Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Percentage Accurate: 93.9% → 98.8%

Time: 25.2s

Alternatives: 16

Speedup: 1.0×

• 31.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / 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 - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / 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 - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.85e-8)
   (+
    (+ 0.91893853320467 (* (log x) -0.5))
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))
     x))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (* z (* z (/ (+ 0.0007936500793651 y) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = (0.91893853320467 + (log(x) * -0.5)) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (z * (z * ((0.0007936500793651 + y) / 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 (x <= 1.85d-8) then
        tmp = (0.91893853320467d0 + (log(x) * (-0.5d0))) + ((0.083333333333333d0 + (z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0))) / x)
    else
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (z * (z * ((0.0007936500793651d0 + y) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = (0.91893853320467 + (Math.log(x) * -0.5)) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	} else {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (z * (z * ((0.0007936500793651 + y) / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.85e-8:
		tmp = (0.91893853320467 + (math.log(x) * -0.5)) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x)
	else:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (z * (z * ((0.0007936500793651 + y) / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.85e-8)
		tmp = Float64(Float64(0.91893853320467 + Float64(log(x) * -0.5)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))) / x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(z * Float64(z * Float64(Float64(0.0007936500793651 + y) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.85e-8)
		tmp = (0.91893853320467 + (log(x) * -0.5)) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	else
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (z * (z * ((0.0007936500793651 + y) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.85e-8], N[(N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(z * N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85e-8

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   4. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 1.85e-8 < x

   1. Initial program 90.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 88.3%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* 93.3%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

      2. unpow2 93.3%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(z \cdot z\right)} \cdot \frac{0.0007936500793651 + y}{x} \]

      3. associate-*l* 98.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]

   5. Applied egg-rr 98.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.0007936500793651 + y}{x}\\ t_1 := t\_0 \cdot {z}^{2}\\ t_2 := \left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot t\_0\right)\\ t_3 := \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;z \leq -1 \cdot 10^{+61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+17}:\\ \;\;\;\;t\_3 + 0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+99}:\\ \;\;\;\;t\_3 + y \cdot \frac{0.083333333333333}{x \cdot y}\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}{x}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ 0.0007936500793651 y) x))
        (t_1 (* t_0 (pow z 2.0)))
        (t_2 (+ (+ 0.91893853320467 (* (log x) -0.5)) (* z (* z t_0))))
        (t_3 (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
   (if (<= z -1e+61)
     t_2
     (if (<= z 4.6e+17)
       (+ t_3 (* 0.083333333333333 (/ 1.0 x)))
       (if (<= z 5.4e+66)
         t_1
         (if (<= z 4e+99)
           (+ t_3 (* y (/ 0.083333333333333 (* x y))))
           (if (<= z 6.1e+115)
             (/ (* (+ 0.0007936500793651 y) (pow z 2.0)) x)
             (if (<= z 6.2e+115)
               (+ (* x (+ (log x) -1.0)) (/ 0.083333333333333 x))
               (if (<= z 4.4e+155) t_1 t_2)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) / x;
	double t_1 = t_0 * pow(z, 2.0);
	double t_2 = (0.91893853320467 + (log(x) * -0.5)) + (z * (z * t_0));
	double t_3 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double tmp;
	if (z <= -1e+61) {
		tmp = t_2;
	} else if (z <= 4.6e+17) {
		tmp = t_3 + (0.083333333333333 * (1.0 / x));
	} else if (z <= 5.4e+66) {
		tmp = t_1;
	} else if (z <= 4e+99) {
		tmp = t_3 + (y * (0.083333333333333 / (x * y)));
	} else if (z <= 6.1e+115) {
		tmp = ((0.0007936500793651 + y) * pow(z, 2.0)) / x;
	} else if (z <= 6.2e+115) {
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
	} else if (z <= 4.4e+155) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (0.0007936500793651d0 + y) / x
    t_1 = t_0 * (z ** 2.0d0)
    t_2 = (0.91893853320467d0 + (log(x) * (-0.5d0))) + (z * (z * t_0))
    t_3 = (((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0
    if (z <= (-1d+61)) then
        tmp = t_2
    else if (z <= 4.6d+17) then
        tmp = t_3 + (0.083333333333333d0 * (1.0d0 / x))
    else if (z <= 5.4d+66) then
        tmp = t_1
    else if (z <= 4d+99) then
        tmp = t_3 + (y * (0.083333333333333d0 / (x * y)))
    else if (z <= 6.1d+115) then
        tmp = ((0.0007936500793651d0 + y) * (z ** 2.0d0)) / x
    else if (z <= 6.2d+115) then
        tmp = (x * (log(x) + (-1.0d0))) + (0.083333333333333d0 / x)
    else if (z <= 4.4d+155) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) / x;
	double t_1 = t_0 * Math.pow(z, 2.0);
	double t_2 = (0.91893853320467 + (Math.log(x) * -0.5)) + (z * (z * t_0));
	double t_3 = (((x - 0.5) * Math.log(x)) - x) + 0.91893853320467;
	double tmp;
	if (z <= -1e+61) {
		tmp = t_2;
	} else if (z <= 4.6e+17) {
		tmp = t_3 + (0.083333333333333 * (1.0 / x));
	} else if (z <= 5.4e+66) {
		tmp = t_1;
	} else if (z <= 4e+99) {
		tmp = t_3 + (y * (0.083333333333333 / (x * y)));
	} else if (z <= 6.1e+115) {
		tmp = ((0.0007936500793651 + y) * Math.pow(z, 2.0)) / x;
	} else if (z <= 6.2e+115) {
		tmp = (x * (Math.log(x) + -1.0)) + (0.083333333333333 / x);
	} else if (z <= 4.4e+155) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.0007936500793651 + y) / x
	t_1 = t_0 * math.pow(z, 2.0)
	t_2 = (0.91893853320467 + (math.log(x) * -0.5)) + (z * (z * t_0))
	t_3 = (((x - 0.5) * math.log(x)) - x) + 0.91893853320467
	tmp = 0
	if z <= -1e+61:
		tmp = t_2
	elif z <= 4.6e+17:
		tmp = t_3 + (0.083333333333333 * (1.0 / x))
	elif z <= 5.4e+66:
		tmp = t_1
	elif z <= 4e+99:
		tmp = t_3 + (y * (0.083333333333333 / (x * y)))
	elif z <= 6.1e+115:
		tmp = ((0.0007936500793651 + y) * math.pow(z, 2.0)) / x
	elif z <= 6.2e+115:
		tmp = (x * (math.log(x) + -1.0)) + (0.083333333333333 / x)
	elif z <= 4.4e+155:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0007936500793651 + y) / x)
	t_1 = Float64(t_0 * (z ^ 2.0))
	t_2 = Float64(Float64(0.91893853320467 + Float64(log(x) * -0.5)) + Float64(z * Float64(z * t_0)))
	t_3 = Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)
	tmp = 0.0
	if (z <= -1e+61)
		tmp = t_2;
	elseif (z <= 4.6e+17)
		tmp = Float64(t_3 + Float64(0.083333333333333 * Float64(1.0 / x)));
	elseif (z <= 5.4e+66)
		tmp = t_1;
	elseif (z <= 4e+99)
		tmp = Float64(t_3 + Float64(y * Float64(0.083333333333333 / Float64(x * y))));
	elseif (z <= 6.1e+115)
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) * (z ^ 2.0)) / x);
	elseif (z <= 6.2e+115)
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(0.083333333333333 / x));
	elseif (z <= 4.4e+155)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.0007936500793651 + y) / x;
	t_1 = t_0 * (z ^ 2.0);
	t_2 = (0.91893853320467 + (log(x) * -0.5)) + (z * (z * t_0));
	t_3 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	tmp = 0.0;
	if (z <= -1e+61)
		tmp = t_2;
	elseif (z <= 4.6e+17)
		tmp = t_3 + (0.083333333333333 * (1.0 / x));
	elseif (z <= 5.4e+66)
		tmp = t_1;
	elseif (z <= 4e+99)
		tmp = t_3 + (y * (0.083333333333333 / (x * y)));
	elseif (z <= 6.1e+115)
		tmp = ((0.0007936500793651 + y) * (z ^ 2.0)) / x;
	elseif (z <= 6.2e+115)
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
	elseif (z <= 4.4e+155)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[z, -1e+61], t$95$2, If[LessEqual[z, 4.6e+17], N[(t$95$3 + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.4e+66], t$95$1, If[LessEqual[z, 4e+99], N[(t$95$3 + N[(y * N[(0.083333333333333 / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.1e+115], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[z, 6.2e+115], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e+155], t$95$1, t$95$2]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if z < -9.99999999999999949e60 or 4.4000000000000005e155 < z

