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

Percentage Accurate: 95.1% → 99.5%

Time: 10.0s

Alternatives: 19

Speedup: 2.8×

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: 95.1% 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: 99.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 1.0)
     (+
      (/ 1.0 (/ (fma 1.1283791670955126 z (fma (- y) x 1.1283791670955126)) y))
      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) <= 1.0) {
		tmp = (1.0 / (fma(1.1283791670955126, z, fma(-y, x, 1.1283791670955126)) / y)) + 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) <= 1.0)
		tmp = Float64(Float64(1.0 / Float64(fma(1.1283791670955126, z, fma(Float64(-y), x, 1.1283791670955126)) / y)) + 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], 1.0], N[(N[(1.0 / N[(N[(1.1283791670955126 * z + N[((-y) * x + 1.1283791670955126), $MachinePrecision]), $MachinePrecision] / y), $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 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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

   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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.9

         \[\leadsto x + \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126 - y \cdot x\right)}{y}}} \]

   7. Applied rewrites 99.9%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(1.1283791670955126, z, \mathsf{fma}\left(-y, x, 1.1283791670955126\right)\right)}{y}}} \]

   if 1 < (exp.f64 z)

   1. Initial program 97.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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%

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

Alternative 2: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)} + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -5.0)
     t_0
     (if (<= t_1 5e+24)
       (+ (/ y (fma 1.1283791670955126 z 1.1283791670955126)) 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 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+24) {
		tmp = (y / fma(1.1283791670955126, z, 1.1283791670955126)) + 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 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 5e+24)
		tmp = Float64(Float64(y / fma(1.1283791670955126, z, 1.1283791670955126)) + x);
	else
		tmp = t_0;
	end
	return 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, -5.0], t$95$0, If[LessEqual[t$95$1, 5e+24], N[(N[(y / N[(1.1283791670955126 * z + 1.1283791670955126), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], 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)))) < -5 or 5.00000000000000045e24 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 95.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 92.0

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

   5. Applied rewrites 92.0%

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

   if -5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 5.00000000000000045e24

   1. Initial program 100.0%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -5:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-27}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot z} + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -5.0)
     t_0
     (if (<= t_1 1e-27) (+ (/ y (* 1.1283791670955126 z)) 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 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 1e-27) {
		tmp = (y / (1.1283791670955126 * z)) + 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 <= (-5.0d0)) then
        tmp = t_0
    else if (t_1 <= 1d-27) then
        tmp = (y / (1.1283791670955126d0 * z)) + 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 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 1e-27) {
		tmp = (y / (1.1283791670955126 * z)) + 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 <= -5.0:
		tmp = t_0
	elif t_1 <= 1e-27:
		tmp = (y / (1.1283791670955126 * z)) + 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 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 1e-27)
		tmp = Float64(Float64(y / Float64(1.1283791670955126 * z)) + 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 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 1e-27)
		tmp = (y / (1.1283791670955126 * z)) + 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, -5.0], t$95$0, If[LessEqual[t$95$1, 1e-27], N[(N[(y / N[(1.1283791670955126 * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], 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)))) < -5 or 1e-27 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 96.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 90.2

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

   5. Applied rewrites 90.2%

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

   if -5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1e-27

   1. Initial program 100.0%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 32.7%

      \[\leadsto x + \frac{y}{1.1283791670955126 \cdot \color{blue}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -5:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 10^{-27}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot z} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+140}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_0 2e+140) t_0 (+ (/ -1.0 x) x))))

C

double code(double x, double y, double z) {
	double t_0 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_0 <= 2e+140) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (y / ((1.1283791670955126d0 * exp(z)) - (y * x))) + x
    if (t_0 <= 2d+140) then
        tmp = t_0
    else
        tmp = ((-1.0d0) / x) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / ((1.1283791670955126 * Math.exp(z)) - (y * x))) + x;
	double tmp;
	if (t_0 <= 2e+140) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / x) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / ((1.1283791670955126 * math.exp(z)) - (y * x))) + x
	tmp = 0
	if t_0 <= 2e+140:
		tmp = t_0
	else:
		tmp = (-1.0 / x) + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_0 <= 2e+140)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / x) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	tmp = 0.0;
	if (t_0 <= 2e+140)
		tmp = t_0;
	else
		tmp = (-1.0 / x) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+140], t$95$0, N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]]]

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)))) < 2.00000000000000012e140

