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

Percentage Accurate: 95.4% → 97.8%

Time: 7.8s

Alternatives: 12

Speedup: 2.7×

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: 95.4% 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: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -80000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-96}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -80000.0)
   (+ x (/ -1.0 x))
   (if (<= z 1.15e-96)
     (+
      x
      (/
       y
       (-
        (fma
         (fma 0.5641895835477563 z 1.1283791670955126)
         z
         1.1283791670955126)
        (* x y))))
     (fma (/ 0.8862269254527579 (exp z)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -80000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.15e-96) {
		tmp = x + (y / (fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - (x * y)));
	} else {
		tmp = fma((0.8862269254527579 / exp(z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -80000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.15e-96)
		tmp = Float64(x + Float64(y / Float64(fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - Float64(x * y))));
	else
		tmp = fma(Float64(0.8862269254527579 / exp(z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -80000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e-96], N[(x + N[(y / N[(N[(N[(0.5641895835477563 * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.8862269254527579 / N[Exp[z], $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8e4

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -8e4 < z < 1.15e-96

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 1.15e-96 < z

   1. Initial program 92.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 74.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_0 \leq -0.5 \lor \neg \left(t\_0 \leq 400\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.8862269254527579 \cdot y}{1 + z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (or (<= t_0 -0.5) (not (<= t_0 400.0)))
     (+ x (/ -1.0 x))
     (/ (* 0.8862269254527579 y) (+ 1.0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	double tmp;
	if ((t_0 <= -0.5) || !(t_0 <= 400.0)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = (0.8862269254527579 * y) / (1.0 + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    if ((t_0 <= (-0.5d0)) .or. (.not. (t_0 <= 400.0d0))) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = (0.8862269254527579d0 * y) / (1.0d0 + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
	double tmp;
	if ((t_0 <= -0.5) || !(t_0 <= 400.0)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = (0.8862269254527579 * y) / (1.0 + z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	tmp = 0
	if (t_0 <= -0.5) or not (t_0 <= 400.0):
		tmp = x + (-1.0 / x)
	else:
		tmp = (0.8862269254527579 * y) / (1.0 + z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
	tmp = 0.0
	if ((t_0 <= -0.5) || !(t_0 <= 400.0))
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(Float64(0.8862269254527579 * y) / Float64(1.0 + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	tmp = 0.0;
	if ((t_0 <= -0.5) || ~((t_0 <= 400.0)))
		tmp = x + (-1.0 / x);
	else
		tmp = (0.8862269254527579 * y) / (1.0 + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.5], N[Not[LessEqual[t$95$0, 400.0]], $MachinePrecision]], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.8862269254527579 * y), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -0.5 or 400 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if -0.5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 400

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 32.8

         \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

   5. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{y}{1 + z} \cdot \frac{5000000000000000}{5641895835477563} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.9%

      \[\leadsto \frac{y}{1 + z} \cdot 0.8862269254527579 \]
   9. Step-by-step derivation

   10. Applied rewrites 30.9%

       \[\leadsto \color{blue}{\frac{0.8862269254527579 \cdot y}{1 + z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq -0.5 \lor \neg \left(x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq 400\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.8862269254527579 \cdot y}{1 + z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_0 \leq -0.5 \lor \neg \left(t\_0 \leq 400\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 + z} \cdot 0.8862269254527579\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (or (<= t_0 -0.5) (not (<= t_0 400.0)))
     (+ x (/ -1.0 x))
     (* (/ y (+ 1.0 z)) 0.8862269254527579))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	double tmp;
	if ((t_0 <= -0.5) || !(t_0 <= 400.0)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = (y / (1.0 + z)) * 0.8862269254527579;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    if ((t_0 <= (-0.5d0)) .or. (.not. (t_0 <= 400.0d0))) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = (y / (1.0d0 + z)) * 0.8862269254527579d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
	double tmp;
	if ((t_0 <= -0.5) || !(t_0 <= 400.0)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = (y / (1.0 + z)) * 0.8862269254527579;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	tmp = 0
	if (t_0 <= -0.5) or not (t_0 <= 400.0):
		tmp = x + (-1.0 / x)
	else:
		tmp = (y / (1.0 + z)) * 0.8862269254527579
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
	tmp = 0.0
	if ((t_0 <= -0.5) || !(t_0 <= 400.0))
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(Float64(y / Float64(1.0 + z)) * 0.8862269254527579);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	tmp = 0.0;
	if ((t_0 <= -0.5) || ~((t_0 <= 400.0)))
		tmp = x + (-1.0 / x);
	else
		tmp = (y / (1.0 + z)) * 0.8862269254527579;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.5], N[Not[LessEqual[t$95$0, 400.0]], $MachinePrecision]], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * 0.8862269254527579), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -0.5 or 400 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if -0.5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 400