   1. Initial program 90.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 83.4%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 83.4%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 83.4%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 83.4%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 83.4%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 83.4%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 83.4%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 83.4%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 85.9%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 85.9%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 85.9%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 85.9%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 84.7%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* 91.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

      2. unpow2 91.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(z \cdot z\right)} \cdot \frac{0.0007936500793651 + y}{x} \]

      3. associate-*l* 99.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]

   10. Applied egg-rr 86.9%

       \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]

   if -9.99999999999999949e60 < z < 4.6e17

   1. Initial program 98.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]
   4. Step-by-step derivation

      1. clear-num 85.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. inv-pow 85.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   5. Applied egg-rr 85.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 85.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. associate-/r/ 85.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   7. Simplified 85.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   if 4.6e17 < z < 5.4e66 or 6.2000000000000001e115 < z < 4.4000000000000005e155

   1. Initial program 95.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 81.9%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 81.9%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 81.9%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 81.9%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 81.9%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 81.9%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 81.9%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 81.9%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 80.6%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 80.6%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 80.6%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 80.6%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 80.6%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 80.8%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   10. Step-by-step derivation

       1. associate-/l* 85.3%

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   11. Simplified 85.3%

       \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   if 5.4e66 < z < 3.9999999999999999e99

   1. Initial program 60.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      2. *-un-lft-identity 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{1 \cdot \left(\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\right)}}{\sqrt{x} \cdot \sqrt{x}} \]

      3. times-frac 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\sqrt{x}} \cdot \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{\sqrt{x}}} \]

      4. *-commutative 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\sqrt{x}} \cdot \frac{\color{blue}{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)} + 0.083333333333333}{\sqrt{x}} \]

      5. fma-undefine 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\sqrt{x}} \cdot \frac{\color{blue}{\mathsf{fma}\left(z, \left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, 0.083333333333333\right)}}{\sqrt{x}} \]

      6. fma-neg 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\sqrt{x}} \cdot \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)}, 0.083333333333333\right)}{\sqrt{x}} \]

      7. metadata-eval 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\sqrt{x}} \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, \color{blue}{-0.0027777777777778}\right), 0.083333333333333\right)}{\sqrt{x}} \]

   4. Applied egg-rr 60.1%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\sqrt{x}} \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt{x}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}}} \]

      2. *-lft-identity 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt{x}}}}{\sqrt{x}} \]

      3. fma-define 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778}, 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}} \]

      4. +-commutative 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{\left(0.0007936500793651 + y\right)} \cdot z + -0.0027777777777778, 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}} \]

      5. *-commutative 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(0.0007936500793651 + y\right)} + -0.0027777777777778, 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}} \]

      6. fma-define 60.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}, 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}} \]

   6. Simplified 60.1%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}}} \]

   7. Taylor expanded in y around inf 99.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{y \cdot \left(\frac{0.083333333333333}{x \cdot y} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x \cdot y} + \frac{{z}^{2}}{x}\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{0.083333333333333}{\color{blue}{y \cdot x}} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x \cdot y} + \frac{{z}^{2}}{x}\right)\right) \]

      2. associate-/r* 99.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\color{blue}{\frac{\frac{0.083333333333333}{y}}{x}} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x \cdot y} + \frac{{z}^{2}}{x}\right)\right) \]

      3. associate-/l* 99.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \left(\color{blue}{z \cdot \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x \cdot y}} + \frac{{z}^{2}}{x}\right)\right) \]

      4. fma-define 99.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \color{blue}{\mathsf{fma}\left(z, \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x \cdot y}, \frac{{z}^{2}}{x}\right)}\right) \]

      5. *-commutative 99.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \mathsf{fma}\left(z, \frac{\color{blue}{z \cdot 0.0007936500793651} - 0.0027777777777778}{x \cdot y}, \frac{{z}^{2}}{x}\right)\right) \]

      6. fma-neg 99.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \mathsf{fma}\left(z, \frac{\color{blue}{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}}{x \cdot y}, \frac{{z}^{2}}{x}\right)\right) \]

      7. metadata-eval 99.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, 0.0007936500793651, \color{blue}{-0.0027777777777778}\right)}{x \cdot y}, \frac{{z}^{2}}{x}\right)\right) \]

   9. Simplified 99.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x \cdot y}, \frac{{z}^{2}}{x}\right)\right)} \]

   10. Taylor expanded in z around 0 79.4%

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \color{blue}{-0.0027777777777778 \cdot \frac{z}{x \cdot y}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 79.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \color{blue}{\frac{-0.0027777777777778 \cdot z}{x \cdot y}}\right) \]

       2. *-commutative 79.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \frac{-0.0027777777777778 \cdot z}{\color{blue}{y \cdot x}}\right) \]

       3. associate-/r* 79.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \color{blue}{\frac{\frac{-0.0027777777777778 \cdot z}{y}}{x}}\right) \]

       4. *-commutative 79.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \frac{\frac{\color{blue}{z \cdot -0.0027777777777778}}{y}}{x}\right) \]

   12. Simplified 79.4%

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \color{blue}{\frac{\frac{z \cdot -0.0027777777777778}{y}}{x}}\right) \]

   13. Taylor expanded in z around 0 99.4%

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \color{blue}{\frac{0.083333333333333}{x \cdot y}} \]

   if 3.9999999999999999e99 < z < 6.09999999999999966e115

   1. Initial program 100.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 100.0%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 100.0%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 100.0%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 100.0%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 100.0%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 100.0%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 100.0%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 100.0%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 100.0%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   if 6.09999999999999966e115 < z < 6.2000000000000001e115

   1. Initial program 100.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. mul-1-neg 100.0%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. log-rec 100.0%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. remove-double-neg 100.0%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 100.0%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. +-commutative 100.0%

