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

Percentage Accurate: 95.7% → 99.4%

Time: 9.7s

Alternatives: 17

Speedup: 3.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 1.0)
     (- x (/ y (fma y x -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) <= 1.0) {
		tmp = x - (y / fma(y, x, -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) <= 1.0)
		tmp = Float64(x - Float64(y / fma(y, x, -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], 1.0], N[(x - N[(y / N[(y * x + -1.1283791670955126), $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 90.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.9%

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 1 < (exp.f64 z)

   1. Initial program 94.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.7853981633974483, y \cdot x, 0.8862269254527579\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* (exp z) 1.1283791670955126) (* y x))) x)))
   (if (<= t_1 -1000000.0)
     t_0
     (if (<= t_1 5.0)
       (fma (fma 0.7853981633974483 (* y x) 0.8862269254527579) y x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((exp(z) * 1.1283791670955126) - (y * x))) + x;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = fma(fma(0.7853981633974483, (y * x), 0.8862269254527579), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = fma(fma(0.7853981633974483, Float64(y * x), 0.8862269254527579), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], t$95$0, If[LessEqual[t$95$1, 5.0], N[(N[(0.7853981633974483 * N[(y * x), $MachinePrecision] + 0.8862269254527579), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -1e6 or 5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if -1e6 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 5

   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. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 53.9

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{5000000000000000}{5641895835477563} + \frac{25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \left(x \cdot y\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.7853981633974483, y \cdot x, 0.8862269254527579\right), \color{blue}{y}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq -1000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.7853981633974483, y \cdot x, 0.8862269254527579\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* (exp z) 1.1283791670955126) (* y x))) x)))
   (if (<= t_1 -1000000.0)
     t_0
     (if (<= t_1 5.0) (- 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 / ((exp(z) * 1.1283791670955126) - (y * x))) + x;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		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 / ((exp(z) * 1.1283791670955126d0) - (y * x))) + x
    if (t_1 <= (-1000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) 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 / ((Math.exp(z) * 1.1283791670955126) - (y * x))) + x;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		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 / ((math.exp(z) * 1.1283791670955126) - (y * x))) + x
	tmp = 0
	if t_1 <= -1000000.0:
		tmp = t_0
	elif t_1 <= 5.0:
		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(exp(z) * 1.1283791670955126) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		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 / ((exp(z) * 1.1283791670955126) - (y * x))) + x;
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		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[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], t$95$0, If[LessEqual[t$95$1, 5.0], 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)))) < -1e6 or 5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if -1e6 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 5

   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. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 53.9

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 52.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq -1000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq 5:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+175}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ y (- (* (exp z) 1.1283791670955126) (* y x))) x)))
   (if (<= t_0 2e+175) t_0 (+ (/ -1.0 x) x))))

C

double code(double x, double y, double z) {
	double t_0 = (y / ((exp(z) * 1.1283791670955126) - (y * x))) + x;
	double tmp;
	if (t_0 <= 2e+175) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / x) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / ((exp(z) * 1.1283791670955126d0) - (y * x))) + x
    if (t_0 <= 2d+175) then
        tmp = t_0
    else
        tmp = ((-1.0d0) / x) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / ((Math.exp(z) * 1.1283791670955126) - (y * x))) + x;
	double tmp;
	if (t_0 <= 2e+175) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / x) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / ((math.exp(z) * 1.1283791670955126) - (y * x))) + x
	tmp = 0
	if t_0 <= 2e+175:
		tmp = t_0
	else:
		tmp = (-1.0 / x) + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_0 <= 2e+175)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / x) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / ((exp(z) * 1.1283791670955126) - (y * x))) + x;
	tmp = 0.0;
	if (t_0 <= 2e+175)
		tmp = t_0;
	else
		tmp = (-1.0 / x) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+175], t$95$0, N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1.9999999999999999e175

   1. Initial program 99.2%

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

   if 1.9999999999999999e175 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 78.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 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 97.8%

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

      1. lift-+.f64 N/A

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

      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 97.8%

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

   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. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 92.3

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

   7. Applied rewrites 92.3%

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

   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%

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

       1. lift-fma.f64 N/A

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

       2. lower-+.f64 N/A

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

   12. Applied rewrites 97.4%

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

4. Final simplification 98.0%

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

Alternative 6: 96.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 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 97.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 95.7%

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

Alternative 7: 96.5% accurate, 0.9× speedup

Math

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

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
      (fma y x -1.1283791670955126)))
    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, fma(y, x, -1.1283791670955126))), 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 / fma(fma(-0.5641895835477563, z, -1.1283791670955126), z, fma(y, x, -1.1283791670955126))), 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[(-0.5641895835477563 * z + -1.1283791670955126), $MachinePrecision] * z + N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 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 97.8%

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

      1. lift-+.f64 N/A

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

      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 97.8%

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

   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. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 92.3

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

   7. Applied rewrites 92.3%

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

4. Final simplification 94.2%

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

Alternative 8: 95.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 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 97.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 93.1%

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

Alternative 9: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 8.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 8.6e-95)
   (+ (/ -1.0 x) x)
   (- x (/ y (fma y x -1.1283791670955126)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 8.6e-95) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 8.6e-95)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 8.6e-95], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 8.59999999999999994e-95

