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

Percentage Accurate: 95.2% → 99.4%

Time: 8.8s

Alternatives: 10

Speedup: 5.8×

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: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.999999998:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.002:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{e^{z}} \cdot -0.8862269254527579\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.999999998)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.002)
     (+ x (/ y (- 1.1283791670955126 (* x y))))
     (- x (* (/ y (exp z)) -0.8862269254527579)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.999999998) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.002) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x - ((y / exp(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) :: tmp
    if (exp(z) <= 0.999999998d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.002d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x - ((y / exp(z)) * (-0.8862269254527579d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.999999998) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 1.002) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x - ((y / Math.exp(z)) * -0.8862269254527579);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.999999998:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.002:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x - ((y / math.exp(z)) * -0.8862269254527579)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.999999998)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.002)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = Float64(x - Float64(Float64(y / exp(z)) * -0.8862269254527579));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.999999998)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 1.002)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x - ((y / exp(z)) * -0.8862269254527579);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.999999998], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.002], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y / N[Exp[z], $MachinePrecision]), $MachinePrecision] * -0.8862269254527579), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.999999997999999946

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 0.999999997999999946 < (exp.f64 z) < 1.002

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. unsub-neg 99.9%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.9%

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

      5. distribute-neg-frac2 99.9%

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

      6. neg-sub0 99.9%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.9%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.9%

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

   if 1.002 < (exp.f64 z)

   1. Initial program 94.3%

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

      1. remove-double-neg 94.3%

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

      2. distribute-frac-neg 94.3%

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

      3. unsub-neg 94.3%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 94.3%

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

      5. distribute-neg-frac2 94.3%

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

      6. neg-sub0 94.3%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 94.3%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 94.3%

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

      9. +-commutative 94.3%

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

      10. fma-define 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

      \[\leadsto x - \color{blue}{-0.8862269254527579 \cdot \frac{y}{e^{z}}} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

      \[\leadsto x - \color{blue}{\frac{y}{e^{z}} \cdot -0.8862269254527579} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.999999998:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.002:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{e^{z}} \cdot -0.8862269254527579\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.999999998:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.999999998)
   (+ x (/ -1.0 x))
   (- x (/ y (fma x y (* (exp z) -1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.999999998) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x - (y / fma(x, y, (exp(z) * -1.1283791670955126)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.999999998)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x - Float64(y / fma(x, y, Float64(exp(z) * -1.1283791670955126))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.999999998], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(x * y + N[(N[Exp[z], $MachinePrecision] * -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.999999997999999946

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 0.999999997999999946 < (exp.f64 z)

   1. Initial program 97.8%

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

      1. remove-double-neg 97.8%

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

      2. distribute-frac-neg 97.8%

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

      3. unsub-neg 97.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 97.8%

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

      5. distribute-neg-frac2 97.8%

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

      6. neg-sub0 97.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 97.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 97.8%

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

      9. +-commutative 97.8%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.999999998:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+240}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_0 5e+240) t_0 (+ x (/ -1.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 5e+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 / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    if (t_0 <= 5d+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 / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 5e+240) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	tmp = 0
	if t_0 <= 5e+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(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_0 <= 5e+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 / ((exp(z) * 1.1283791670955126) - (x * y)));
	tmp = 0.0;
	if (t_0 <= 5e+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[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+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 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))) < 5.0000000000000003e240

   1. Initial program 98.3%

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

   if 5.0000000000000003e240 < (+.f64 x (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))

   1. Initial program 58.4%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.999999998:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - \left(x \cdot y + z \cdot -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.999999998)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 2.0)
     (+ x (/ y (- 1.1283791670955126 (+ (* x y) (* z -1.1283791670955126)))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.999999998) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 2.0) {
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * -1.1283791670955126))));
	} else {
		tmp = 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) <= 0.999999998d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 2.0d0) then
        tmp = x + (y / (1.1283791670955126d0 - ((x * y) + (z * (-1.1283791670955126d0)))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.999999998) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 2.0) {
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * -1.1283791670955126))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.999999998:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 2.0:
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * -1.1283791670955126))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.999999998)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 2.0)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(Float64(x * y) + Float64(z * -1.1283791670955126)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.999999998)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 2.0)
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * -1.1283791670955126))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.999999998], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(x + N[(y / N[(1.1283791670955126 - N[(N[(x * y), $MachinePrecision] + N[(z * -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.999999997999999946

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 0.999999997999999946 < (exp.f64 z) < 2

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. unsub-neg 99.9%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.9%

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

      5. distribute-neg-frac2 99.9%

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

      6. neg-sub0 99.9%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.9%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.4%

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

   if 2 < (exp.f64 z)

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.999999998:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - \left(x \cdot y + z \cdot -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -850000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 126:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -850000.0)
   (+ x (/ -1.0 x))
   (if (<= z 126.0) (+ x (/ 1.0 (- (/ 1.1283791670955126 y) x))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -850000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 126.0) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = 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 (z <= (-850000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 126.0d0) then
        tmp = x + (1.0d0 / ((1.1283791670955126d0 / y) - x))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -850000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 126.0) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -850000.0:
		tmp = x + (-1.0 / x)
	elif z <= 126.0:
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -850000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 126.0)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(1.1283791670955126 / y) - x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -850000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 126.0)
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -850000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 126.0], N[(x + N[(1.0 / N[(N[(1.1283791670955126 / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -8.5e5

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf 100.0%

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

   if -8.5e5 < z < 126

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. unsub-neg 99.9%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.9%

