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

Percentage Accurate: 95.6% → 99.6%

Time: 8.0s

Alternatives: 13

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.6% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.05:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-y, x, \mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)\right)}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579 \cdot y, e^{-z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 1.05)
     (+
      (/ 1.0 (/ (fma (- y) x (fma 1.1283791670955126 z 1.1283791670955126)) y))
      x)
     (fma (* 0.8862269254527579 y) (exp (- z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 1.05) {
		tmp = (1.0 / (fma(-y, x, fma(1.1283791670955126, z, 1.1283791670955126)) / y)) + x;
	} else {
		tmp = fma((0.8862269254527579 * y), exp(-z), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 1.05)
		tmp = Float64(Float64(1.0 / Float64(fma(Float64(-y), x, fma(1.1283791670955126, z, 1.1283791670955126)) / y)) + x);
	else
		tmp = fma(Float64(0.8862269254527579 * y), exp(Float64(-z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.05], N[(N[(1.0 / N[(N[((-y) * x + N[(1.1283791670955126 * z + 1.1283791670955126), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.8862269254527579 * y), $MachinePrecision] * N[Exp[(-z)], $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 82.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 1.05000000000000004

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.8

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.05000000000000004 < (exp.f64 z)

   1. Initial program 94.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{e^{z}}, 0.8862269254527579, x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.05:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-y, x, \mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)\right)}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579 \cdot y, e^{-z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot x, 0.7853981633974483, 0.8862269254527579\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_1 -5000.0)
     t_0
     (if (<= t_1 0.001)
       (fma (fma (* y x) 0.7853981633974483 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 / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.001) {
		tmp = fma(fma((y * x), 0.7853981633974483, 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(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_0;
	elseif (t_1 <= 0.001)
		tmp = fma(fma(Float64(y * x), 0.7853981633974483, 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[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$0, If[LessEqual[t$95$1, 0.001], N[(N[(N[(y * x), $MachinePrecision] * 0.7853981633974483 + 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)))) < -5e3 or 1e-3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 91.7

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

   5. Applied rewrites 91.7%

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

   if -5e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1e-3

   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 61.6

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(1 + \frac{25000000000000000000000000000000}{31830988618379068626528276418969} \cdot {y}^{2}\right) - \color{blue}{\frac{-5000000000000000}{5641895835477563} \cdot y} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -5000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot x, 0.7853981633974483, 0.8862269254527579\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.001:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_1 -5000.0)
     t_0
     (if (<= t_1 0.001) (- x (/ y -1.1283791670955126)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.001) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / x) + x
    t_1 = (y / ((1.1283791670955126d0 * exp(z)) - (y * x))) + x
    if (t_1 <= (-5000.0d0)) then
        tmp = t_0
    else if (t_1 <= 0.001d0) then
        tmp = x - (y / (-1.1283791670955126d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * Math.exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.001) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 / x) + x
	t_1 = (y / ((1.1283791670955126 * math.exp(z)) - (y * x))) + x
	tmp = 0
	if t_1 <= -5000.0:
		tmp = t_0
	elif t_1 <= 0.001:
		tmp = x - (y / -1.1283791670955126)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_0;
	elseif (t_1 <= 0.001)
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 / x) + x;
	t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	tmp = 0.0;
	if (t_1 <= -5000.0)
		tmp = t_0;
	elseif (t_1 <= 0.001)
		tmp = x - (y / -1.1283791670955126);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$0, If[LessEqual[t$95$1, 0.001], 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)))) < -5e3 or 1e-3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 91.7

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

   5. Applied rewrites 91.7%

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

   if -5e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1e-3

   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 61.6

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 61.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -5000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 0.001:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (+
      (/
       y
       (-
        (fma
         (fma
          (fma 0.18806319451591877 z 0.5641895835477563)
          z
          1.1283791670955126)
         z
         1.1283791670955126)
        (* y x)))
      x)
     (/ 1.0 (/ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2.0) {
		tmp = (y / (fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = 1.0 / (1.0 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2.0)
		tmp = Float64(Float64(y / Float64(fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = Float64(1.0 / Float64(1.0 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(N[(y / N[(N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 82.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 2 < (exp.f64 z)