   1. Initial program 99.4%

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

   if 2.00000000000000012e140 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 83.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.005:\\ \;\;\;\;\frac{1}{\frac{1.1283791670955126}{y} - x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1.1283791670955126, z, \left(-x\right) \cdot y\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 1.005)
     (+ (/ 1.0 (- (/ 1.1283791670955126 y) x)) x)
     (+ (/ y (fma 1.1283791670955126 z (* (- 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 if (exp(z) <= 1.005) {
		tmp = (1.0 / ((1.1283791670955126 / y) - x)) + x;
	} else {
		tmp = (y / fma(1.1283791670955126, z, (-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);
	elseif (exp(z) <= 1.005)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.1283791670955126 / y) - x)) + x);
	else
		tmp = Float64(Float64(y / fma(1.1283791670955126, z, Float64(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], If[LessEqual[N[Exp[z], $MachinePrecision], 1.005], N[(N[(1.0 / N[(N[(1.1283791670955126 / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(1.1283791670955126 * z + N[((-x) * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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) < 1.0049999999999999

   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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.5

         \[\leadsto x + \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126 - y \cdot x\right)}{y}}} \]

   7. Applied rewrites 99.5%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(1.1283791670955126, z, \mathsf{fma}\left(-y, x, 1.1283791670955126\right)\right)}{y}}} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{5641895835477563}{5000000000000000} - x \cdot y}{y}}} \]
   9. Step-by-step derivation

      1. div-sub N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\frac{5641895835477563}{5000000000000000}}}{y} - x} \]

      10. lower-/.f64 98.5

          \[\leadsto x + \frac{1}{\color{blue}{\frac{1.1283791670955126}{y}} - x} \]

   10. Applied rewrites 98.5%

       \[\leadsto x + \frac{1}{\color{blue}{\frac{1.1283791670955126}{y} - x}} \]

   if 1.0049999999999999 < (exp.f64 z)

   1. Initial program 97.0%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(\frac{5641895835477563}{5000000000000000}, z, -1 \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.005:\\ \;\;\;\;\frac{1}{\frac{1.1283791670955126}{y} - x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1.1283791670955126, z, \left(-x\right) \cdot y\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.005:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-y, x, 1.1283791670955126\right)} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1.1283791670955126, z, \left(-x\right) \cdot y\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 1.005)
     (+ (/ y (fma (- y) x 1.1283791670955126)) x)
     (+ (/ y (fma 1.1283791670955126 z (* (- 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 if (exp(z) <= 1.005) {
		tmp = (y / fma(-y, x, 1.1283791670955126)) + x;
	} else {
		tmp = (y / fma(1.1283791670955126, z, (-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);
	elseif (exp(z) <= 1.005)
		tmp = Float64(Float64(y / fma(Float64(-y), x, 1.1283791670955126)) + x);
	else
		tmp = Float64(Float64(y / fma(1.1283791670955126, z, Float64(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], If[LessEqual[N[Exp[z], $MachinePrecision], 1.005], N[(N[(y / N[((-y) * x + 1.1283791670955126), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(1.1283791670955126 * z + N[((-x) * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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) < 1.0049999999999999

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 98.5%

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

   if 1.0049999999999999 < (exp.f64 z)

   1. Initial program 97.0%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(\frac{5641895835477563}{5000000000000000}, z, -1 \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.005:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-y, x, 1.1283791670955126\right)} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1.1283791670955126, z, \left(-x\right) \cdot y\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+ (/ y (fma (- y) x (* 1.1283791670955126 (exp z)))) 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(-y, x, (1.1283791670955126 * exp(z)))) + 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 / fma(Float64(-y), x, Float64(1.1283791670955126 * exp(z)))) + 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[((-y) * x + N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

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

Alternative 8: 97.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/
     y
     (fma
      (- y)
      x
      (fma
       (fma
        (fma 0.18806319451591877 z 0.5641895835477563)
        z
        1.1283791670955126)
       z
       1.1283791670955126)))
    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(-y, x, fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126))) + 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 / fma(Float64(-y), x, fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126))) + 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[((-y) * x + N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 96.4

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

   7. Applied rewrites 96.4%

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

4. Final simplification 97.2%

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

Alternative 9: 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 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 96.8%

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

Alternative 10: 95.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/
     y
     (-
      (fma (fma 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(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(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[(0.5641895835477563 * 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 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 93.6

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

   5. Applied rewrites 93.6%

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

4. Final simplification 95.2%

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

Alternative 11: 95.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/ y (- (fma (* 0.5641895835477563 z) 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((0.5641895835477563 * z), 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(Float64(0.5641895835477563 * z), 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[(0.5641895835477563 * z), $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 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 93.6

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(\frac{5641895835477563}{10000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.9%

