Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A

Percentage Accurate: 95.7% → 99.3%

Time: 10.6s

Alternatives: 11

Speedup: 0.5×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126 - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.0)
     (+
      x
      (/
       y
       (fma
        z
        (fma
         z
         (fma z 0.18806319451591877 0.5641895835477563)
         1.1283791670955126)
        (- 1.1283791670955126 (* x y)))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.0) {
		tmp = x + (y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), (1.1283791670955126 - (x * y))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.0)
		tmp = Float64(x + Float64(y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), Float64(1.1283791670955126 - Float64(x * y)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0], N[(x + N[(y / N[(z * N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] + N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 84.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right) + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. associate--l+ N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right) + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right), \frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}, \frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

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

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z, \frac{5641895835477563}{5000000000000000}\right)}, \frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

      6. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{5641895835477563}{30000000000000000} \cdot z + \frac{5641895835477563}{10000000000000000}}, \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

      7. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{5641895835477563}{30000000000000000}} + \frac{5641895835477563}{10000000000000000}, \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

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

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right)}, \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

      9. --lowering--.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]

      10. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}\right)} \]

      11. *-lowering-*.f64 99.7

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126 - \color{blue}{y \cdot x}\right)} \]

   5. Simplified 99.7%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126 - y \cdot x\right)}} \]

   if 1 < (exp.f64 z)

   1. Initial program 96.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126 - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-223}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_1 -2e-25)
     t_0
     (if (<= t_1 2e-223)
       x
       (if (<= t_1 0.05)
         (+ x (/ y (fma 1.1283791670955126 z 1.1283791670955126)))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_1 <= -2e-25) {
		tmp = t_0;
	} else if (t_1 <= 2e-223) {
		tmp = x;
	} else if (t_1 <= 0.05) {
		tmp = x + (y / fma(1.1283791670955126, z, 1.1283791670955126));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -2e-25)
		tmp = t_0;
	elseif (t_1 <= 2e-223)
		tmp = x;
	elseif (t_1 <= 0.05)
		tmp = Float64(x + Float64(y / fma(1.1283791670955126, z, 1.1283791670955126)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-25], t$95$0, If[LessEqual[t$95$1, 2e-223], x, If[LessEqual[t$95$1, 0.05], N[(x + N[(y / N[(1.1283791670955126 * z + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -2.00000000000000008e-25 or 0.050000000000000003 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 93.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 91.9

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

   5. Simplified 91.9%

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

   if -2.00000000000000008e-25 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1.9999999999999999e-223

   1. Initial program 100.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 90.5%

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

   if 1.9999999999999999e-223 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.050000000000000003

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{y \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}\right)} \]

      10. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{z \cdot \frac{5641895835477563}{5000000000000000}} + \frac{5641895835477563}{5000000000000000}\right)} \]

      11. accelerator-lowering-fma.f64 94.7

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \color{blue}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)}\right)} \]

   5. Simplified 94.7%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z}} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z}} \]

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

         \[\leadsto x + \color{blue}{\frac{y}{\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z}} \]

      3. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}} \]

      4. accelerator-lowering-fma.f64 94.6

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)}} \]

   8. Simplified 94.6%

      \[\leadsto \color{blue}{x + \frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -2 \cdot 10^{-25}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 2 \cdot 10^{-223}:\\ \;\;\;\;x\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 0.05:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-85}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_1 -2e-25)
     t_0
     (if (<= t_1 5e-85)
       x
       (if (<= t_1 0.05) (+ x (* y 0.8862269254527579)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_1 <= -2e-25) {
		tmp = t_0;
	} else if (t_1 <= 5e-85) {
		tmp = x;
	} else if (t_1 <= 0.05) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x + (y / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    if (t_1 <= (-2d-25)) then
        tmp = t_0
    else if (t_1 <= 5d-85) then
        tmp = x
    else if (t_1 <= 0.05d0) then
        tmp = x + (y * 0.8862269254527579d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_1 <= -2e-25) {
		tmp = t_0;
	} else if (t_1 <= 5e-85) {
		tmp = x;
	} else if (t_1 <= 0.05) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	tmp = 0
	if t_1 <= -2e-25:
		tmp = t_0
	elif t_1 <= 5e-85:
		tmp = x
	elif t_1 <= 0.05:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -2e-25)
		tmp = t_0;
	elseif (t_1 <= 5e-85)
		tmp = x;
	elseif (t_1 <= 0.05)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	tmp = 0.0;
	if (t_1 <= -2e-25)
		tmp = t_0;
	elseif (t_1 <= 5e-85)
		tmp = x;
	elseif (t_1 <= 0.05)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-25], t$95$0, If[LessEqual[t$95$1, 5e-85], x, If[LessEqual[t$95$1, 0.05], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -2.00000000000000008e-25 or 0.050000000000000003 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 93.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 91.9