         \[\leadsto x \cdot \color{blue}{\left(-1 + \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} + \frac{0.083333333333333}{x} \]
3. Recombined 6 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+66}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+99}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \frac{0.083333333333333}{x \cdot y}\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}{x}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{-100}:\\ \;\;\;\;t\_0 + \frac{y \cdot \left(z \cdot \left(z + \frac{z \cdot 0.0007936500793651 - 0.0027777777777778}{y}\right)\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(z \cdot \left(z \cdot \left(0.0007936500793651 \cdot \frac{1}{x} + \frac{y}{x}\right) + 0.0027777777777778 \cdot \frac{-1}{x}\right) + 0.083333333333333 \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
   (if (<= x 2.6e-100)
     (+
      t_0
      (/
       (+
        (* y (* z (+ z (/ (- (* z 0.0007936500793651) 0.0027777777777778) y))))
        0.083333333333333)
       x))
     (+
      t_0
      (+
       (*
        z
        (+
         (* z (+ (* 0.0007936500793651 (/ 1.0 x)) (/ y x)))
         (* 0.0027777777777778 (/ -1.0 x))))
       (* 0.083333333333333 (/ 1.0 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double tmp;
	if (x <= 2.6e-100) {
		tmp = t_0 + (((y * (z * (z + (((z * 0.0007936500793651) - 0.0027777777777778) / y)))) + 0.083333333333333) / x);
	} else {
		tmp = t_0 + ((z * ((z * ((0.0007936500793651 * (1.0 / x)) + (y / x))) + (0.0027777777777778 * (-1.0 / x)))) + (0.083333333333333 * (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 - 0.5d0) * log(x)) - x) + 0.91893853320467d0
    if (x <= 2.6d-100) then
        tmp = t_0 + (((y * (z * (z + (((z * 0.0007936500793651d0) - 0.0027777777777778d0) / y)))) + 0.083333333333333d0) / x)
    else
        tmp = t_0 + ((z * ((z * ((0.0007936500793651d0 * (1.0d0 / x)) + (y / x))) + (0.0027777777777778d0 * ((-1.0d0) / x)))) + (0.083333333333333d0 * (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * Math.log(x)) - x) + 0.91893853320467;
	double tmp;
	if (x <= 2.6e-100) {
		tmp = t_0 + (((y * (z * (z + (((z * 0.0007936500793651) - 0.0027777777777778) / y)))) + 0.083333333333333) / x);
	} else {
		tmp = t_0 + ((z * ((z * ((0.0007936500793651 * (1.0 / x)) + (y / x))) + (0.0027777777777778 * (-1.0 / x)))) + (0.083333333333333 * (1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x - 0.5) * math.log(x)) - x) + 0.91893853320467
	tmp = 0
	if x <= 2.6e-100:
		tmp = t_0 + (((y * (z * (z + (((z * 0.0007936500793651) - 0.0027777777777778) / y)))) + 0.083333333333333) / x)
	else:
		tmp = t_0 + ((z * ((z * ((0.0007936500793651 * (1.0 / x)) + (y / x))) + (0.0027777777777778 * (-1.0 / x)))) + (0.083333333333333 * (1.0 / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)
	tmp = 0.0
	if (x <= 2.6e-100)
		tmp = Float64(t_0 + Float64(Float64(Float64(y * Float64(z * Float64(z + Float64(Float64(Float64(z * 0.0007936500793651) - 0.0027777777777778) / y)))) + 0.083333333333333) / x));
	else
		tmp = Float64(t_0 + Float64(Float64(z * Float64(Float64(z * Float64(Float64(0.0007936500793651 * Float64(1.0 / x)) + Float64(y / x))) + Float64(0.0027777777777778 * Float64(-1.0 / x)))) + Float64(0.083333333333333 * Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	tmp = 0.0;
	if (x <= 2.6e-100)
		tmp = t_0 + (((y * (z * (z + (((z * 0.0007936500793651) - 0.0027777777777778) / y)))) + 0.083333333333333) / x);
	else
		tmp = t_0 + ((z * ((z * ((0.0007936500793651 * (1.0 / x)) + (y / x))) + (0.0027777777777778 * (-1.0 / x)))) + (0.083333333333333 * (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[x, 2.6e-100], N[(t$95$0 + N[(N[(N[(y * N[(z * N[(z + N[(N[(N[(z * 0.0007936500793651), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(z * N[(N[(z * N[(N[(0.0007936500793651 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0027777777777778 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5999999999999998e-100

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 99.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      2. *-un-lft-identity 99.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{1 \cdot \left(\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\right)}}{\sqrt{x} \cdot \sqrt{x}} \]

      3. times-frac 99.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\sqrt{x}} \cdot \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{\sqrt{x}}} \]

      4. *-commutative 99.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\sqrt{x}} \cdot \frac{\color{blue}{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)} + 0.083333333333333}{\sqrt{x}} \]

      5. fma-undefine 99.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\sqrt{x}} \cdot \frac{\color{blue}{\mathsf{fma}\left(z, \left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, 0.083333333333333\right)}}{\sqrt{x}} \]

      6. fma-neg 99.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\sqrt{x}} \cdot \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)}, 0.083333333333333\right)}{\sqrt{x}} \]

      7. metadata-eval 99.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\sqrt{x}} \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, \color{blue}{-0.0027777777777778}\right), 0.083333333333333\right)}{\sqrt{x}} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\sqrt{x}} \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt{x}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}}} \]

      2. *-lft-identity 99.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt{x}}}}{\sqrt{x}} \]

      3. fma-define 99.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778}, 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}} \]

      4. +-commutative 99.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{\left(0.0007936500793651 + y\right)} \cdot z + -0.0027777777777778, 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}} \]

      5. *-commutative 99.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(0.0007936500793651 + y\right)} + -0.0027777777777778, 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}} \]

      6. fma-define 99.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}, 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}} \]

   6. Simplified 99.6%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt{x}}}{\sqrt{x}}} \]

   7. Taylor expanded in y around inf 57.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{y \cdot \left(\frac{0.083333333333333}{x \cdot y} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x \cdot y} + \frac{{z}^{2}}{x}\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 57.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{0.083333333333333}{\color{blue}{y \cdot x}} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x \cdot y} + \frac{{z}^{2}}{x}\right)\right) \]

      2. associate-/r* 57.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\color{blue}{\frac{\frac{0.083333333333333}{y}}{x}} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x \cdot y} + \frac{{z}^{2}}{x}\right)\right) \]

      3. associate-/l* 60.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \left(\color{blue}{z \cdot \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x \cdot y}} + \frac{{z}^{2}}{x}\right)\right) \]

      4. fma-define 63.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \color{blue}{\mathsf{fma}\left(z, \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x \cdot y}, \frac{{z}^{2}}{x}\right)}\right) \]

      5. *-commutative 63.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \mathsf{fma}\left(z, \frac{\color{blue}{z \cdot 0.0007936500793651} - 0.0027777777777778}{x \cdot y}, \frac{{z}^{2}}{x}\right)\right) \]

      6. fma-neg 63.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \mathsf{fma}\left(z, \frac{\color{blue}{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}}{x \cdot y}, \frac{{z}^{2}}{x}\right)\right) \]

      7. metadata-eval 63.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, 0.0007936500793651, \color{blue}{-0.0027777777777778}\right)}{x \cdot y}, \frac{{z}^{2}}{x}\right)\right) \]

   9. Simplified 63.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{y \cdot \left(\frac{\frac{0.083333333333333}{y}}{x} + \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x \cdot y}, \frac{{z}^{2}}{x}\right)\right)} \]

   10. Taylor expanded in x around 0 81.9%

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{y \cdot \left(0.083333333333333 \cdot \frac{1}{y} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{y} + {z}^{2}\right)\right)}{x}} \]
   11. Step-by-step derivation

       1. *-commutative 81.9%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(0.083333333333333 \cdot \frac{1}{y} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{y} + {z}^{2}\right)\right) \cdot y}}{x} \]

       2. associate-/l* 66.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(0.083333333333333 \cdot \frac{1}{y} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{y} + {z}^{2}\right)\right) \cdot \frac{y}{x}} \]

       3. associate-*r/ 66.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{y}} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{y} + {z}^{2}\right)\right) \cdot \frac{y}{x} \]

       4. metadata-eval 66.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{\color{blue}{0.083333333333333}}{y} + \left(\frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{y} + {z}^{2}\right)\right) \cdot \frac{y}{x} \]

       5. +-commutative 66.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{0.083333333333333}{y} + \color{blue}{\left({z}^{2} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{y}\right)}\right) \cdot \frac{y}{x} \]

       6. unpow2 66.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{0.083333333333333}{y} + \left(\color{blue}{z \cdot z} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{y}\right)\right) \cdot \frac{y}{x} \]

       7. associate-/l* 71.7%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{0.083333333333333}{y} + \left(z \cdot z + \color{blue}{z \cdot \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{y}}\right)\right) \cdot \frac{y}{x} \]

       8. distribute-lft-out 84.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{0.083333333333333}{y} + \color{blue}{z \cdot \left(z + \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{y}\right)}\right) \cdot \frac{y}{x} \]