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

4. Final simplification 94.6%

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

Alternative 12: 93.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+ (/ y (fma (- y) x (* (+ 1.0 z) 1.1283791670955126))) 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(-y, x, ((1.0 + z) * 1.1283791670955126))) + 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 / fma(Float64(-y), x, Float64(Float64(1.0 + z) * 1.1283791670955126))) + 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[((-y) * x + N[(N[(1.0 + z), $MachinePrecision] * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 87.2

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(-y, x, \color{blue}{\left(1 + z\right)} \cdot 1.1283791670955126\right)} \]

   7. Applied rewrites 87.2%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-y, x, \color{blue}{\left(1 + z\right)} \cdot 1.1283791670955126\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

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

Alternative 13: 93.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+ (/ y (- (* (+ 1.0 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 / (((1.0 + z) * 1.1283791670955126) - (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 / (((1.0d0 + z) * 1.1283791670955126d0) - (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 / (((1.0 + z) * 1.1283791670955126) - (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 / (((1.0 + 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(Float64(Float64(1.0 + z) * 1.1283791670955126) - 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 / (((1.0 + z) * 1.1283791670955126) - (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[(N[(1.0 + z), $MachinePrecision] * 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 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{\left(1 + z\right)} - x \cdot y} \]
   4. Step-by-step derivation

      1. lower-+.f64 87.2

         \[\leadsto x + \frac{y}{1.1283791670955126 \cdot \color{blue}{\left(1 + z\right)} - x \cdot y} \]

   5. Applied rewrites 87.2%

      \[\leadsto x + \frac{y}{1.1283791670955126 \cdot \color{blue}{\left(1 + z\right)} - x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

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

Alternative 14: 93.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+ (/ y (fma 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(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 / fma(1.1283791670955126, z, Float64(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[(1.1283791670955126 * z + N[(1.1283791670955126 - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.9%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

4. Final simplification 90.3%

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

Alternative 15: 97.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -180:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+68}:\\ \;\;\;\;\frac{y}{\left(\frac{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)}{x} - y\right) \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-y, x, \mathsf{fma}\left(\left(0.18806319451591877 \cdot z\right) \cdot z, z, 1.1283791670955126\right)\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -180.0)
   (+ (/ -1.0 x) x)
   (if (<= z 1.55e+68)
     (+
      (/ y (* (- (/ (fma 1.1283791670955126 z 1.1283791670955126) x) y) x))
      x)
     (+
      (/
       y
       (fma
        (- y)
        x
        (fma (* (* 0.18806319451591877 z) z) z 1.1283791670955126)))
      x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -180.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 1.55e+68) {
		tmp = (y / (((fma(1.1283791670955126, z, 1.1283791670955126) / x) - y) * x)) + x;
	} else {
		tmp = (y / fma(-y, x, fma(((0.18806319451591877 * z) * z), z, 1.1283791670955126))) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -180.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 1.55e+68)
		tmp = Float64(Float64(y / Float64(Float64(Float64(fma(1.1283791670955126, z, 1.1283791670955126) / x) - y) * x)) + x);
	else
		tmp = Float64(Float64(y / fma(Float64(-y), x, fma(Float64(Float64(0.18806319451591877 * z) * z), z, 1.1283791670955126))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -180.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.55e+68], N[(N[(y / N[(N[(N[(N[(1.1283791670955126 * z + 1.1283791670955126), $MachinePrecision] / x), $MachinePrecision] - y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[((-y) * x + N[(N[(N[(0.18806319451591877 * z), $MachinePrecision] * z), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -180

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 -180 < z < 1.5499999999999999e68

   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. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in x around -inf

      \[\leadsto x + \frac{y}{-1 \cdot \color{blue}{\left(x \cdot \left(-1 \cdot \frac{\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z}{x} - -1 \cdot y\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.9%

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

   if 1.5499999999999999e68 < z

   1. Initial program 95.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 100.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-y, x, \mathsf{fma}\left(\frac{5641895835477563}{30000000000000000} \cdot {z}^{2}, z, \frac{5641895835477563}{5000000000000000}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto x + \frac{y}{\mathsf{fma}\left(-y, x, \mathsf{fma}\left(\left(z \cdot z\right) \cdot 0.18806319451591877, z, 1.1283791670955126\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -180:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+68}:\\ \;\;\;\;\frac{y}{\left(\frac{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)}{x} - y\right) \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-y, x, \mathsf{fma}\left(\left(0.18806319451591877 \cdot z\right) \cdot z, z, 1.1283791670955126\right)\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -180:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-y, x, \mathsf{fma}\left(\left(0.18806319451591877 \cdot z\right) \cdot z, z, 1.1283791670955126\right)\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -180.0)
   (+ (/ -1.0 x) x)
   (+
    (/
     y
     (fma (- y) x (fma (* (* 0.18806319451591877 z) z) z 1.1283791670955126)))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -180.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / fma(-y, x, fma(((0.18806319451591877 * z) * z), z, 1.1283791670955126))) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -180.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / fma(Float64(-y), x, fma(Float64(Float64(0.18806319451591877 * z) * z), z, 1.1283791670955126))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -180.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[((-y) * x + N[(N[(N[(0.18806319451591877 * z), $MachinePrecision] * z), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -180

