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

Percentage Accurate: 95.7% → 99.7%

Time: 8.8s

Alternatives: 12

Speedup: 0.9×

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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.002:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 1.002)
     (+
      (/
       y
       (-
        (fma
         (fma
          (fma 0.18806319451591877 z 0.5641895835477563)
          z
          1.1283791670955126)
         z
         1.1283791670955126)
        (* y x)))
      x)
     (fma (/ 0.8862269254527579 (exp z)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 1.002) {
		tmp = (y / (fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = fma((0.8862269254527579 / exp(z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 1.002)
		tmp = Float64(Float64(y / Float64(fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = fma(Float64(0.8862269254527579 / exp(z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.002], N[(N[(y / N[(N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.8862269254527579 / N[Exp[z], $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 1.002

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.002 < (exp.f64 z)

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

         \[\leadsto \frac{5000000000000000}{5641895835477563} \cdot \frac{\color{blue}{1 \cdot y}}{e^{z}} + x \]

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.002:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_1 -1000000000.0) t_0 (if (<= t_1 5e-5) (- (- x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-5) {
		tmp = -(-x);
	} 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 = ((-1.0d0) / x) + x
    t_1 = (y / ((1.1283791670955126d0 * exp(z)) - (y * x))) + x
    if (t_1 <= (-1000000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 5d-5) then
        tmp = -(-x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * Math.exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-5) {
		tmp = -(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 / x) + x
	t_1 = (y / ((1.1283791670955126 * math.exp(z)) - (y * x))) + x
	tmp = 0
	if t_1 <= -1000000000.0:
		tmp = t_0
	elif t_1 <= 5e-5:
		tmp = -(-x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = t_0;
	elseif (t_1 <= 5e-5)
		tmp = Float64(-Float64(-x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 / x) + x;
	t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	tmp = 0.0;
	if (t_1 <= -1000000000.0)
		tmp = t_0;
	elseif (t_1 <= 5e-5)
		tmp = -(-x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000.0], t$95$0, If[LessEqual[t$95$1, 5e-5], (-(-x)), t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -1e9 or 5.00000000000000024e-5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 97.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if -1e9 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 5.00000000000000024e-5

   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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lower-neg.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 77.6%

      \[\leadsto -\left(-x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -1000000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (+
      (/
       y
       (-
        (fma
         (fma 0.5641895835477563 z 1.1283791670955126)
         z
         1.1283791670955126)
        (* y x)))
      x)
     (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2.0) {
		tmp = (y / (fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2.0)
		tmp = Float64(Float64(y / Float64(fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(N[(y / N[(N[(N[(0.5641895835477563 * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 2 < (exp.f64 z)

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lower-neg.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

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

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

      7. distribute-lft-neg-out N/A

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

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

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

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

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-in N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

      18. cancel-sign-sub-inv N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{e^{z}}{y}, 1.1283791670955126, -x\right) \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

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

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (+ (/ y (- (fma z 1.1283791670955126 1.1283791670955126) (* y x))) x)
     (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2.0) {
		tmp = (y / (fma(z, 1.1283791670955126, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2.0)
		tmp = Float64(Float64(y / Float64(fma(z, 1.1283791670955126, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(N[(y / N[(N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 2 < (exp.f64 z)

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lower-neg.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0) (+ (/ y (- 1.1283791670955126 (* y x))) x) (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2.0) {
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	} 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 (exp(z) <= 0.0d0) then
        tmp = ((-1.0d0) / x) + x
    else if (exp(z) <= 2.0d0) then
        tmp = (y / (1.1283791670955126d0 - (y * x))) + x
    else
        tmp = -(-x)
    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 = (-1.0 / x) + x;
	} else if (Math.exp(z) <= 2.0) {
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = (-1.0 / x) + x
	elif math.exp(z) <= 2.0:
		tmp = (y / (1.1283791670955126 - (y * x))) + x
	else:
		tmp = -(-x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2.0)
		tmp = Float64(Float64(y / Float64(1.1283791670955126 - Float64(y * x))) + x);
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = (-1.0 / x) + x;
	elseif (exp(z) <= 2.0)
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	else
		tmp = -(-x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(N[(y / N[(1.1283791670955126 - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 98.8%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]

   if 2 < (exp.f64 z)