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

   5. Simplified 91.9%

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

   if -2.00000000000000008e-25 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 5.0000000000000002e-85

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 87.8%

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

   if 5.0000000000000002e-85 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.050000000000000003

   1. Initial program 99.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{y \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}\right)} \]

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

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{5641895835477563}{10000000000000000}} + \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      13. accelerator-lowering-fma.f64 94.3

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right)}, 1.1283791670955126\right)\right)} \]

   5. Simplified 94.3%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right)\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\frac{5641895835477563}{5000000000000000}}\right)} \]
   7. Step-by-step derivation

   8. Simplified 88.8%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \color{blue}{1.1283791670955126}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 89.1

          \[\leadsto x + \color{blue}{0.8862269254527579 \cdot y} \]

   11. Simplified 89.1%

       \[\leadsto x + \color{blue}{0.8862269254527579 \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -2 \cdot 10^{-25}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 5 \cdot 10^{-85}:\\ \;\;\;\;x\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 0.05:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.0)
     (+
      x
      (/
       y
       (fma
        y
        (- x)
        (fma
         z
         (fma z 0.5641895835477563 1.1283791670955126)
         1.1283791670955126))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.0) {
		tmp = x + (y / fma(y, -x, fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.0)
		tmp = Float64(x + Float64(y / fma(y, Float64(-x), fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0], N[(x + N[(y / N[(y * (-x) + N[(z * N[(z * 0.5641895835477563 + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 84.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{y \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}\right)} \]

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

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{5641895835477563}{10000000000000000}} + \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      13. accelerator-lowering-fma.f64 99.6

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right)}, 1.1283791670955126\right)\right)} \]

   5. Simplified 99.6%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right)\right)}} \]

   if 1 < (exp.f64 z)

   1. Initial program 96.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 5: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.0)
     (+ x (/ y (fma y (- x) (fma z 1.1283791670955126 1.1283791670955126))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.0) {
		tmp = x + (y / fma(y, -x, fma(z, 1.1283791670955126, 1.1283791670955126)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.0)
		tmp = Float64(x + Float64(y / fma(y, Float64(-x), fma(z, 1.1283791670955126, 1.1283791670955126))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0], N[(x + N[(y / N[(y * (-x) + N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 84.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{y \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}\right)} \]

      10. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{z \cdot \frac{5641895835477563}{5000000000000000}} + \frac{5641895835477563}{5000000000000000}\right)} \]

      11. accelerator-lowering-fma.f64 99.4

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \color{blue}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)}\right)} \]

   5. Simplified 99.4%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)\right)}} \]

   if 1 < (exp.f64 z)

   1. Initial program 96.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 6: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = x + (y / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	else:
		tmp = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	else
		tmp = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 84.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.0) (- x (/ y (fma x y -1.1283791670955126))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.0) {
		tmp = x - (y / fma(x, y, -1.1283791670955126));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.0)
		tmp = Float64(x - Float64(y / fma(x, y, -1.1283791670955126)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0], N[(x - N[(y / N[(x * y + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 84.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{y \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}\right)} \]

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

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{5641895835477563}{10000000000000000}} + \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      13. accelerator-lowering-fma.f64 99.6

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right)}, 1.1283791670955126\right)\right)} \]

   5. Simplified 99.6%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right)\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\frac{5641895835477563}{5000000000000000}}\right)} \]
   7. Step-by-step derivation