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 -180 < z

   1. Initial program 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 96.4

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

   7. Applied rewrites 96.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-y, x, \mathsf{fma}\left(\frac{5641895835477563}{30000000000000000} \cdot {z}^{2}, z, \frac{5641895835477563}{5000000000000000}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.5%

       \[\leadsto x + \frac{y}{\mathsf{fma}\left(-y, x, \mathsf{fma}\left(\left(z \cdot z\right) \cdot 0.18806319451591877, z, 1.1283791670955126\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 95.5%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -180:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-y, x, \mathsf{fma}\left(\left(0.18806319451591877 \cdot z\right) \cdot z, z, 1.1283791670955126\right)\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 92.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -180:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-y, x, 1.1283791670955126\right)} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot z} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -180.0)
   (+ (/ -1.0 x) x)
   (if (<= z 8e+79)
     (+ (/ y (fma (- y) x 1.1283791670955126)) x)
     (+ (/ y (* 1.1283791670955126 z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -180.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 8e+79) {
		tmp = (y / fma(-y, x, 1.1283791670955126)) + x;
	} else {
		tmp = (y / (1.1283791670955126 * z)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -180.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 8e+79)
		tmp = Float64(Float64(y / fma(Float64(-y), x, 1.1283791670955126)) + x);
	else
		tmp = Float64(Float64(y / Float64(1.1283791670955126 * z)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -180.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 8e+79], N[(N[(y / N[((-y) * x + 1.1283791670955126), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(1.1283791670955126 * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -180

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 -180 < z < 7.99999999999999974e79

   1. Initial program 99.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 93.6%

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

   if 7.99999999999999974e79 < z

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   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. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 58.5%

      \[\leadsto x + \frac{y}{1.1283791670955126 \cdot \color{blue}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -180:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-y, x, 1.1283791670955126\right)} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot z} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 92.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -180:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot z} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -180.0)
   (+ (/ -1.0 x) x)
   (if (<= z 8e+79)
     (+ (/ y (- 1.1283791670955126 (* y x))) x)
     (+ (/ y (* 1.1283791670955126 z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -180.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 8e+79) {
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	} else {
		tmp = (y / (1.1283791670955126 * z)) + 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 <= (-180.0d0)) then
        tmp = ((-1.0d0) / x) + x
    else if (z <= 8d+79) then
        tmp = (y / (1.1283791670955126d0 - (y * x))) + x
    else
        tmp = (y / (1.1283791670955126d0 * z)) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -180.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 8e+79) {
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	} else {
		tmp = (y / (1.1283791670955126 * z)) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -180.0:
		tmp = (-1.0 / x) + x
	elif z <= 8e+79:
		tmp = (y / (1.1283791670955126 - (y * x))) + x
	else:
		tmp = (y / (1.1283791670955126 * z)) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -180.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 8e+79)
		tmp = Float64(Float64(y / Float64(1.1283791670955126 - Float64(y * x))) + x);
	else
		tmp = Float64(Float64(y / Float64(1.1283791670955126 * z)) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -180.0)
		tmp = (-1.0 / x) + x;
	elseif (z <= 8e+79)
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	else
		tmp = (y / (1.1283791670955126 * z)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -180.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 8e+79], N[(N[(y / N[(1.1283791670955126 - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(1.1283791670955126 * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -180

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 -180 < z < 7.99999999999999974e79

   1. Initial program 99.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 93.6%

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

   if 7.99999999999999974e79 < z

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   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. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 58.5%

      \[\leadsto x + \frac{y}{1.1283791670955126 \cdot \color{blue}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -180:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot z} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 68.8% accurate, 8.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (-1.0 / x) + 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 = ((-1.0d0) / x) + x
end function

Java

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

Python

def code(x, y, z):
	return (-1.0 / x) + x

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 65.7

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

5. Applied rewrites 65.7%

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

6. Final simplification 65.7%

   \[\leadsto \frac{-1}{x} + x \]
7. 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 2024276 
(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)))))