       9. *-commutative 84.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{0.083333333333333}{y} + z \cdot \left(z + \frac{\color{blue}{z \cdot 0.0007936500793651} - 0.0027777777777778}{y}\right)\right) \cdot \frac{y}{x} \]

       10. fma-neg 84.4%

           \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{0.083333333333333}{y} + z \cdot \left(z + \frac{\color{blue}{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}}{y}\right)\right) \cdot \frac{y}{x} \]

       11. metadata-eval 84.4%

           \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{0.083333333333333}{y} + z \cdot \left(z + \frac{\mathsf{fma}\left(z, 0.0007936500793651, \color{blue}{-0.0027777777777778}\right)}{y}\right)\right) \cdot \frac{y}{x} \]

   12. Simplified 84.4%

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\frac{0.083333333333333}{y} + z \cdot \left(z + \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{y}\right)\right) \cdot \frac{y}{x}} \]

   13. Taylor expanded in x around 0 99.6%

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{y \cdot \left(z \cdot \left(\left(z + 0.0007936500793651 \cdot \frac{z}{y}\right) - 0.0027777777777778 \cdot \frac{1}{y}\right) + 0.083333333333333 \cdot \frac{1}{y}\right)}{x}} \]
   14. Step-by-step derivation

       1. distribute-rgt-in 99.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(z \cdot \left(\left(z + 0.0007936500793651 \cdot \frac{z}{y}\right) - 0.0027777777777778 \cdot \frac{1}{y}\right)\right) \cdot y + \left(0.083333333333333 \cdot \frac{1}{y}\right) \cdot y}}{x} \]

       2. associate--l+ 99.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot \color{blue}{\left(z + \left(0.0007936500793651 \cdot \frac{z}{y} - 0.0027777777777778 \cdot \frac{1}{y}\right)\right)}\right) \cdot y + \left(0.083333333333333 \cdot \frac{1}{y}\right) \cdot y}{x} \]

       3. associate-*r/ 99.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot \left(z + \left(\color{blue}{\frac{0.0007936500793651 \cdot z}{y}} - 0.0027777777777778 \cdot \frac{1}{y}\right)\right)\right) \cdot y + \left(0.083333333333333 \cdot \frac{1}{y}\right) \cdot y}{x} \]

       4. un-div-inv 99.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot \left(z + \left(\frac{0.0007936500793651 \cdot z}{y} - \color{blue}{\frac{0.0027777777777778}{y}}\right)\right)\right) \cdot y + \left(0.083333333333333 \cdot \frac{1}{y}\right) \cdot y}{x} \]

       5. sub-div 99.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot \left(z + \color{blue}{\frac{0.0007936500793651 \cdot z - 0.0027777777777778}{y}}\right)\right) \cdot y + \left(0.083333333333333 \cdot \frac{1}{y}\right) \cdot y}{x} \]

       6. un-div-inv 99.7%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot \left(z + \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{y}\right)\right) \cdot y + \color{blue}{\frac{0.083333333333333}{y}} \cdot y}{x} \]

   15. Applied egg-rr 99.7%

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(z \cdot \left(z + \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{y}\right)\right) \cdot y + \frac{0.083333333333333}{y} \cdot y}}{x} \]

   16. Taylor expanded in y around 0 99.7%

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot \left(z + \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{y}\right)\right) \cdot y + \color{blue}{0.083333333333333}}{x} \]

   if 2.5999999999999998e-100 < x

   1. Initial program 92.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 98.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(z \cdot \left(z \cdot \left(0.0007936500793651 \cdot \frac{1}{x} + \frac{y}{x}\right) - 0.0027777777777778 \cdot \frac{1}{x}\right) + 0.083333333333333 \cdot \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-100}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{y \cdot \left(z \cdot \left(z + \frac{z \cdot 0.0007936500793651 - 0.0027777777777778}{y}\right)\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(z \cdot \left(z \cdot \left(0.0007936500793651 \cdot \frac{1}{x} + \frac{y}{x}\right) + 0.0027777777777778 \cdot \frac{-1}{x}\right) + 0.083333333333333 \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.0007936500793651 + y}{x}\\ t_1 := \left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot t\_0\right)\\ \mathbf{if}\;z \leq -1.06 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+67}:\\ \;\;\;\;t\_0 \cdot {z}^{2}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ 0.0007936500793651 y) x))
        (t_1 (+ (+ 0.91893853320467 (* (log x) -0.5)) (* z (* z t_0)))))
   (if (<= z -1.06e+61)
     t_1
     (if (<= z 4.1e+16)
       (+
        (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
        (* 0.083333333333333 (/ 1.0 x)))
       (if (<= z 7.5e+67)
         (* t_0 (pow z 2.0))
         (if (<= z 1.05e+68)
           (+ (* x (+ (log x) -1.0)) (/ 0.083333333333333 x))
           t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) / x;
	double t_1 = (0.91893853320467 + (log(x) * -0.5)) + (z * (z * t_0));
	double tmp;
	if (z <= -1.06e+61) {
		tmp = t_1;
	} else if (z <= 4.1e+16) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 * (1.0 / x));
	} else if (z <= 7.5e+67) {
		tmp = t_0 * pow(z, 2.0);
	} else if (z <= 1.05e+68) {
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
	} else {
		tmp = t_1;
	}
	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 = (0.0007936500793651d0 + y) / x
    t_1 = (0.91893853320467d0 + (log(x) * (-0.5d0))) + (z * (z * t_0))
    if (z <= (-1.06d+61)) then
        tmp = t_1
    else if (z <= 4.1d+16) then
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (0.083333333333333d0 * (1.0d0 / x))
    else if (z <= 7.5d+67) then
        tmp = t_0 * (z ** 2.0d0)
    else if (z <= 1.05d+68) then
        tmp = (x * (log(x) + (-1.0d0))) + (0.083333333333333d0 / x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) / x;
	double t_1 = (0.91893853320467 + (Math.log(x) * -0.5)) + (z * (z * t_0));
	double tmp;
	if (z <= -1.06e+61) {
		tmp = t_1;
	} else if (z <= 4.1e+16) {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 * (1.0 / x));
	} else if (z <= 7.5e+67) {
		tmp = t_0 * Math.pow(z, 2.0);
	} else if (z <= 1.05e+68) {
		tmp = (x * (Math.log(x) + -1.0)) + (0.083333333333333 / x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.0007936500793651 + y) / x
	t_1 = (0.91893853320467 + (math.log(x) * -0.5)) + (z * (z * t_0))
	tmp = 0
	if z <= -1.06e+61:
		tmp = t_1
	elif z <= 4.1e+16:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 * (1.0 / x))
	elif z <= 7.5e+67:
		tmp = t_0 * math.pow(z, 2.0)
	elif z <= 1.05e+68:
		tmp = (x * (math.log(x) + -1.0)) + (0.083333333333333 / x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0007936500793651 + y) / x)
	t_1 = Float64(Float64(0.91893853320467 + Float64(log(x) * -0.5)) + Float64(z * Float64(z * t_0)))
	tmp = 0.0
	if (z <= -1.06e+61)
		tmp = t_1;
	elseif (z <= 4.1e+16)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(0.083333333333333 * Float64(1.0 / x)));
	elseif (z <= 7.5e+67)
		tmp = Float64(t_0 * (z ^ 2.0));
	elseif (z <= 1.05e+68)
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(0.083333333333333 / x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.0007936500793651 + y) / x;
	t_1 = (0.91893853320467 + (log(x) * -0.5)) + (z * (z * t_0));
	tmp = 0.0;
	if (z <= -1.06e+61)
		tmp = t_1;
	elseif (z <= 4.1e+16)
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 * (1.0 / x));
	elseif (z <= 7.5e+67)
		tmp = t_0 * (z ^ 2.0);
	elseif (z <= 1.05e+68)
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.06e+61], t$95$1, If[LessEqual[z, 4.1e+16], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.5e+67], N[(t$95$0 * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+68], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.0599999999999999e61 or 1.05e68 < z

