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

Percentage Accurate: 95.6% → 99.9%

Time: 10.3s

Alternatives: 16

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.9%

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

   3. 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 99.9%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.9

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 1 < (exp.f64 z)

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

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

Alternative 3: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{\left(1 + z\right) \cdot 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1000.0)
     t_0
     (if (<= t_1 2e-13) (/ y (* (+ 1.0 z) 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 <= -1000.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-13) {
		tmp = y / ((1.0 + z) * 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 <= (-1000.0d0)) then
        tmp = t_0
    else if (t_1 <= 2d-13) then
        tmp = y / ((1.0d0 + z) * 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 <= -1000.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-13) {
		tmp = y / ((1.0 + z) * 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 <= -1000.0:
		tmp = t_0
	elif t_1 <= 2e-13:
		tmp = y / ((1.0 + z) * 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 <= -1000.0)
		tmp = t_0;
	elseif (t_1 <= 2e-13)
		tmp = Float64(y / Float64(Float64(1.0 + z) * 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 <= -1000.0)
		tmp = t_0;
	elseif (t_1 <= 2e-13)
		tmp = y / ((1.0 + z) * 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, -1000.0], t$95$0, If[LessEqual[t$95$1, 2e-13], N[(y / N[(N[(1.0 + z), $MachinePrecision] * 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)))) < -1e3 or 2.0000000000000001e-13 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 95.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -1e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 2.0000000000000001e-13

   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 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 35.0%

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

   10. Applied rewrites 35.0%

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

   12. Applied rewrites 35.2%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -1000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{\left(1 + z\right) \cdot 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.8862269254527579}{1 + z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1000.0)
     t_0
     (if (<= t_1 2e-13) (* (/ 0.8862269254527579 (+ 1.0 z)) y) 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 <= -1000.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-13) {
		tmp = (0.8862269254527579 / (1.0 + z)) * y;
	} 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 <= (-1000.0d0)) then
        tmp = t_0
    else if (t_1 <= 2d-13) then
        tmp = (0.8862269254527579d0 / (1.0d0 + z)) * y
    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 <= -1000.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-13) {
		tmp = (0.8862269254527579 / (1.0 + z)) * y;
	} 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 <= -1000.0:
		tmp = t_0
	elif t_1 <= 2e-13:
		tmp = (0.8862269254527579 / (1.0 + z)) * y
	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 <= -1000.0)
		tmp = t_0;
	elseif (t_1 <= 2e-13)
		tmp = Float64(Float64(0.8862269254527579 / Float64(1.0 + z)) * y);
	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 <= -1000.0)
		tmp = t_0;
	elseif (t_1 <= 2e-13)
		tmp = (0.8862269254527579 / (1.0 + z)) * y;
	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, -1000.0], t$95$0, If[LessEqual[t$95$1, 2e-13], N[(N[(0.8862269254527579 / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * y), $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)))) < -1e3 or 2.0000000000000001e-13 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 95.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -1e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 2.0000000000000001e-13

   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 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 35.0%

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

   10. Applied rewrites 35.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -1000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.8862269254527579}{1 + z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1000.0) t_0 (if (<= t_1 2e-13) (* 0.8862269254527579 y) 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 <= -1000.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-13) {
		tmp = 0.8862269254527579 * y;
	} 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 <= (-1000.0d0)) then
        tmp = t_0
    else if (t_1 <= 2d-13) then
        tmp = 0.8862269254527579d0 * y
    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 <= -1000.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-13) {
		tmp = 0.8862269254527579 * y;
	} 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 <= -1000.0:
		tmp = t_0
	elif t_1 <= 2e-13:
		tmp = 0.8862269254527579 * y
	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 <= -1000.0)
		tmp = t_0;
	elseif (t_1 <= 2e-13)
		tmp = Float64(0.8862269254527579 * y);
	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 <= -1000.0)
		tmp = t_0;
	elseif (t_1 <= 2e-13)
		tmp = 0.8862269254527579 * y;
	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, -1000.0], t$95$0, If[LessEqual[t$95$1, 2e-13], N[(0.8862269254527579 * y), $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)))) < -1e3 or 2.0000000000000001e-13 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 95.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -1e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 2.0000000000000001e-13