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

      5. distribute-neg-frac2 99.9%

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

      6. neg-sub0 99.9%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.9%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.3%

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

      1. clear-num 99.3%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{x \cdot y - 1.1283791670955126}{y}}} \]

      2. inv-pow 99.3%

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

      3. *-commutative 99.3%

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

      4. div-sub 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/l* 99.3%

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

      7. *-inverses 99.3%

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

      8. *-commutative 99.3%

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

      9. *-un-lft-identity 99.3%

         \[\leadsto x - {\left(\color{blue}{x} - \frac{1.1283791670955126}{y}\right)}^{-1} \]

   7. Applied egg-rr 99.3%

      \[\leadsto x - \color{blue}{{\left(x - \frac{1.1283791670955126}{y}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 99.3%

         \[\leadsto x - \color{blue}{\frac{1}{x - \frac{1.1283791670955126}{y}}} \]

   9. Simplified 99.3%

      \[\leadsto x - \color{blue}{\frac{1}{x - \frac{1.1283791670955126}{y}}} \]

   if 126 < z

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -850000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 126:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 104:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.85e-9)
   (+ x (/ -1.0 x))
   (if (<= z 104.0) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.85e-9) {
		tmp = x + (-1.0 / x);
	} else if (z <= 104.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = 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 (z <= (-2.85d-9)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 104.0d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.85e-9) {
		tmp = x + (-1.0 / x);
	} else if (z <= 104.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.85e-9:
		tmp = x + (-1.0 / x)
	elif z <= 104.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.85e-9)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 104.0)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.85e-9)
		tmp = x + (-1.0 / x);
	elseif (z <= 104.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.85e-9], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 104.0], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8499999999999999e-9

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf 100.0%

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

   if -2.8499999999999999e-9 < z < 104

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. unsub-neg 99.9%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.9%

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

      5. distribute-neg-frac2 99.9%

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

      6. neg-sub0 99.9%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.9%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.3%

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

   if 104 < z

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 104:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-200}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.5e-13) x (if (<= z 7e-200) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e-13) {
		tmp = x;
	} else if (z <= 7e-200) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = 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 (z <= (-7.5d-13)) then
        tmp = x
    else if (z <= 7d-200) then
        tmp = x - (y * (-0.8862269254527579d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e-13) {
		tmp = x;
	} else if (z <= 7e-200) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.5e-13:
		tmp = x
	elif z <= 7e-200:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.5e-13)
		tmp = x;
	elseif (z <= 7e-200)
		tmp = Float64(x - Float64(y * -0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.5e-13)
		tmp = x;
	elseif (z <= 7e-200)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.5e-13], x, If[LessEqual[z, 7e-200], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5000000000000004e-13 or 7.00000000000000045e-200 < z

   1. Initial program 93.2%

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

   3. Taylor expanded in x around inf 77.9%

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

   if -7.5000000000000004e-13 < z < 7.00000000000000045e-200

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.8%

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

   6. Taylor expanded in y around 0 85.2%

      \[\leadsto x - \color{blue}{-0.8862269254527579 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 85.2%

         \[\leadsto x - \color{blue}{y \cdot -0.8862269254527579} \]

   8. Simplified 85.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-200}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-13}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-200}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.4e-13)
   (+ x (/ -1.0 x))
   (if (<= z 1.15e-200) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.4e-13) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.15e-200) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = 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 (z <= (-1.4d-13)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.15d-200) then
        tmp = x - (y * (-0.8862269254527579d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.4e-13) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.15e-200) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.4e-13:
		tmp = x + (-1.0 / x)
	elif z <= 1.15e-200:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.4e-13)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.15e-200)
		tmp = Float64(x - Float64(y * -0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.4e-13)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.15e-200)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.4e-13], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e-200], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4000000000000001e-13

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf 100.0%

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

   if -1.4000000000000001e-13 < z < 1.15000000000000004e-200

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.8%

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

   6. Taylor expanded in y around 0 85.2%

      \[\leadsto x - \color{blue}{-0.8862269254527579 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 85.2%

         \[\leadsto x - \color{blue}{y \cdot -0.8862269254527579} \]

   8. Simplified 85.2%

      \[\leadsto x - \color{blue}{y \cdot -0.8862269254527579} \]

   if 1.15000000000000004e-200 < z

   1. Initial program 96.3%

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

   3. Taylor expanded in x around inf 92.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-13}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-200}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-200}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5e-15)
   (+ x (/ -1.0 x))
   (if (<= z 1.2e-200) (- x (/ y -1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-15) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.2e-200) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = 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 (z <= (-5d-15)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.2d-200) then
        tmp = x - (y / (-1.1283791670955126d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-15) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.2e-200) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5e-15:
		tmp = x + (-1.0 / x)
	elif z <= 1.2e-200:
		tmp = x - (y / -1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e-15)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.2e-200)
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e-15)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.2e-200)
		tmp = x - (y / -1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5e-15], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.2e-200], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -4.99999999999999999e-15

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf 100.0%

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

   if -4.99999999999999999e-15 < z < 1.20000000000000001e-200

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.8%

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

   6. Taylor expanded in x around 0 85.2%

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

   if 1.20000000000000001e-200 < z

   1. Initial program 96.3%

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

   3. Taylor expanded in x around inf 92.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-200}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.1% accurate, 111.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return 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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in x around inf 76.2%

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

4. Final simplification 76.2%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 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 2024043 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))