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

Percentage Accurate: 96.0% → 99.8%

Time: 8.3s

Alternatives: 12

Speedup: 3.1×

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: 96.0% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)
     (fma (/ 0.8862269254527579 (exp z)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2.0) {
		tmp = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	} else {
		tmp = fma((0.8862269254527579 / exp(z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2.0)
		tmp = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x);
	else
		tmp = fma(Float64(0.8862269254527579 / exp(z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.8862269254527579 / N[Exp[z], $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

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

   if 2 < (exp.f64 z)

   1. Initial program 93.1%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(\mathsf{fma}\left(z, -0.5641895835477563, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (-
      x
      (/
       y
       (fma
        y
        x
        (fma
         (fma z -0.5641895835477563 -1.1283791670955126)
         z
         -1.1283791670955126))))
     (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) <= 2.0) {
		tmp = x - (y / fma(y, x, fma(fma(z, -0.5641895835477563, -1.1283791670955126), z, -1.1283791670955126)));
	} 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) <= 2.0)
		tmp = Float64(x - Float64(y / fma(y, x, fma(fma(z, -0.5641895835477563, -1.1283791670955126), z, -1.1283791670955126))));
	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], 2.0], N[(x - N[(y / N[(y * x + N[(N[(z * -0.5641895835477563 + -1.1283791670955126), $MachinePrecision] * z + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 86.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 99.4

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \color{blue}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\right)}, y, x\right) \]

   7. Applied rewrites 99.4%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \mathsf{fma}\left(x, y, -1.1283791670955126\right)\right)}}, y, x\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. neg-mul-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.4

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

   9. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{-y}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \mathsf{fma}\left(x, y, -1.1283791670955126\right)\right)} + x} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 N/A

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

       5. distribute-frac-neg N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 N/A

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

       8. lower-/.f64 99.4

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

   11. Applied rewrites 99.4%

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

   if 2 < (exp.f64 z)

   1. Initial program 93.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(\mathsf{fma}\left(z, -0.5641895835477563, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 0.0001) (- x (/ y -1.1283791670955126)) 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 <= 0.0001) {
		tmp = x - (y / -1.1283791670955126);
	} 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 <= 0.0001d0) then
        tmp = x - (y / (-1.1283791670955126d0))
    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 <= 0.0001) {
		tmp = x - (y / -1.1283791670955126);
	} 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 <= 0.0001:
		tmp = x - (y / -1.1283791670955126)
	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 <= 0.0001)
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	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 <= 0.0001)
		tmp = x - (y / -1.1283791670955126);
	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, 0.0001], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $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 1.00000000000000005e-4 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 92.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if -5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1.00000000000000005e-4

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 60.6

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

   7. Applied rewrites 60.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto x - \frac{y}{\frac{-5641895835477563}{5000000000000000}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.9%

       \[\leadsto x - \frac{y}{-1.1283791670955126} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -5:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 0.0001:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 86.6%

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

   3. Taylor expanded in x 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 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 95.6

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \color{blue}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\right)}, y, x\right) \]

   7. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \mathsf{fma}\left(x, y, -1.1283791670955126\right)\right)}}, y, x\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 97.3%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, -1.1283791670955126\right)}{x} + y\right) \cdot \color{blue}{x}}, y, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

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

Alternative 5: 96.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = x - (y / fma(y, x, fma(fma(z, -0.5641895835477563, -1.1283791670955126), z, -1.1283791670955126)));
	}
	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(x - Float64(y / fma(y, x, fma(fma(z, -0.5641895835477563, -1.1283791670955126), z, -1.1283791670955126))));
	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[(x - N[(y / N[(y * x + N[(N[(z * -0.5641895835477563 + -1.1283791670955126), $MachinePrecision] * z + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 86.6%

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

   3. Taylor expanded in x 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 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 95.6

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \color{blue}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\right)}, y, x\right) \]

   7. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \mathsf{fma}\left(x, y, -1.1283791670955126\right)\right)}}, y, x\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. neg-mul-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 95.7

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

   9. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\frac{-y}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \mathsf{fma}\left(x, y, -1.1283791670955126\right)\right)} + x} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 N/A

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

       5. distribute-frac-neg N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 N/A