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lower-neg.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + 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 (exp(z) <= 0.0d0) then
        tmp = ((-1.0d0) / x) + x
    else
        tmp = (y / ((1.1283791670955126d0 * exp(z)) - (y * x))) + x
    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 = (-1.0 / x) + x;
	} else {
		tmp = (y / ((1.1283791670955126 * Math.exp(z)) - (y * x))) + x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 99.4%

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

4. Final simplification 99.5%

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

Alternative 8: 96.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 99.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.9

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

   5. Applied rewrites 97.9%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right)} - x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

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

Alternative 9: 71.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\left(-x\right)\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-200}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-198}:\\ \;\;\;\;\mathsf{fma}\left(-0.8862269254527579, z, 0.8862269254527579\right) \cdot y\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- x))))
   (if (<= x -1.85e-200)
     t_0
     (if (<= x 2e-198)
       (* (fma -0.8862269254527579 z 0.8862269254527579) y)
       (if (<= x 2.3e-82) (/ -1.0 x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -(-x);
	double tmp;
	if (x <= -1.85e-200) {
		tmp = t_0;
	} else if (x <= 2e-198) {
		tmp = fma(-0.8862269254527579, z, 0.8862269254527579) * y;
	} else if (x <= 2.3e-82) {
		tmp = -1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(-x))
	tmp = 0.0
	if (x <= -1.85e-200)
		tmp = t_0;
	elseif (x <= 2e-198)
		tmp = Float64(fma(-0.8862269254527579, z, 0.8862269254527579) * y);
	elseif (x <= 2.3e-82)
		tmp = Float64(-1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-(-x))}, If[LessEqual[x, -1.85e-200], t$95$0, If[LessEqual[x, 2e-198], N[(N[(-0.8862269254527579 * z + 0.8862269254527579), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 2.3e-82], N[(-1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.85000000000000005e-200 or 2.29999999999999997e-82 < x

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lower-neg.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 86.5%

      \[\leadsto -\left(-x\right) \]

   if -1.85000000000000005e-200 < x < 1.9999999999999998e-198

   1. Initial program 90.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}}} \cdot \frac{5000000000000000}{5641895835477563} \]

      4. lower-exp.f64 50.2

         \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 49.3%

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

   if 1.9999999999999998e-198 < x < 2.29999999999999997e-82

   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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lower-neg.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 34.3

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-200}:\\ \;\;\;\;-\left(-x\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-198}:\\ \;\;\;\;\mathsf{fma}\left(-0.8862269254527579, z, 0.8862269254527579\right) \cdot y\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\left(-x\right)\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-200}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-198}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- x))))
   (if (<= x -1.85e-200)
     t_0
     (if (<= x 2e-198)
       (* 0.8862269254527579 y)
       (if (<= x 2.3e-82) (/ -1.0 x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -(-x);
	double tmp;
	if (x <= -1.85e-200) {
		tmp = t_0;
	} else if (x <= 2e-198) {
		tmp = 0.8862269254527579 * y;
	} else if (x <= 2.3e-82) {
		tmp = -1.0 / x;
	} 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) :: tmp
    t_0 = -(-x)
    if (x <= (-1.85d-200)) then
        tmp = t_0
    else if (x <= 2d-198) then
        tmp = 0.8862269254527579d0 * y
    else if (x <= 2.3d-82) 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 z) {
	double t_0 = -(-x);
	double tmp;
	if (x <= -1.85e-200) {
		tmp = t_0;
	} else if (x <= 2e-198) {
		tmp = 0.8862269254527579 * y;
	} else if (x <= 2.3e-82) {
		tmp = -1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(-x)
	tmp = 0
	if x <= -1.85e-200:
		tmp = t_0
	elif x <= 2e-198:
		tmp = 0.8862269254527579 * y
	elif x <= 2.3e-82:
		tmp = -1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(-x))
	tmp = 0.0
	if (x <= -1.85e-200)
		tmp = t_0;
	elseif (x <= 2e-198)
		tmp = Float64(0.8862269254527579 * y);
	elseif (x <= 2.3e-82)
		tmp = Float64(-1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(-x);
	tmp = 0.0;
	if (x <= -1.85e-200)
		tmp = t_0;
	elseif (x <= 2e-198)
		tmp = 0.8862269254527579 * y;
	elseif (x <= 2.3e-82)
		tmp = -1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-(-x))}, If[LessEqual[x, -1.85e-200], t$95$0, If[LessEqual[x, 2e-198], N[(0.8862269254527579 * y), $MachinePrecision], If[LessEqual[x, 2.3e-82], N[(-1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.85000000000000005e-200 or 2.29999999999999997e-82 < x