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 32.8

         \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

   5. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{y}{1 + z} \cdot \frac{5000000000000000}{5641895835477563} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.9%

      \[\leadsto \frac{y}{1 + z} \cdot 0.8862269254527579 \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq -0.5 \lor \neg \left(x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq 400\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 + z} \cdot 0.8862269254527579\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_0 \leq -0.5 \lor \neg \left(t\_0 \leq 400\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(-0.8862269254527579, z, 0.8862269254527579\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (or (<= t_0 -0.5) (not (<= t_0 400.0)))
     (+ x (/ -1.0 x))
     (* y (fma -0.8862269254527579 z 0.8862269254527579)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	double tmp;
	if ((t_0 <= -0.5) || !(t_0 <= 400.0)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = y * fma(-0.8862269254527579, z, 0.8862269254527579);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
	tmp = 0.0
	if ((t_0 <= -0.5) || !(t_0 <= 400.0))
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(y * fma(-0.8862269254527579, z, 0.8862269254527579));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.5], N[Not[LessEqual[t$95$0, 400.0]], $MachinePrecision]], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.8862269254527579 * z + 0.8862269254527579), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -0.5 or 400 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if -0.5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 400

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 32.8

         \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

   5. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 30.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq -0.5 \lor \neg \left(x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq 400\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(-0.8862269254527579, z, 0.8862269254527579\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_0 \leq 10^{+240}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (<= t_0 1e+240) t_0 (+ x (/ -1.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	double tmp;
	if (t_0 <= 1e+240) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    if (t_0 <= 1d+240) then
        tmp = t_0
    else
        tmp = x + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
	double tmp;
	if (t_0 <= 1e+240) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	tmp = 0
	if t_0 <= 1e+240:
		tmp = t_0
	else:
		tmp = x + (-1.0 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
	tmp = 0.0
	if (t_0 <= 1e+240)
		tmp = t_0;
	else
		tmp = Float64(x + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	tmp = 0.0;
	if (t_0 <= 1e+240)
		tmp = t_0;
	else
		tmp = x + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+240], t$95$0, N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1.00000000000000001e240

   1. Initial program 99.1%

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

   if 1.00000000000000001e240 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 68.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 2 \cdot 10^{-72}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\left(z \cdot z\right) \cdot 0.18806319451591877, z, 1.1283791670955126\right) - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 2e-72)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (- (fma (* (* z z) 0.18806319451591877) z 1.1283791670955126) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 2e-72) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (fma(((z * z) * 0.18806319451591877), z, 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 2e-72)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(fma(Float64(Float64(z * z) * 0.18806319451591877), z, 1.1283791670955126) - Float64(x * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 2e-72], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(N[(N[(z * z), $MachinePrecision] * 0.18806319451591877), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 1.9999999999999999e-72

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 1.9999999999999999e-72 < (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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 94.1

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 93.3%

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

Alternative 7: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -120:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right) - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -120.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (-
      (fma
       (fma
        (fma 0.18806319451591877 z 0.5641895835477563)
        z
        1.1283791670955126)
       z
       1.1283791670955126)
      (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -120.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -120.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - Float64(x * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -120.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -120

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -120 < 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 94.1

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

   5. Applied rewrites 94.1%

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

Alternative 8: 93.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -80000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot 1.1283791670955126 - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -80000.0)
   (+ x (/ -1.0 x))
   (if (<= z 1.0)
     (+ x (/ y (- 1.1283791670955126 (* x y))))
     (+ x (/ y (- (* z 1.1283791670955126) (* x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -80000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x + (y / ((z * 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-80000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.0d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x + (y / ((z * 1.1283791670955126d0) - (x * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -80000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x + (y / ((z * 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -80000.0:
		tmp = x + (-1.0 / x)
	elif z <= 1.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x + (y / ((z * 1.1283791670955126) - (x * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -80000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.0)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = Float64(x + Float64(y / Float64(Float64(z * 1.1283791670955126) - Float64(x * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -80000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x + (y / ((z * 1.1283791670955126) - (x * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -80000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(z * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8e4

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -8e4 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 98.0%

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

   if 1 < z

   1. Initial program 90.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 75.6

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

   5. Applied rewrites 75.6%

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

   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 75.6%

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

Alternative 9: 93.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -120:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right) - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -120.0)
   (+ x (/ -1.0 x))
   (+ x (/ y (- (fma 1.1283791670955126 z 1.1283791670955126) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -120.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (fma(1.1283791670955126, z, 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -120.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(fma(1.1283791670955126, z, 1.1283791670955126) - Float64(x * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -120.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(1.1283791670955126 * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -120

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -120 < 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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 92.2

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

   5. Applied rewrites 92.2%

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

Alternative 10: 90.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -80000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -80000.0)
   (+ x (/ -1.0 x))
   (+ x (/ y (- 1.1283791670955126 (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -80000.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-80000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -80000.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -80000.0:
		tmp = x + (-1.0 / x)
	else:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -80000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -80000.0)
		tmp = x + (-1.0 / x);
	else
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -80000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8e4

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -8e4 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 88.7%

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

Alternative 11: 15.0% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-exp.f64 17.1

      \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

5. Applied rewrites 17.1%

   \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

6. Taylor expanded in z around 0

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

8. Applied rewrites 16.6%

   \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(-0.8862269254527579, z, 0.8862269254527579\right)} \]
9. Add Preprocessing

Alternative 12: 15.0% 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 96.4%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-exp.f64 17.1

      \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

5. Applied rewrites 17.1%

   \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

6. Taylor expanded in z around 0

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

8. Applied rewrites 16.0%

   \[\leadsto 0.8862269254527579 \cdot \color{blue}{y} \]
9. 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 2024324 
(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)))))