   1. Initial program 93.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 82.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   if 2 < (exp.f64 z)

   1. Initial program 93.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.6%

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

Alternative 6: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 82.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 2 < (exp.f64 z)

   1. Initial program 93.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.6%

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

Alternative 7: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 82.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if 2 < (exp.f64 z)

   1. Initial program 93.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.5%

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

Alternative 8: 92.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -255:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+71}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{0.8862269254527579}{1 + z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -255.0)
   (+ (/ -1.0 x) x)
   (if (<= z 9e+71)
     (- x (/ y (fma y x -1.1283791670955126)))
     (fma y (/ 0.8862269254527579 (+ 1.0 z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -255.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 9e+71) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = fma(y, (0.8862269254527579 / (1.0 + z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -255.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 9e+71)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = fma(y, Float64(0.8862269254527579 / Float64(1.0 + z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -255.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 9e+71], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.8862269254527579 / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -255

   1. Initial program 82.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -255 < z < 9.00000000000000087e71

   1. Initial program 98.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 93.0

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

   7. Applied rewrites 93.0%

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

   if 9.00000000000000087e71 < z

   1. Initial program 95.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 75.9%

       \[\leadsto \mathsf{fma}\left(\frac{y}{1 + z}, 0.8862269254527579, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 75.9%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{0.8862269254527579}{z + 1}}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -255:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+71}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{0.8862269254527579}{1 + z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -255:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -255.0) (+ (/ -1.0 x) x) (- x (/ y (fma y x -1.1283791670955126)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -255.0) {
		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 (z <= -255.0)
		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[z, -255.0], 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 z < -255

   1. Initial program 82.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -255 < z

   1. Initial program 97.6%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 84.8

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

   7. Applied rewrites 84.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -255:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{-x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3e+66) (/ 1.0 (- x)) (- x (/ y -1.1283791670955126))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+66) {
		tmp = 1.0 / -x;
	} else {
		tmp = x - (y / -1.1283791670955126);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3d+66)) then
        tmp = 1.0d0 / -x
    else
        tmp = x - (y / (-1.1283791670955126d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+66) {
		tmp = 1.0 / -x;
	} else {
		tmp = x - (y / -1.1283791670955126);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3e+66:
		tmp = 1.0 / -x
	else:
		tmp = x - (y / -1.1283791670955126)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3e+66)
		tmp = Float64(1.0 / Float64(-x));
	else
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3e+66)
		tmp = 1.0 / -x;
	else
		tmp = x - (y / -1.1283791670955126);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3e+66], N[(1.0 / (-x)), $MachinePrecision], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.00000000000000002e66

   1. Initial program 81.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 55.6%

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

   if -3.00000000000000002e66 < z

   1. Initial program 97.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.5%

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

Alternative 11: 62.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3e+66) (/ 1.0 (- x)) (fma 0.8862269254527579 y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+66) {
		tmp = 1.0 / -x;
	} else {
		tmp = fma(0.8862269254527579, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3e+66)
		tmp = Float64(1.0 / Float64(-x));
	else
		tmp = fma(0.8862269254527579, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3e+66], N[(1.0 / (-x)), $MachinePrecision], N[(0.8862269254527579 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.00000000000000002e66

   1. Initial program 81.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 55.6%

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

   if -3.00000000000000002e66 < z

   1. Initial program 97.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 65.5%

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

Alternative 12: 60.0% 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 92.7%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
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. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

   5. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

   6. flip3-+ N/A

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

4. Applied rewrites 92.7%

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

5. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 78.5

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

7. Applied rewrites 78.5%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 57.4%

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

Alternative 13: 14.4% 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 92.7%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
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. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}}} \]

   5. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}^{3}}{x \cdot x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y} - x \cdot \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}\right)}}}} \]

   6. flip3-+ N/A

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

4. Applied rewrites 92.7%

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

5. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 78.5

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

7. Applied rewrites 78.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 15.2%

    \[\leadsto 0.8862269254527579 \cdot \color{blue}{y} \]
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 2024331 
(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)))))