   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 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 34.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -1000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+205}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_0 2e+205) t_0 (+ (/ -1.0 x) x))))

C

double code(double x, double y, double z) {
	double t_0 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_0 <= 2e+205) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / x) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / ((1.1283791670955126d0 * exp(z)) - (y * x))) + x
    if (t_0 <= 2d+205) then
        tmp = t_0
    else
        tmp = ((-1.0d0) / x) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / ((1.1283791670955126 * Math.exp(z)) - (y * x))) + x;
	double tmp;
	if (t_0 <= 2e+205) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / x) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / ((1.1283791670955126 * math.exp(z)) - (y * x))) + x
	tmp = 0
	if t_0 <= 2e+205:
		tmp = t_0
	else:
		tmp = (-1.0 / x) + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_0 <= 2e+205)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / x) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	tmp = 0.0;
	if (t_0 <= 2e+205)
		tmp = t_0;
	else
		tmp = (-1.0 / x) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+205], t$95$0, N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 2.00000000000000003e205

   1. Initial program 98.6%

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

   if 2.00000000000000003e205 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 67.5%

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

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 2 \cdot 10^{+205}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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) < 2

   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 99.4%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.4

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if 2 < (exp.f64 z)

   1. Initial program 93.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 57.3%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 57.3

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 57.3

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

   7. Applied rewrites 57.3%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 72.0

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

   10. Applied rewrites 72.0%

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

   11. Taylor expanded in z around inf

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

   13. Applied rewrites 72.0%

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

4. Final simplification 93.4%

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

Alternative 8: 97.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/
     y
     (fma
      (- y)
      x
      (fma
       (fma
        (fma 0.18806319451591877 z 0.5641895835477563)
        z
        1.1283791670955126)
       z
       1.1283791670955126)))
    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(-y, x, fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126))) + 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 / fma(Float64(-y), x, fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126))) + 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[((-y) * x + N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 96.9

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

   7. Applied rewrites 96.9%

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

4. Final simplification 97.6%

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

Alternative 9: 97.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/
     y
     (fma (- y) x (fma (* (* z z) 0.18806319451591877) z 1.1283791670955126)))
    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(-y, x, fma(((z * z) * 0.18806319451591877), z, 1.1283791670955126))) + 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 / fma(Float64(-y), x, fma(Float64(Float64(z * z) * 0.18806319451591877), z, 1.1283791670955126))) + 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[((-y) * x + N[(N[(N[(z * z), $MachinePrecision] * 0.18806319451591877), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 96.9

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 96.8%

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

4. Final simplification 97.4%

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

Alternative 10: 97.0% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 95.8

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

   7. Applied rewrites 95.8%

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

4. Final simplification 96.7%

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

Alternative 11: 96.2% 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 90.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.0%

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

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

   5. Applied rewrites 94.4%

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

4. Final simplification 95.5%

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

Alternative 12: 94.0% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 91.8

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

   7. Applied rewrites 91.8%

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

4. Final simplification 93.5%

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

Alternative 13: 94.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+ (/ y (- (fma z 1.1283791670955126 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(z, 1.1283791670955126, 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(z, 1.1283791670955126, 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[(z * 1.1283791670955126 + 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 90.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 93.5%

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

Alternative 14: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 87.6%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 87.6

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 87.6

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

   7. Applied rewrites 87.6%

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

4. Final simplification 90.2%

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

Alternative 15: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+ (/ y (- 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 / (1.1283791670955126 - (y * x))) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = ((-1.0d0) / x) + x
    else
        tmp = (y / (1.1283791670955126d0 - (y * x))) + x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = (-1.0 / x) + x
	else:
		tmp = (y / (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(1.1283791670955126 - Float64(y * x))) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = (-1.0 / x) + x;
	else
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	end
	tmp_2 = 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[(1.1283791670955126 - 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 90.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 87.6%

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

4. Final simplification 90.2%

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

Alternative 16: 14.0% 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 17.6

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

5. Applied rewrites 17.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 17.3%

   \[\leadsto 0.8862269254527579 \cdot \color{blue}{y} \]
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 2024270 
(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)))))