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

Percentage Accurate: 95.6% → 98.3%

Time: 9.0s

Alternatives: 12

Speedup: 3.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 96.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.4%

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

4. Final simplification 99.5%

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

Alternative 2: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.7853981633974483, y \cdot x, 0.8862269254527579\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_1 -5.0)
     t_0
     (if (<= t_1 0.01)
       (fma (fma 0.7853981633974483 (* y x) 0.8862269254527579) y x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 0.01) {
		tmp = fma(fma(0.7853981633974483, (y * x), 0.8862269254527579), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 0.01)
		tmp = fma(fma(0.7853981633974483, Float64(y * x), 0.8862269254527579), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5.0], t$95$0, If[LessEqual[t$95$1, 0.01], N[(N[(0.7853981633974483 * N[(y * x), $MachinePrecision] + 0.8862269254527579), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -5 or 0.0100000000000000002 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 93.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.0100000000000000002

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 56.9

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.1%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -5:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.7853981633974483, y \cdot x, 0.8862269254527579\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_1 -5.0)
     t_0
     (if (<= t_1 0.01) (- x (/ y -1.1283791670955126)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 0.01) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / x) + x
    t_1 = (y / ((1.1283791670955126d0 * exp(z)) - (y * x))) + x
    if (t_1 <= (-5.0d0)) then
        tmp = t_0
    else if (t_1 <= 0.01d0) then
        tmp = x - (y / (-1.1283791670955126d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * Math.exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 0.01) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 / x) + x
	t_1 = (y / ((1.1283791670955126 * math.exp(z)) - (y * x))) + x
	tmp = 0
	if t_1 <= -5.0:
		tmp = t_0
	elif t_1 <= 0.01:
		tmp = x - (y / -1.1283791670955126)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 0.01)
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 / x) + x;
	t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	tmp = 0.0;
	if (t_1 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 0.01)
		tmp = x - (y / -1.1283791670955126);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5.0], t$95$0, If[LessEqual[t$95$1, 0.01], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -5 or 0.0100000000000000002 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 93.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.0100000000000000002

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 56.9

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -5:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 0.01:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 96.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

4. Final simplification 98.0%

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

Alternative 5: 96.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 96.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 97.2%

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

Alternative 6: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 6.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 6.2e-92)
   (+ (/ -1.0 x) x)
   (- x (/ y (fma y x -1.1283791670955126)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 6.2e-92) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 6.2e-92)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 6.2e-92], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 6.2000000000000002e-92

   1. Initial program 90.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 6.2000000000000002e-92 < (exp.f64 z)

   1. Initial program 96.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 86.6

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

   7. Applied rewrites 86.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 6.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+59}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\left(z \cdot z\right) \cdot -0.5641895835477563}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e+24)
   (+ (/ -1.0 x) x)
   (if (<= z 6.1e+59)
     (+ (/ y (- (fma z 1.1283791670955126 1.1283791670955126) (* y x))) x)
     (fma (/ -1.0 (* (* z z) -0.5641895835477563)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+24) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 6.1e+59) {
		tmp = (y / (fma(z, 1.1283791670955126, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = fma((-1.0 / ((z * z) * -0.5641895835477563)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e+24)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 6.1e+59)
		tmp = Float64(Float64(y / Float64(fma(z, 1.1283791670955126, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = fma(Float64(-1.0 / Float64(Float64(z * z) * -0.5641895835477563)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e+24], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 6.1e+59], N[(N[(y / N[(N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(-1.0 / N[(N[(z * z), $MachinePrecision] * -0.5641895835477563), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1000000000000001e24

   1. Initial program 90.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.1000000000000001e24 < z < 6.09999999999999973e59

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 6.09999999999999973e59 < z

   1. Initial program 94.5%

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

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 94.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 94.7

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 94.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+59}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\left(z \cdot z\right) \cdot -0.5641895835477563}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+54}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\left(z \cdot z\right) \cdot -0.5641895835477563}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e+24)
   (+ (/ -1.0 x) x)
   (if (<= z 4.2e+54)
     (- x (/ y (fma y x -1.1283791670955126)))
     (fma (/ -1.0 (* (* z z) -0.5641895835477563)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+24) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 4.2e+54) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = fma((-1.0 / ((z * z) * -0.5641895835477563)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e+24)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 4.2e+54)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = fma(Float64(-1.0 / Float64(Float64(z * z) * -0.5641895835477563)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e+24], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.2e+54], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(N[(z * z), $MachinePrecision] * -0.5641895835477563), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1000000000000001e24

   1. Initial program 90.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.1000000000000001e24 < z < 4.19999999999999972e54

   1. Initial program 97.7%

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

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 4.19999999999999972e54 < z

   1. Initial program 94.5%

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

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 94.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 94.7

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 94.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+54}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\left(z \cdot z\right) \cdot -0.5641895835477563}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot z - y \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e+24)
   (+ (/ -1.0 x) x)
   (if (<= z 1.75e-6)
     (- x (/ y (fma y x -1.1283791670955126)))
     (+ (/ y (- (* 1.1283791670955126 z) (* y x))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+24) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 1.75e-6) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = (y / ((1.1283791670955126 * z) - (y * x))) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e+24)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 1.75e-6)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = Float64(Float64(y / Float64(Float64(1.1283791670955126 * z) - Float64(y * x))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e+24], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.75e-6], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(1.1283791670955126 * z), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1000000000000001e24

   1. Initial program 90.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.1000000000000001e24 < z < 1.74999999999999997e-6

   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. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 1.74999999999999997e-6 < z

   1. Initial program 91.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot z - y \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.2% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

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

   3. lift-/.f64 N/A

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

   4. *-lft-identity N/A

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

   5. associate-*l/ N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. clear-num N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 95.3%

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

5. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 81.0

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

7. Applied rewrites 81.0%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 60.9%

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

Alternative 11: 60.2% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

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

   3. lift-/.f64 N/A

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

   4. *-lft-identity N/A

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

   5. associate-*l/ N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. clear-num N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 95.3%

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

5. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 81.0

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

7. Applied rewrites 81.0%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 60.9%

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

Alternative 12: 14.6% accurate, 21.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.8862269254527579d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

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

   3. lift-/.f64 N/A

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

   4. *-lft-identity N/A

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

   5. associate-*l/ N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. clear-num N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 95.3%

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

5. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 81.0

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

7. Applied rewrites 81.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 15.4%

    \[\leadsto 0.8862269254527579 \cdot \color{blue}{y} \]
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 2024235 
(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)))))