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

       8. lower-/.f64 95.7

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

   11. Applied rewrites 95.7%

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

4. Final simplification 97.1%

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

Alternative 6: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 2.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 2.6e-159)
   (+ (/ -1.0 x) x)
   (- x (/ y (fma x y -1.1283791670955126)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 2.6e-159) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = x - (y / fma(x, y, -1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 2.6e-159)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(x - Float64(y / fma(x, y, -1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 2.6e-159], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(x * y + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 2.5999999999999998e-159

   1. Initial program 86.6%

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

   3. Taylor expanded in x 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 2.5999999999999998e-159 < (exp.f64 z)

   1. Initial program 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 86.8

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

   7. Applied rewrites 86.8%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 2.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -140:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+89}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right) \cdot z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -140.0)
   (+ (/ -1.0 x) x)
   (if (<= z 5.1e+89)
     (+ (/ y (- (fma 1.1283791670955126 z 1.1283791670955126) (* y x))) x)
     (fma
      (/ -1.0 (* (fma -0.5641895835477563 z -1.1283791670955126) z))
      y
      x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -140.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 5.1e+89) {
		tmp = (y / (fma(1.1283791670955126, z, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = fma((-1.0 / (fma(-0.5641895835477563, z, -1.1283791670955126) * z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -140.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 5.1e+89)
		tmp = Float64(Float64(y / Float64(fma(1.1283791670955126, z, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = fma(Float64(-1.0 / Float64(fma(-0.5641895835477563, z, -1.1283791670955126) * z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -140.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.1e+89], N[(N[(y / N[(N[(1.1283791670955126 * z + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(-1.0 / N[(N[(-0.5641895835477563 * z + -1.1283791670955126), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -140

   1. Initial program 86.6%

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

   3. Taylor expanded in x 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 -140 < z < 5.10000000000000027e89

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} - x \cdot y} \]
   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. lower-fma.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if 5.10000000000000027e89 < z

   1. Initial program 94.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 95.1

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \color{blue}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\right)}, y, x\right) \]

   7. Applied rewrites 95.1%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \mathsf{fma}\left(x, y, -1.1283791670955126\right)\right)}}, y, x\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, -1.1283791670955126\right)}{x} + y\right) \cdot \color{blue}{x}}, y, x\right) \]

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(\frac{-1}{-1 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot \frac{1}{z}\right)}\right)}, y, x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 95.1%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right) \cdot z}, y, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -140:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+89}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right) \cdot z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -140:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+89}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(z \cdot z\right) \cdot -0.5641895835477563}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -140.0)
   (+ (/ -1.0 x) x)
   (if (<= z 5.1e+89)
     (+ (/ y (- (fma 1.1283791670955126 z 1.1283791670955126) (* y x))) x)
     (- x (/ y (* (* z z) -0.5641895835477563))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -140.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 5.1e+89) {
		tmp = (y / (fma(1.1283791670955126, z, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = x - (y / ((z * z) * -0.5641895835477563));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -140.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 5.1e+89)
		tmp = Float64(Float64(y / Float64(fma(1.1283791670955126, z, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = Float64(x - Float64(y / Float64(Float64(z * z) * -0.5641895835477563)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -140.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.1e+89], N[(N[(y / N[(N[(1.1283791670955126 * z + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(N[(z * z), $MachinePrecision] * -0.5641895835477563), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -140

   1. Initial program 86.6%

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

   3. Taylor expanded in x 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 -140 < z < 5.10000000000000027e89

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} - x \cdot y} \]
   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. lower-fma.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if 5.10000000000000027e89 < z

   1. Initial program 94.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 95.1

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \color{blue}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\right)}, y, x\right) \]

   7. Applied rewrites 95.1%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \mathsf{fma}\left(x, y, -1.1283791670955126\right)\right)}}, y, x\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\frac{-5641895835477563}{10000000000000000} \cdot \color{blue}{{z}^{2}}}, y, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.1%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{\left(z \cdot z\right) \cdot \color{blue}{-0.5641895835477563}}, y, x\right) \]
   11. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. neg-mul-1 N/A