   1. Initial program 90.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 8.59999999999999994e-95 < (exp.f64 z)

   1. Initial program 97.8%

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

      1. lift-+.f64 N/A

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

      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 97.8%

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

   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. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 81.4

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

   7. Applied rewrites 81.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 8.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1600:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 68000000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\left(z \cdot z\right) \cdot -0.5641895835477563}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1600.0)
   (+ (/ -1.0 x) x)
   (if (<= z 68000000000.0)
     (+ (/ y (- (fma z 1.1283791670955126 1.1283791670955126) (* y x))) x)
     (fma (/ -1.0 (* (* z z) -0.5641895835477563)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1600.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 68000000000.0) {
		tmp = (y / (fma(z, 1.1283791670955126, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = fma((-1.0 / ((z * z) * -0.5641895835477563)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1600.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 68000000000.0)
		tmp = Float64(Float64(y / Float64(fma(z, 1.1283791670955126, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = fma(Float64(-1.0 / Float64(Float64(z * z) * -0.5641895835477563)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1600.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 68000000000.0], N[(N[(y / N[(N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(-1.0 / N[(N[(z * z), $MachinePrecision] * -0.5641895835477563), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1600

   1. Initial program 89.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1600 < z < 6.8e10

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 6.8e10 < z

   1. Initial program 94.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 94.4%

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

   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. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 79.9

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

   7. Applied rewrites 79.9%

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

   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 79.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1600:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 68000000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\left(z \cdot z\right) \cdot -0.5641895835477563}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1600:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 57000000000:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\left(z \cdot z\right) \cdot -0.5641895835477563}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1600.0)
   (+ (/ -1.0 x) x)
   (if (<= z 57000000000.0)
     (- x (/ y (fma y x -1.1283791670955126)))
     (fma (/ -1.0 (* (* z z) -0.5641895835477563)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1600.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 57000000000.0) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = fma((-1.0 / ((z * z) * -0.5641895835477563)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1600.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 57000000000.0)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = fma(Float64(-1.0 / Float64(Float64(z * z) * -0.5641895835477563)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1600.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 57000000000.0], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(N[(z * z), $MachinePrecision] * -0.5641895835477563), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1600

   1. Initial program 89.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1600 < z < 5.7e10

   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. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 5.7e10 < z

   1. Initial program 94.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 94.4%

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

   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. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 79.9

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

   7. Applied rewrites 79.9%

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

   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 79.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1600:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 57000000000:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\left(z \cdot z\right) \cdot -0.5641895835477563}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1600:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot 1.1283791670955126 - y \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1600.0)
   (+ (/ -1.0 x) x)
   (if (<= z 4.3e-10)
     (- x (/ y (fma y x -1.1283791670955126)))
     (+ (/ y (- (* z 1.1283791670955126) (* y x))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1600.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 4.3e-10) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = (y / ((z * 1.1283791670955126) - (y * x))) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1600.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 4.3e-10)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = Float64(Float64(y / Float64(Float64(z * 1.1283791670955126) - Float64(y * x))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1600.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.3e-10], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(z * 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1600

   1. Initial program 89.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1600 < z < 4.30000000000000014e-10

   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. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 4.30000000000000014e-10 < z

   1. Initial program 94.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 66.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1600:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot 1.1283791670955126 - y \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1600:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 57000000000:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(z, -0.5641895835477563, -1.1283791670955126\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1600.0)
   (+ (/ -1.0 x) x)
   (if (<= z 57000000000.0)
     (- x (/ y (fma y x -1.1283791670955126)))
     (- x (/ y (* (fma z -0.5641895835477563 -1.1283791670955126) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1600.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 57000000000.0) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = x - (y / (fma(z, -0.5641895835477563, -1.1283791670955126) * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1600.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 57000000000.0)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = Float64(x - Float64(y / Float64(fma(z, -0.5641895835477563, -1.1283791670955126) * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1600.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 57000000000.0], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(z * -0.5641895835477563 + -1.1283791670955126), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1600

   1. Initial program 89.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1600 < z < 5.7e10

   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. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 5.7e10 < z

   1. Initial program 94.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 94.4%

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

   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. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 79.9

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 79.9%

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

       1. lift-fma.f64 N/A

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

       2. lower-+.f64 N/A

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

   12. Applied rewrites 79.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1600:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 57000000000:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(z, -0.5641895835477563, -1.1283791670955126\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.7% 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 95.8%

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

   1. lift-+.f64 N/A

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

   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 95.4%

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

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. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 75.5

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

7. Applied rewrites 75.5%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 61.7%

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

Alternative 15: 58.7% accurate, 14.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

   1. lift-+.f64 N/A

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

   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 95.4%

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

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. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 75.5

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

7. Applied rewrites 75.5%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 61.6%

    \[\leadsto x - -0.8862269254527579 \cdot \color{blue}{y} \]
11. Add Preprocessing

Alternative 16: 58.7% 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 95.8%

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

   1. lift-+.f64 N/A

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

   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 95.4%

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

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. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 75.5

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

7. Applied rewrites 75.5%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 61.6%

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

Alternative 17: 14.2% 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 95.8%

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

   1. lift-+.f64 N/A

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

   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 95.4%

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

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. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 75.5

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

7. Applied rewrites 75.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 16.6%

    \[\leadsto 0.8862269254527579 \cdot \color{blue}{y} \]
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 2024249 
(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)))))