   1. Initial program 89.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 81.1%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 81.1%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 81.1%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 81.1%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 81.1%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 81.1%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 81.1%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 81.1%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 83.0%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 83.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 83.0%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 83.0%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 82.1%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* 93.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

      2. unpow2 93.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(z \cdot z\right)} \cdot \frac{0.0007936500793651 + y}{x} \]

      3. associate-*l* 99.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]

   10. Applied egg-rr 84.7%

       \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]

   if -1.0599999999999999e61 < z < 4.1e16

   1. Initial program 98.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]
   4. Step-by-step derivation

      1. clear-num 85.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. inv-pow 85.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   5. Applied egg-rr 85.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 85.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. associate-/r/ 85.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   7. Simplified 85.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   if 4.1e16 < z < 7.5000000000000005e67

   1. Initial program 99.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 79.2%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 79.2%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 79.2%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 79.2%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 79.2%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 79.2%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 79.2%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 79.2%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 76.0%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 76.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 76.0%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 76.0%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 76.2%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 76.5%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   10. Step-by-step derivation

       1. associate-/l* 76.6%

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   11. Simplified 76.6%

       \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   if 7.5000000000000005e67 < z < 1.05e68

   1. Initial program 100.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. mul-1-neg 100.0%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. log-rec 100.0%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. remove-double-neg 100.0%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 100.0%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. +-commutative 100.0%

         \[\leadsto x \cdot \color{blue}{\left(-1 + \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} + \frac{0.083333333333333}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+61}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.0007936500793651 + y}{x}\\ t_1 := \left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot t\_0\right)\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+66}:\\ \;\;\;\;t\_0 \cdot {z}^{2}\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{+81}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ 0.0007936500793651 y) x))
        (t_1 (+ (+ 0.91893853320467 (* (log x) -0.5)) (* z (* z t_0)))))
   (if (<= z -4.5e+61)
     t_1
     (if (<= z 5e+16)
       (+
        (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
        (/ 0.083333333333333 x))
       (if (<= z 5.4e+66)
         (* t_0 (pow z 2.0))
         (if (<= z 7.3e+81)
           (+ (- (* x (log x)) x) (/ 0.083333333333333 x))
           t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) / x;
	double t_1 = (0.91893853320467 + (log(x) * -0.5)) + (z * (z * t_0));
	double tmp;
	if (z <= -4.5e+61) {
		tmp = t_1;
	} else if (z <= 5e+16) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	} else if (z <= 5.4e+66) {
		tmp = t_0 * pow(z, 2.0);
	} else if (z <= 7.3e+81) {
		tmp = ((x * log(x)) - x) + (0.083333333333333 / x);
	} else {
		tmp = t_1;
	}
	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 = (0.0007936500793651d0 + y) / x
    t_1 = (0.91893853320467d0 + (log(x) * (-0.5d0))) + (z * (z * t_0))
    if (z <= (-4.5d+61)) then
        tmp = t_1
    else if (z <= 5d+16) then
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (0.083333333333333d0 / x)
    else if (z <= 5.4d+66) then
        tmp = t_0 * (z ** 2.0d0)
    else if (z <= 7.3d+81) then
        tmp = ((x * log(x)) - x) + (0.083333333333333d0 / x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) / x;
	double t_1 = (0.91893853320467 + (Math.log(x) * -0.5)) + (z * (z * t_0));
	double tmp;
	if (z <= -4.5e+61) {
		tmp = t_1;
	} else if (z <= 5e+16) {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	} else if (z <= 5.4e+66) {
		tmp = t_0 * Math.pow(z, 2.0);
	} else if (z <= 7.3e+81) {
		tmp = ((x * Math.log(x)) - x) + (0.083333333333333 / x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.0007936500793651 + y) / x
	t_1 = (0.91893853320467 + (math.log(x) * -0.5)) + (z * (z * t_0))
	tmp = 0
	if z <= -4.5e+61:
		tmp = t_1
	elif z <= 5e+16:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x)
	elif z <= 5.4e+66:
		tmp = t_0 * math.pow(z, 2.0)
	elif z <= 7.3e+81:
		tmp = ((x * math.log(x)) - x) + (0.083333333333333 / x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0007936500793651 + y) / x)
	t_1 = Float64(Float64(0.91893853320467 + Float64(log(x) * -0.5)) + Float64(z * Float64(z * t_0)))
	tmp = 0.0
	if (z <= -4.5e+61)
		tmp = t_1;
	elseif (z <= 5e+16)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(0.083333333333333 / x));
	elseif (z <= 5.4e+66)
		tmp = Float64(t_0 * (z ^ 2.0));
	elseif (z <= 7.3e+81)
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(0.083333333333333 / x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.0007936500793651 + y) / x;
	t_1 = (0.91893853320467 + (log(x) * -0.5)) + (z * (z * t_0));
	tmp = 0.0;
	if (z <= -4.5e+61)
		tmp = t_1;
	elseif (z <= 5e+16)
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	elseif (z <= 5.4e+66)
		tmp = t_0 * (z ^ 2.0);
	elseif (z <= 7.3e+81)
		tmp = ((x * log(x)) - x) + (0.083333333333333 / x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.5e+61], t$95$1, If[LessEqual[z, 5e+16], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.4e+66], N[(t$95$0 * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.3e+81], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.5e61 or 7.2999999999999997e81 < z

   1. Initial program 89.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 81.9%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 81.9%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 81.9%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 81.9%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 81.9%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 81.9%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 81.9%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 81.9%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 83.8%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 83.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 83.8%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 82.9%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* 93.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

      2. unpow2 93.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(z \cdot z\right)} \cdot \frac{0.0007936500793651 + y}{x} \]

      3. associate-*l* 99.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]

   10. Applied egg-rr 85.5%

       \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]

   if -4.5e61 < z < 5e16

   1. Initial program 98.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]

   if 5e16 < z < 5.4e66

   1. Initial program 99.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 79.2%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 79.2%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 79.2%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 79.2%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 79.2%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 79.2%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 79.2%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 79.2%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 76.0%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 76.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 76.0%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 76.0%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 76.2%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 76.5%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   10. Step-by-step derivation

       1. associate-/l* 76.6%

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   11. Simplified 76.6%

       \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   if 5.4e66 < z < 7.2999999999999997e81

   1. Initial program 100.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]

   4. Taylor expanded in x around inf 99.2%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. sub-neg 99.2%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. mul-1-neg 99.2%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. log-rec 99.2%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. remove-double-neg 99.2%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 99.2%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. +-commutative 99.2%

         \[\leadsto x \cdot \color{blue}{\left(-1 + \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   6. Simplified 99.2%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} + \frac{0.083333333333333}{x} \]
   7. Step-by-step derivation

      1. +-commutative 99.2%

         \[\leadsto x \cdot \color{blue}{\left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

      2. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \log x + x \cdot -1\right)} + \frac{0.083333333333333}{x} \]

      3. *-commutative 100.0%

         \[\leadsto \left(x \cdot \log x + \color{blue}{-1 \cdot x}\right) + \frac{0.083333333333333}{x} \]

      4. neg-mul-1 100.0%

         \[\leadsto \left(x \cdot \log x + \color{blue}{\left(-x\right)}\right) + \frac{0.083333333333333}{x} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \log x + \left(-x\right)\right)} + \frac{0.083333333333333}{x} \]
   9. Step-by-step derivation

