22)
       (/ (fma y (* -0.16666666666666666 (* y y)) 1.0) x)
       (if (<= x 2.1e-6) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / fma(x, fma(y, (y * 0.5), y), x);
	double tmp;
	if (x <= -5e+227) {
		tmp = t_0;
	} else if (x <= -3.5e+22) {
		tmp = fma(y, (-0.16666666666666666 * (y * y)), 1.0) / x;
	} else if (x <= 2.1e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / fma(x, fma(y, Float64(y * 0.5), y), x))
	tmp = 0.0
	if (x <= -5e+227)
		tmp = t_0;
	elseif (x <= -3.5e+22)
		tmp = Float64(fma(y, Float64(-0.16666666666666666 * Float64(y * y)), 1.0) / x);
	elseif (x <= 2.1e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(y * N[(y * 0.5), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+227], t$95$0, If[LessEqual[x, -3.5e+22], N[(N[(y * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.1e-6], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999996e227 or 2.0999999999999998e-6 < x

   1. Initial program 62.4%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y\right)} \cdot \frac{1}{x}} \]

      2. exp-neg N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-out N/A

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

      9. *-rgt-identity N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, y \cdot \left(1 + \frac{1}{2} \cdot y\right), x\right)}} \]

      15. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, y \cdot \color{blue}{\left(\frac{1}{2} \cdot y + 1\right)}, x\right)} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{y \cdot \left(\frac{1}{2} \cdot y\right) + y \cdot 1}, x\right)} \]

      17. *-rgt-identity N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, y \cdot \left(\frac{1}{2} \cdot y\right) + \color{blue}{y}, x\right)} \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} \cdot y, y\right)}, x\right)} \]

      19. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{2}}, y\right), x\right)} \]

      20. *-lowering-*.f64 72.7

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.5}, y\right), x\right)} \]

   10. Simplified 72.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot 0.5, y\right), x\right)}} \]

   if -4.9999999999999996e227 < x < -3.5e22

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right) + 1}}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1, 1\right)}}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \color{blue}{-1}, 1\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} + \frac{-1}{6} \cdot y, -1\right)}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{-1}{6} \cdot y + \frac{1}{2}}, -1\right), 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{x} \]

      8. accelerator-lowering-fma.f64 78.2

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{x} \]

   8. Simplified 78.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{x} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\frac{-1}{6} \cdot {y}^{2}}, 1\right)}{x} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\frac{-1}{6} \cdot {y}^{2}}, 1\right)}{x} \]

       2. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{-1}{6} \cdot \color{blue}{\left(y \cdot y\right)}, 1\right)}{x} \]

       3. *-lowering-*.f64 77.5

          \[\leadsto \frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}, 1\right)}{x} \]

   11. Simplified 77.5%

       \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{-0.16666666666666666 \cdot \left(y \cdot y\right)}, 1\right)}{x} \]

   if -3.5e22 < x < 2.0999999999999998e-6

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.9%

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

Alternative 8: 83.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(x, y, x\right)}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+245}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot y\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma x y x))))
   (if (<= x -3.6e+245)
     t_0
     (if (<= x -3.5e+22)
       (/ (fma y (* -0.16666666666666666 (* y y)) 1.0) x)
       (if (<= x 2.1e-6) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / fma(x, y, x);
	double tmp;
	if (x <= -3.6e+245) {
		tmp = t_0;
	} else if (x <= -3.5e+22) {
		tmp = fma(y, (-0.16666666666666666 * (y * y)), 1.0) / x;
	} else if (x <= 2.1e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / fma(x, y, x))
	tmp = 0.0
	if (x <= -3.6e+245)
		tmp = t_0;
	elseif (x <= -3.5e+22)
		tmp = Float64(fma(y, Float64(-0.16666666666666666 * Float64(y * y)), 1.0) / x);
	elseif (x <= 2.1e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.6e+245], t$95$0, If[LessEqual[x, -3.5e+22], N[(N[(y * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.1e-6], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6000000000000002e245 or 2.0999999999999998e-6 < x