   8. Simplified 99.2%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \color{blue}{1.1283791670955126}\right)} \]
   9. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(y\right)}{\mathsf{neg}\left(\left(y \cdot \left(\mathsf{neg}\left(x\right)\right) + \frac{5641895835477563}{5000000000000000}\right)\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y\right)}{y \cdot \left(\mathsf{neg}\left(x\right)\right) + \frac{5641895835477563}{5000000000000000}}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\mathsf{neg}\left(y\right)}{y \cdot \left(\mathsf{neg}\left(x\right)\right) + \frac{5641895835477563}{5000000000000000}}} \]

      4. distribute-frac-neg N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\frac{y}{y \cdot \left(\mathsf{neg}\left(x\right)\right) + \frac{5641895835477563}{5000000000000000}}\right)\right)} \]

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

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{y}{y \cdot \left(\mathsf{neg}\left(x\right)\right) + \frac{5641895835477563}{5000000000000000}}\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto x - \color{blue}{\frac{y}{\mathsf{neg}\left(\left(y \cdot \left(\mathsf{neg}\left(x\right)\right) + \frac{5641895835477563}{5000000000000000}\right)\right)}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y}{\mathsf{neg}\left(\left(y \cdot \left(\mathsf{neg}\left(x\right)\right) + \frac{5641895835477563}{5000000000000000}\right)\right)}} \]

      8. distribute-neg-in N/A

         \[\leadsto x - \frac{y}{\color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(\frac{5641895835477563}{5000000000000000}\right)\right)}} \]

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

         \[\leadsto x - \frac{y}{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{5641895835477563}{5000000000000000}\right)\right)} \]

      10. remove-double-neg N/A

          \[\leadsto x - \frac{y}{\color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{5641895835477563}{5000000000000000}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto x - \frac{y}{\color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{5641895835477563}{5000000000000000}\right)\right)} \]

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

          \[\leadsto x - \frac{y}{\color{blue}{\mathsf{fma}\left(x, y, \mathsf{neg}\left(\frac{5641895835477563}{5000000000000000}\right)\right)}} \]

      13. metadata-eval 99.2

          \[\leadsto x - \frac{y}{\mathsf{fma}\left(x, y, \color{blue}{-1.1283791670955126}\right)} \]

   10. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}} \]

   if 1 < (exp.f64 z)

   1. Initial program 96.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 8: 74.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+207}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-31}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7e+207)
   (/ -1.0 x)
   (if (<= z -1.32e-18)
     x
     (if (<= z 2.6e-31) (+ x (* y 0.8862269254527579)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e+207) {
		tmp = -1.0 / x;
	} else if (z <= -1.32e-18) {
		tmp = x;
	} else if (z <= 2.6e-31) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-7d+207)) then
        tmp = (-1.0d0) / x
    else if (z <= (-1.32d-18)) then
        tmp = x
    else if (z <= 2.6d-31) then
        tmp = x + (y * 0.8862269254527579d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e+207) {
		tmp = -1.0 / x;
	} else if (z <= -1.32e-18) {
		tmp = x;
	} else if (z <= 2.6e-31) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7e+207:
		tmp = -1.0 / x
	elif z <= -1.32e-18:
		tmp = x
	elif z <= 2.6e-31:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7e+207)
		tmp = Float64(-1.0 / x);
	elseif (z <= -1.32e-18)
		tmp = x;
	elseif (z <= 2.6e-31)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7e+207)
		tmp = -1.0 / x;
	elseif (z <= -1.32e-18)
		tmp = x;
	elseif (z <= 2.6e-31)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7e+207], N[(-1.0 / x), $MachinePrecision], If[LessEqual[z, -1.32e-18], x, If[LessEqual[z, 2.6e-31], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.00000000000000056e207

   1. Initial program 72.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 73.9

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

   5. Simplified 73.9%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 64.2

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

   8. Simplified 64.2%

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

   if -7.00000000000000056e207 < z < -1.3199999999999999e-18 or 2.59999999999999995e-31 < z