      1. unsub-neg 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \log x - x\right)} + \frac{0.083333333333333}{x} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(x \cdot \log x - x\right)} + \frac{0.083333333333333}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+66}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{+81}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}{x}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+67} \lor \neg \left(z \leq 1.7 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.35e+61)
   (/ (* (+ 0.0007936500793651 y) (pow z 2.0)) x)
   (if (<= z 6e+17)
     (+
      (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
      (/ 0.083333333333333 x))
     (if (or (<= z 7.5e+67) (not (<= z 1.7e+92)))
       (* (/ (+ 0.0007936500793651 y) x) (pow z 2.0))
       (+ (- (* x (log x)) x) (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e+61) {
		tmp = ((0.0007936500793651 + y) * pow(z, 2.0)) / x;
	} else if (z <= 6e+17) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	} else if ((z <= 7.5e+67) || !(z <= 1.7e+92)) {
		tmp = ((0.0007936500793651 + y) / x) * pow(z, 2.0);
	} else {
		tmp = ((x * log(x)) - x) + (0.083333333333333 / 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 <= (-1.35d+61)) then
        tmp = ((0.0007936500793651d0 + y) * (z ** 2.0d0)) / x
    else if (z <= 6d+17) then
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (0.083333333333333d0 / x)
    else if ((z <= 7.5d+67) .or. (.not. (z <= 1.7d+92))) then
        tmp = ((0.0007936500793651d0 + y) / x) * (z ** 2.0d0)
    else
        tmp = ((x * log(x)) - x) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e+61) {
		tmp = ((0.0007936500793651 + y) * Math.pow(z, 2.0)) / x;
	} else if (z <= 6e+17) {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	} else if ((z <= 7.5e+67) || !(z <= 1.7e+92)) {
		tmp = ((0.0007936500793651 + y) / x) * Math.pow(z, 2.0);
	} else {
		tmp = ((x * Math.log(x)) - x) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.35e+61:
		tmp = ((0.0007936500793651 + y) * math.pow(z, 2.0)) / x
	elif z <= 6e+17:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x)
	elif (z <= 7.5e+67) or not (z <= 1.7e+92):
		tmp = ((0.0007936500793651 + y) / x) * math.pow(z, 2.0)
	else:
		tmp = ((x * math.log(x)) - x) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.35e+61)
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) * (z ^ 2.0)) / x);
	elseif (z <= 6e+17)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(0.083333333333333 / x));
	elseif ((z <= 7.5e+67) || !(z <= 1.7e+92))
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) / x) * (z ^ 2.0));
	else
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.35e+61)
		tmp = ((0.0007936500793651 + y) * (z ^ 2.0)) / x;
	elseif (z <= 6e+17)
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	elseif ((z <= 7.5e+67) || ~((z <= 1.7e+92)))
		tmp = ((0.0007936500793651 + y) / x) * (z ^ 2.0);
	else
		tmp = ((x * log(x)) - x) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.35e+61], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[z, 6e+17], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 7.5e+67], N[Not[LessEqual[z, 1.7e+92]], $MachinePrecision]], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.3500000000000001e61

   1. Initial program 91.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 82.1%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 82.1%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 82.1%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 82.1%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 83.9%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 83.9%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 83.9%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 83.9%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 82.2%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 82.2%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   if -1.3500000000000001e61 < z < 6e17

   1. Initial program 98.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]

   if 6e17 < z < 7.5000000000000005e67 or 1.6999999999999999e92 < z

   1. Initial program 91.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 82.7%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 82.7%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 82.7%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 82.7%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 82.7%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 82.7%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 82.7%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 82.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 83.9%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 83.9%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 83.9%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 83.9%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 84.0%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 84.0%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   10. Step-by-step derivation

       1. associate-/l* 85.7%

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   11. Simplified 85.7%

       \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   if 7.5000000000000005e67 < z < 1.6999999999999999e92

   1. Initial program 66.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 99.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]

   4. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. sub-neg 66.2%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. mul-1-neg 66.2%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. log-rec 66.2%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. remove-double-neg 66.2%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 66.2%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. +-commutative 66.2%

         \[\leadsto x \cdot \color{blue}{\left(-1 + \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   6. Simplified 99.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} + \frac{0.083333333333333}{x} \]
   7. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto x \cdot \color{blue}{\left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

      2. distribute-lft-in 99.5%

         \[\leadsto \color{blue}{\left(x \cdot \log x + x \cdot -1\right)} + \frac{0.083333333333333}{x} \]

      3. *-commutative 99.5%

         \[\leadsto \left(x \cdot \log x + \color{blue}{-1 \cdot x}\right) + \frac{0.083333333333333}{x} \]

      4. neg-mul-1 99.5%

         \[\leadsto \left(x \cdot \log x + \color{blue}{\left(-x\right)}\right) + \frac{0.083333333333333}{x} \]

   8. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(x \cdot \log x + \left(-x\right)\right)} + \frac{0.083333333333333}{x} \]
   9. Step-by-step derivation

      1. unsub-neg 99.5%

         \[\leadsto \color{blue}{\left(x \cdot \log x - x\right)} + \frac{0.083333333333333}{x} \]

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\left(x \cdot \log x - x\right)} + \frac{0.083333333333333}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}{x}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+67} \lor \neg \left(z \leq 1.7 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}{x}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+16} \lor \neg \left(z \leq 7.5 \cdot 10^{+67}\right) \land z \leq 3.45 \cdot 10^{+71}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+61)
   (/ (* (+ 0.0007936500793651 y) (pow z 2.0)) x)
   (if (or (<= z 6.8e+16) (and (not (<= z 7.5e+67)) (<= z 3.45e+71)))
     (+ (- (* x (log x)) x) (/ 0.083333333333333 x))
     (* (/ (+ 0.0007936500793651 y) x) (pow z 2.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+61) {
		tmp = ((0.0007936500793651 + y) * pow(z, 2.0)) / x;
	} else if ((z <= 6.8e+16) || (!(z <= 7.5e+67) && (z <= 3.45e+71))) {
		tmp = ((x * log(x)) - x) + (0.083333333333333 / x);
	} else {
		tmp = ((0.0007936500793651 + y) / x) * pow(z, 2.0);
	}
	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) :: tmp
    if (z <= (-1d+61)) then
        tmp = ((0.0007936500793651d0 + y) * (z ** 2.0d0)) / x
    else if ((z <= 6.8d+16) .or. (.not. (z <= 7.5d+67)) .and. (z <= 3.45d+71)) then
        tmp = ((x * log(x)) - x) + (0.083333333333333d0 / x)
    else
        tmp = ((0.0007936500793651d0 + y) / x) * (z ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+61) {
		tmp = ((0.0007936500793651 + y) * Math.pow(z, 2.0)) / x;
	} else if ((z <= 6.8e+16) || (!(z <= 7.5e+67) && (z <= 3.45e+71))) {
		tmp = ((x * Math.log(x)) - x) + (0.083333333333333 / x);
	} else {
		tmp = ((0.0007936500793651 + y) / x) * Math.pow(z, 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1e+61:
		tmp = ((0.0007936500793651 + y) * math.pow(z, 2.0)) / x
	elif (z <= 6.8e+16) or (not (z <= 7.5e+67) and (z <= 3.45e+71)):
		tmp = ((x * math.log(x)) - x) + (0.083333333333333 / x)
	else:
		tmp = ((0.0007936500793651 + y) / x) * math.pow(z, 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e+61)
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) * (z ^ 2.0)) / x);
	elseif ((z <= 6.8e+16) || (!(z <= 7.5e+67) && (z <= 3.45e+71)))
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(0.083333333333333 / x));
	else
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) / x) * (z ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e+61)
		tmp = ((0.0007936500793651 + y) * (z ^ 2.0)) / x;
	elseif ((z <= 6.8e+16) || (~((z <= 7.5e+67)) && (z <= 3.45e+71)))
		tmp = ((x * log(x)) - x) + (0.083333333333333 / x);
	else
		tmp = ((0.0007936500793651 + y) / x) * (z ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e+61], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[z, 6.8e+16], And[N[Not[LessEqual[z, 7.5e+67]], $MachinePrecision], LessEqual[z, 3.45e+71]]], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.99999999999999949e60