   1. Initial program 64.1%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y\right)} \cdot \frac{1}{x}} \]

      2. exp-neg N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 68.8

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

   10. Simplified 68.8%

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

   if -3.6000000000000002e245 < x < -3.5e22

   1. Initial program 73.6%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right) + 1}}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1, 1\right)}}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \color{blue}{-1}, 1\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} + \frac{-1}{6} \cdot y, -1\right)}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{-1}{6} \cdot y + \frac{1}{2}}, -1\right), 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{x} \]

      8. accelerator-lowering-fma.f64 75.6

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{x} \]

   8. Simplified 75.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{x} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\frac{-1}{6} \cdot {y}^{2}}, 1\right)}{x} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\frac{-1}{6} \cdot {y}^{2}}, 1\right)}{x} \]

       2. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{-1}{6} \cdot \color{blue}{\left(y \cdot y\right)}, 1\right)}{x} \]

       3. *-lowering-*.f64 75.0

          \[\leadsto \frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}, 1\right)}{x} \]

   11. Simplified 75.0%

       \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{-0.16666666666666666 \cdot \left(y \cdot y\right)}, 1\right)}{x} \]

   if -3.5e22 < x < 2.0999999999999998e-6

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.9%

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

Alternative 9: 82.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e+22)
   (/ (fma y (fma y 0.5 -1.0) 1.0) x)
   (if (<= x 2.1e-6) (/ 1.0 x) (/ 1.0 (fma x y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e+22) {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	} else if (x <= 2.1e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(x, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.5e+22)
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x);
	elseif (x <= 2.1e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(x, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e+22], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.1e-6], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e22

   1. Initial program 69.8%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-sub N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{2} \cdot y - 1}{x}} + \frac{1}{x} \]

      3. associate-*r/ N/A

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

      4. sub-neg N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)}}{x} + \frac{1}{x} \]

      5. metadata-eval N/A

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

      6. distribute-lft-neg-in N/A

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. distribute-frac-neg N/A

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

      11. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(0 - \frac{y \cdot \left(1 + \frac{-1}{2} \cdot y\right)}{x}\right)} + \frac{1}{x} \]

      12. associate--r- N/A

          \[\leadsto \color{blue}{0 - \left(\frac{y \cdot \left(1 + \frac{-1}{2} \cdot y\right)}{x} - \frac{1}{x}\right)} \]

      13. div-sub N/A

          \[\leadsto 0 - \color{blue}{\frac{y \cdot \left(1 + \frac{-1}{2} \cdot y\right) - 1}{x}} \]

      14. neg-sub0 N/A

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

      15. distribute-neg-frac N/A

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

   8. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}} \]

   if -3.5e22 < x < 2.0999999999999998e-6

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.9%

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

   if 2.0999999999999998e-6 < x

   1. Initial program 65.9%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y\right)} \cdot \frac{1}{x}} \]

      2. exp-neg N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 65.8

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

   10. Simplified 65.8%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, y, x\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 80.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(x, y, x\right)}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+170}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma x y x))))
   (if (<= x -2.8e+170) t_0 (if (<= x 2.1e-6) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 / fma(x, y, x);
	double tmp;
	if (x <= -2.8e+170) {
		tmp = t_0;
	} else if (x <= 2.1e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / fma(x, y, x))
	tmp = 0.0
	if (x <= -2.8e+170)
		tmp = t_0;
	elseif (x <= 2.1e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e+170], t$95$0, If[LessEqual[x, 2.1e-6], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.80000000000000015e170 or 2.0999999999999998e-6 < x

   1. Initial program 61.0%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y\right)} \cdot \frac{1}{x}} \]

      2. exp-neg N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 65.1

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

   10. Simplified 65.1%

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

   if -2.80000000000000015e170 < x < 2.0999999999999998e-6

   1. Initial program 85.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 83.4%

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

Alternative 11: 74.0% accurate, 19.3× 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 74.4%

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

3. Taylor expanded in x around 0

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

5. Simplified 68.2%

   \[\leadsto \frac{\color{blue}{1}}{x} \]
6. Add Preprocessing

Developer Target 1: 77.1% 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 2024199 
(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))