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

       6. lift-neg.f64 N/A

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

       7. lower-/.f64 95.1

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

   12. Applied rewrites 95.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -140:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+89}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(z \cdot z\right) \cdot -0.5641895835477563}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -140:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+89}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(z \cdot z\right) \cdot -0.5641895835477563}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -140.0)
   (+ (/ -1.0 x) x)
   (if (<= z 5e+89)
     (- x (/ y (fma x y -1.1283791670955126)))
     (- x (/ y (* (* z z) -0.5641895835477563))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -140.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 5e+89) {
		tmp = x - (y / fma(x, y, -1.1283791670955126));
	} else {
		tmp = x - (y / ((z * z) * -0.5641895835477563));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -140.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 5e+89)
		tmp = Float64(x - Float64(y / fma(x, y, -1.1283791670955126)));
	else
		tmp = Float64(x - Float64(y / Float64(Float64(z * z) * -0.5641895835477563)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -140.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5e+89], N[(x - N[(y / N[(x * y + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(z * z), $MachinePrecision] * -0.5641895835477563), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -140

   1. Initial program 86.6%

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

   3. Taylor expanded in x 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 -140 < z < 4.99999999999999983e89

   1. Initial program 98.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 95.4

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

   7. Applied rewrites 95.4%

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

   if 4.99999999999999983e89 < z

   1. Initial program 94.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 95.1

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \color{blue}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\right)}, y, x\right) \]

   7. Applied rewrites 95.1%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, \mathsf{fma}\left(x, y, -1.1283791670955126\right)\right)}}, y, x\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\frac{-5641895835477563}{10000000000000000} \cdot \color{blue}{{z}^{2}}}, y, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.1%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{\left(z \cdot z\right) \cdot \color{blue}{-0.5641895835477563}}, y, x\right) \]
   11. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. neg-mul-1 N/A

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

       6. lift-neg.f64 N/A

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

       7. lower-/.f64 95.1

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

   12. Applied rewrites 95.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -140:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+89}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(z \cdot z\right) \cdot -0.5641895835477563}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{-1.1283791670955126} \end{array} \]

FPCore

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

C

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

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))
end function

Java

public static double code(double x, double y, double z) {
	return x - (y / -1.1283791670955126);
}

Python

def code(x, y, z):
	return x - (y / -1.1283791670955126)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. *-lft-identity N/A

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

   5. associate-*l/ N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. clear-num N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 94.0%

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

5. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 78.7

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

7. Applied rewrites 78.7%

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

8. Taylor expanded in x around 0

   \[\leadsto x - \frac{y}{\frac{-5641895835477563}{5000000000000000}} \]
9. Step-by-step derivation

10. Applied rewrites 55.6%

    \[\leadsto x - \frac{y}{-1.1283791670955126} \]
11. Add Preprocessing

Alternative 11: 59.6% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.8862269254527579, y, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma 0.8862269254527579 y x))

C

double code(double x, double y, double z) {
	return fma(0.8862269254527579, y, x);
}

Julia

function code(x, y, z)
	return fma(0.8862269254527579, y, x)
end

Wolfram

code[x_, y_, z_] := N[(0.8862269254527579 * y + x), $MachinePrecision]

Derivation

1. Initial program 94.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. *-lft-identity N/A

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

   5. associate-*l/ N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. clear-num N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 94.0%

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

5. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 78.7

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

7. Applied rewrites 78.7%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 55.5%

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

Alternative 12: 14.6% accurate, 21.3× speedup

Math

\[\begin{array}{l} \\ 0.8862269254527579 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.8862269254527579 y))

C

double code(double x, double y, double z) {
	return 0.8862269254527579 * 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 = 0.8862269254527579d0 * y
end function

Java

public static double code(double x, double y, double z) {
	return 0.8862269254527579 * y;
}

Python

def code(x, y, z):
	return 0.8862269254527579 * y

Julia

function code(x, y, z)
	return Float64(0.8862269254527579 * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.8862269254527579 * y;
end

Wolfram

code[x_, y_, z_] := N[(0.8862269254527579 * y), $MachinePrecision]

Derivation

1. Initial program 94.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. *-lft-identity N/A

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

   5. associate-*l/ N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. clear-num N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 94.0%

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

5. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 78.7

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

7. Applied rewrites 78.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 15.3%

    \[\leadsto 0.8862269254527579 \cdot \color{blue}{y} \]
11. 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 2024332 
(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)))))