   1. Initial program 91.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 82.1%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 82.1%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 82.1%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 82.1%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 83.9%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 83.9%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 83.9%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 83.9%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 82.2%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 82.2%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   if -9.99999999999999949e60 < z < 6.8e16 or 7.5000000000000005e67 < z < 3.44999999999999987e71

   1. Initial program 98.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.2%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]

   4. Taylor expanded in x around inf 82.8%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. sub-neg 95.5%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. mul-1-neg 95.5%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. log-rec 95.5%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. remove-double-neg 95.5%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 95.5%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. +-commutative 95.5%

         \[\leadsto x \cdot \color{blue}{\left(-1 + \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   6. Simplified 82.8%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} + \frac{0.083333333333333}{x} \]
   7. Step-by-step derivation

      1. +-commutative 82.8%

         \[\leadsto x \cdot \color{blue}{\left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

      2. distribute-lft-in 82.8%

         \[\leadsto \color{blue}{\left(x \cdot \log x + x \cdot -1\right)} + \frac{0.083333333333333}{x} \]

      3. *-commutative 82.8%

         \[\leadsto \left(x \cdot \log x + \color{blue}{-1 \cdot x}\right) + \frac{0.083333333333333}{x} \]

      4. neg-mul-1 82.8%

         \[\leadsto \left(x \cdot \log x + \color{blue}{\left(-x\right)}\right) + \frac{0.083333333333333}{x} \]

   8. Applied egg-rr 82.8%

      \[\leadsto \color{blue}{\left(x \cdot \log x + \left(-x\right)\right)} + \frac{0.083333333333333}{x} \]
   9. Step-by-step derivation

      1. unsub-neg 82.8%

         \[\leadsto \color{blue}{\left(x \cdot \log x - x\right)} + \frac{0.083333333333333}{x} \]

   10. Applied egg-rr 82.8%

       \[\leadsto \color{blue}{\left(x \cdot \log x - x\right)} + \frac{0.083333333333333}{x} \]

   if 6.8e16 < z < 7.5000000000000005e67 or 3.44999999999999987e71 < z

   1. Initial program 89.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 81.3%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 81.3%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 81.3%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 81.3%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 81.3%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 81.3%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 81.3%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 81.3%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 82.4%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 82.4%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 82.4%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 82.4%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 82.5%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 82.5%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   10. Step-by-step derivation

       1. associate-/l* 84.2%

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   11. Simplified 84.2%

       \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}{x}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+16} \lor \neg \left(z \leq 7.5 \cdot 10^{+67}\right) \land z \leq 3.45 \cdot 10^{+71}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}{x}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+16} \lor \neg \left(z \leq 7.5 \cdot 10^{+67}\right) \land z \leq 3.45 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+61)
   (/ (* (+ 0.0007936500793651 y) (pow z 2.0)) x)
   (if (or (<= z 4.5e+16) (and (not (<= z 7.5e+67)) (<= z 3.45e+71)))
     (+ (* x (+ (log x) -1.0)) (/ 0.083333333333333 x))
     (* (/ (+ 0.0007936500793651 y) x) (pow z 2.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+61) {
		tmp = ((0.0007936500793651 + y) * pow(z, 2.0)) / x;
	} else if ((z <= 4.5e+16) || (!(z <= 7.5e+67) && (z <= 3.45e+71))) {
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
	} else {
		tmp = ((0.0007936500793651 + y) / x) * pow(z, 2.0);
	}
	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) :: tmp
    if (z <= (-1d+61)) then
        tmp = ((0.0007936500793651d0 + y) * (z ** 2.0d0)) / x
    else if ((z <= 4.5d+16) .or. (.not. (z <= 7.5d+67)) .and. (z <= 3.45d+71)) then
        tmp = (x * (log(x) + (-1.0d0))) + (0.083333333333333d0 / x)
    else
        tmp = ((0.0007936500793651d0 + y) / x) * (z ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+61) {
		tmp = ((0.0007936500793651 + y) * Math.pow(z, 2.0)) / x;
	} else if ((z <= 4.5e+16) || (!(z <= 7.5e+67) && (z <= 3.45e+71))) {
		tmp = (x * (Math.log(x) + -1.0)) + (0.083333333333333 / x);
	} else {
		tmp = ((0.0007936500793651 + y) / x) * Math.pow(z, 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1e+61:
		tmp = ((0.0007936500793651 + y) * math.pow(z, 2.0)) / x
	elif (z <= 4.5e+16) or (not (z <= 7.5e+67) and (z <= 3.45e+71)):
		tmp = (x * (math.log(x) + -1.0)) + (0.083333333333333 / x)
	else:
		tmp = ((0.0007936500793651 + y) / x) * math.pow(z, 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e+61)
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) * (z ^ 2.0)) / x);
	elseif ((z <= 4.5e+16) || (!(z <= 7.5e+67) && (z <= 3.45e+71)))
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(0.083333333333333 / x));
	else
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) / x) * (z ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e+61)
		tmp = ((0.0007936500793651 + y) * (z ^ 2.0)) / x;
	elseif ((z <= 4.5e+16) || (~((z <= 7.5e+67)) && (z <= 3.45e+71)))
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
	else
		tmp = ((0.0007936500793651 + y) / x) * (z ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e+61], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[z, 4.5e+16], And[N[Not[LessEqual[z, 7.5e+67]], $MachinePrecision], LessEqual[z, 3.45e+71]]], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.99999999999999949e60

   1. Initial program 91.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 82.1%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 82.1%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 82.1%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 82.1%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 82.1%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 83.9%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 83.9%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 83.9%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 83.9%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 82.2%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 82.2%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   if -9.99999999999999949e60 < z < 4.5e16 or 7.5000000000000005e67 < z < 3.44999999999999987e71

   1. Initial program 98.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.2%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]

   4. Taylor expanded in x around inf 82.8%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. sub-neg 95.5%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. mul-1-neg 95.5%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. log-rec 95.5%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. remove-double-neg 95.5%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 95.5%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. +-commutative 95.5%

         \[\leadsto x \cdot \color{blue}{\left(-1 + \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   6. Simplified 82.8%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} + \frac{0.083333333333333}{x} \]

   if 4.5e16 < z < 7.5000000000000005e67 or 3.44999999999999987e71 < z

   1. Initial program 89.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 81.3%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 81.3%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 81.3%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 81.3%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 81.3%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 81.3%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 81.3%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 81.3%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 82.4%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 82.4%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 82.4%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 82.4%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 82.5%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 82.5%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   10. Step-by-step derivation

       1. associate-/l* 84.2%

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   11. Simplified 84.2%

       \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}{x}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+16} \lor \neg \left(z \leq 7.5 \cdot 10^{+67}\right) \land z \leq 3.45 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.9 \cdot 10^{-12}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + \left(\frac{z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} + 0.083333333333333 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.9e-12)
   (+
    (+ 0.91893853320467 (* (log x) -0.5))
    (+
     (/ (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)) x)
     (* 0.083333333333333 (/ 1.0 x))))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (* z (* z (/ (+ 0.0007936500793651 y) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.9e-12) {
		tmp = (0.91893853320467 + (log(x) * -0.5)) + (((z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)) / x) + (0.083333333333333 * (1.0 / x)));
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (z * (z * ((0.0007936500793651 + y) / 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 (x <= 5.9d-12) then
        tmp = (0.91893853320467d0 + (log(x) * (-0.5d0))) + (((z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0)) / x) + (0.083333333333333d0 * (1.0d0 / x)))
    else
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (z * (z * ((0.0007936500793651d0 + y) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.9e-12) {
		tmp = (0.91893853320467 + (Math.log(x) * -0.5)) + (((z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)) / x) + (0.083333333333333 * (1.0 / x)));
	} else {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (z * (z * ((0.0007936500793651 + y) / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 5.9e-12:
		tmp = (0.91893853320467 + (math.log(x) * -0.5)) + (((z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)) / x) + (0.083333333333333 * (1.0 / x)))
	else:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (z * (z * ((0.0007936500793651 + y) / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.9e-12)
		tmp = Float64(Float64(0.91893853320467 + Float64(log(x) * -0.5)) + Float64(Float64(Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778)) / x) + Float64(0.083333333333333 * Float64(1.0 / x))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(z * Float64(z * Float64(Float64(0.0007936500793651 + y) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5.9e-12)
		tmp = (0.91893853320467 + (log(x) * -0.5)) + (((z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)) / x) + (0.083333333333333 * (1.0 / x)));
	else
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (z * (z * ((0.0007936500793651 + y) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.9e-12], N[(N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(z * N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.9e-12