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lower-neg.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 86.5%

      \[\leadsto -\left(-x\right) \]

   if -1.85000000000000005e-200 < x < 1.9999999999999998e-198

   1. Initial program 90.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}}} \cdot \frac{5000000000000000}{5641895835477563} \]

      4. lower-exp.f64 50.2

         \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 49.2%

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

   if 1.9999999999999998e-198 < x < 2.29999999999999997e-82

   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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lower-neg.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 34.3

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.5%

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

Alternative 11: 71.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\left(-x\right)\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-200}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-198}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- x))))
   (if (<= x -1.85e-200)
     t_0
     (if (<= x 1.55e-198) (* 0.8862269254527579 y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(-x);
	double tmp;
	if (x <= -1.85e-200) {
		tmp = t_0;
	} else if (x <= 1.55e-198) {
		tmp = 0.8862269254527579 * y;
	} 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) :: tmp
    t_0 = -(-x)
    if (x <= (-1.85d-200)) then
        tmp = t_0
    else if (x <= 1.55d-198) then
        tmp = 0.8862269254527579d0 * y
    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);
	double tmp;
	if (x <= -1.85e-200) {
		tmp = t_0;
	} else if (x <= 1.55e-198) {
		tmp = 0.8862269254527579 * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(-x)
	tmp = 0
	if x <= -1.85e-200:
		tmp = t_0
	elif x <= 1.55e-198:
		tmp = 0.8862269254527579 * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(-x))
	tmp = 0.0
	if (x <= -1.85e-200)
		tmp = t_0;
	elseif (x <= 1.55e-198)
		tmp = Float64(0.8862269254527579 * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(-x);
	tmp = 0.0;
	if (x <= -1.85e-200)
		tmp = t_0;
	elseif (x <= 1.55e-198)
		tmp = 0.8862269254527579 * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-(-x))}, If[LessEqual[x, -1.85e-200], t$95$0, If[LessEqual[x, 1.55e-198], N[(0.8862269254527579 * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85000000000000005e-200 or 1.5499999999999999e-198 < x

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lower-neg.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 80.4%

      \[\leadsto -\left(-x\right) \]

   if -1.85000000000000005e-200 < x < 1.5499999999999999e-198

   1. Initial program 90.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}}} \cdot \frac{5000000000000000}{5641895835477563} \]

      4. lower-exp.f64 50.2

         \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 49.2%

      \[\leadsto 0.8862269254527579 \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.5% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ -\left(-x\right) \end{array} \]

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 Float64(-Float64(-x))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := (-(-x))

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. neg-sub0 N/A

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

   4. associate-+l- N/A

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

   5. neg-sub0 N/A

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

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

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

   7. lower-neg.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-in N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 N/A

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

   14. associate-*l/ N/A

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

   15. *-lft-identity N/A

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

   16. lower-/.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 60.1

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in x around inf

   \[\leadsto --1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 72.0%

   \[\leadsto -\left(-x\right) \]
9. 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 2024277 
(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)))))