   1. Initial program 95.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 90.1%

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

   if -1.3199999999999999e-18 < z < 2.59999999999999995e-31

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{y \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}\right)} \]

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

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{5641895835477563}{10000000000000000}} + \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      13. accelerator-lowering-fma.f64 99.8

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right)}, 1.1283791670955126\right)\right)} \]

   5. Simplified 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right)\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\frac{5641895835477563}{5000000000000000}}\right)} \]
   7. Step-by-step derivation

   8. Simplified 99.8%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \color{blue}{1.1283791670955126}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 76.4

          \[\leadsto x + \color{blue}{0.8862269254527579 \cdot y} \]

   11. Simplified 76.4%

       \[\leadsto x + \color{blue}{0.8862269254527579 \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+207}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-31}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-31}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.55e-18) x (if (<= z 4e-31) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.55e-18) {
		tmp = x;
	} else if (z <= 4e-31) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.55d-18)) then
        tmp = x
    else if (z <= 4d-31) then
        tmp = x + (y * 0.8862269254527579d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.55e-18) {
		tmp = x;
	} else if (z <= 4e-31) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.55e-18:
		tmp = x
	elif z <= 4e-31:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.55e-18)
		tmp = x;
	elseif (z <= 4e-31)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.55e-18)
		tmp = x;
	elseif (z <= 4e-31)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.55e-18], x, If[LessEqual[z, 4e-31], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55000000000000003e-18 or 4e-31 < z

   1. Initial program 91.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 81.8%

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

   if -1.55000000000000003e-18 < z < 4e-31

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{y \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}\right)} \]

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

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{5641895835477563}{10000000000000000}} + \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      13. accelerator-lowering-fma.f64 99.8

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right)}, 1.1283791670955126\right)\right)} \]

   5. Simplified 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right)\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\frac{5641895835477563}{5000000000000000}}\right)} \]
   7. Step-by-step derivation

   8. Simplified 99.8%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \color{blue}{1.1283791670955126}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 76.4

          \[\leadsto x + \color{blue}{0.8862269254527579 \cdot y} \]

   11. Simplified 76.4%

       \[\leadsto x + \color{blue}{0.8862269254527579 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-31}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.6e-19) x (if (<= z 3.4e-32) (fma 0.8862269254527579 y x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.6e-19) {
		tmp = x;
	} else if (z <= 3.4e-32) {
		tmp = fma(0.8862269254527579, y, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.6e-19)
		tmp = x;
	elseif (z <= 3.4e-32)
		tmp = fma(0.8862269254527579, y, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.6e-19], x, If[LessEqual[z, 3.4e-32], N[(0.8862269254527579 * y + x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -3.6000000000000001e-19 or 3.39999999999999978e-32 < z

   1. Initial program 91.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 81.8%

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

   if -3.6000000000000001e-19 < z < 3.39999999999999978e-32

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{y \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}\right)} \]

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

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{5641895835477563}{10000000000000000}} + \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      13. accelerator-lowering-fma.f64 99.8

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right)}, 1.1283791670955126\right)\right)} \]

   5. Simplified 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right)\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\frac{5641895835477563}{5000000000000000}}\right)} \]
   7. Step-by-step derivation

   8. Simplified 99.8%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \color{blue}{1.1283791670955126}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{5000000000000000}{5641895835477563} \cdot y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y + x} \]

       2. accelerator-lowering-fma.f64 76.3

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.8862269254527579, y, x\right)} \]

   11. Simplified 76.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.8862269254527579, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 68.8% accurate, 128.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 95.3%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Simplified 73.6%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x))))

C

double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / (((1.1283791670955126d0 / y) * exp(z)) - x))
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * Math.exp(z)) - x));
}

Python

def code(x, y, z):
	return x + (1.0 / (((1.1283791670955126 / y) * math.exp(z)) - x))

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / Float64(Float64(Float64(1.1283791670955126 / y) * exp(z)) - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(1.0 / N[(N[(N[(1.1283791670955126 / y), $MachinePrecision] * N[Exp[z], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024205 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ 1 (- (* (/ 5641895835477563/5000000000000000 y) (exp z)) x))))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))