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 99.6%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 99.6%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 99.6%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 99.6%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around 0 86.9%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\left(z \cdot \left(z \cdot \left(0.0007936500793651 \cdot \frac{1}{x} + \frac{y}{x}\right) - 0.0027777777777778 \cdot \frac{1}{x}\right) + 0.083333333333333 \cdot \frac{1}{x}\right)} \]

   9. Taylor expanded in x around 0 99.7%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \left(\color{blue}{\frac{z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} + 0.083333333333333 \cdot \frac{1}{x}\right) \]

   if 5.9e-12 < x

   1. Initial program 90.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 87.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* 92.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

      2. unpow2 92.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(z \cdot z\right)} \cdot \frac{0.0007936500793651 + y}{x} \]

      3. associate-*l* 97.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]

   5. Applied egg-rr 97.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.9 \cdot 10^{-12}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + \left(\frac{z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} + 0.083333333333333 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+209}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6e+209)
   (+
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))
     x)
    (* x (+ (log x) -1.0)))
   (+ (- (* x (log x)) x) (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6e+209) {
		tmp = ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x) + (x * (log(x) + -1.0));
	} else {
		tmp = ((x * log(x)) - x) + (0.083333333333333 / 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 (x <= 6d+209) then
        tmp = ((0.083333333333333d0 + (z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0))) / x) + (x * (log(x) + (-1.0d0)))
    else
        tmp = ((x * log(x)) - x) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 6e+209) {
		tmp = ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x) + (x * (Math.log(x) + -1.0));
	} else {
		tmp = ((x * Math.log(x)) - x) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 6e+209:
		tmp = ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x) + (x * (math.log(x) + -1.0))
	else:
		tmp = ((x * math.log(x)) - x) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6e+209)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))) / x) + Float64(x * Float64(log(x) + -1.0)));
	else
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 6e+209)
		tmp = ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x) + (x * (log(x) + -1.0));
	else
		tmp = ((x * log(x)) - x) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6e+209], N[(N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999971e209

   1. Initial program 98.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 96.5%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   4. Step-by-step derivation

      1. sub-neg 96.5%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. mul-1-neg 96.5%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. log-rec 96.5%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. remove-double-neg 96.5%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 96.5%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. +-commutative 96.5%

         \[\leadsto x \cdot \color{blue}{\left(-1 + \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Simplified 96.5%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 5.99999999999999971e209 < x

   1. Initial program 79.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 89.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]

   4. Taylor expanded in x around inf 89.3%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. sub-neg 79.4%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. mul-1-neg 79.4%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. log-rec 79.4%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. remove-double-neg 79.4%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 79.4%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. +-commutative 79.4%

         \[\leadsto x \cdot \color{blue}{\left(-1 + \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   6. Simplified 89.3%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} + \frac{0.083333333333333}{x} \]
   7. Step-by-step derivation

      1. +-commutative 89.3%

         \[\leadsto x \cdot \color{blue}{\left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

      2. distribute-lft-in 89.4%

         \[\leadsto \color{blue}{\left(x \cdot \log x + x \cdot -1\right)} + \frac{0.083333333333333}{x} \]

      3. *-commutative 89.4%

         \[\leadsto \left(x \cdot \log x + \color{blue}{-1 \cdot x}\right) + \frac{0.083333333333333}{x} \]

      4. neg-mul-1 89.4%

         \[\leadsto \left(x \cdot \log x + \color{blue}{\left(-x\right)}\right) + \frac{0.083333333333333}{x} \]

   8. Applied egg-rr 89.4%

      \[\leadsto \color{blue}{\left(x \cdot \log x + \left(-x\right)\right)} + \frac{0.083333333333333}{x} \]
   9. Step-by-step derivation

      1. unsub-neg 89.4%

         \[\leadsto \color{blue}{\left(x \cdot \log x - x\right)} + \frac{0.083333333333333}{x} \]

   10. Applied egg-rr 89.4%

       \[\leadsto \color{blue}{\left(x \cdot \log x - x\right)} + \frac{0.083333333333333}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+209}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1800000 \lor \neg \left(z \leq 5.2 \cdot 10^{-137}\right):\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1800000.0) (not (<= z 5.2e-137)))
   (* (/ (+ 0.0007936500793651 y) x) (pow z 2.0))
   (* 0.083333333333333 (/ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1800000.0) || !(z <= 5.2e-137)) {
		tmp = ((0.0007936500793651 + y) / x) * pow(z, 2.0);
	} else {
		tmp = 0.083333333333333 * (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) :: tmp
    if ((z <= (-1800000.0d0)) .or. (.not. (z <= 5.2d-137))) then
        tmp = ((0.0007936500793651d0 + y) / x) * (z ** 2.0d0)
    else
        tmp = 0.083333333333333d0 * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1800000.0) || !(z <= 5.2e-137)) {
		tmp = ((0.0007936500793651 + y) / x) * Math.pow(z, 2.0);
	} else {
		tmp = 0.083333333333333 * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1800000.0) or not (z <= 5.2e-137):
		tmp = ((0.0007936500793651 + y) / x) * math.pow(z, 2.0)
	else:
		tmp = 0.083333333333333 * (1.0 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1800000.0) || !(z <= 5.2e-137))
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) / x) * (z ^ 2.0));
	else
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1800000.0) || ~((z <= 5.2e-137)))
		tmp = ((0.0007936500793651 + y) / x) * (z ^ 2.0);
	else
		tmp = 0.083333333333333 * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1800000.0], N[Not[LessEqual[z, 5.2e-137]], $MachinePrecision]], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e6 or 5.1999999999999999e-137 < z

   1. Initial program 91.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 79.6%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 79.6%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 79.6%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 79.6%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 79.6%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 79.6%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 79.6%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 79.6%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 71.4%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 71.4%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 71.4%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 71.4%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 68.4%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 68.7%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   10. Step-by-step derivation

       1. associate-/l* 68.6%

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   11. Simplified 68.6%

       \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   if -1.8e6 < z < 5.1999999999999999e-137

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 92.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]
   4. Step-by-step derivation

      1. clear-num 92.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. inv-pow 92.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   5. Applied egg-rr 92.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 92.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. associate-/r/ 92.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   7. Simplified 92.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   8. Taylor expanded in x around 0 46.2%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{1}{x} \cdot 0.083333333333333 \]
   9. Step-by-step derivation

      1. +-commutative 53.1%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   10. Simplified 46.2%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{1}{x} \cdot 0.083333333333333 \]

   11. Taylor expanded in x around 0 46.2%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num 92.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

       2. inv-pow 92.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   13. Applied egg-rr 46.2%

       \[\leadsto \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   14. Step-by-step derivation

       1. unpow-1 92.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

       2. associate-/r/ 92.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   15. Simplified 46.3%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1800000 \lor \neg \left(z \leq 5.2 \cdot 10^{-137}\right):\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-137}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.6e-13)
   (* (/ (+ 0.0007936500793651 y) x) (pow z 2.0))
   (if (<= z 5.2e-137)
     (* 0.083333333333333 (/ 1.0 x))
     (/ (* (+ 0.0007936500793651 y) (pow z 2.0)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.6e-13) {
		tmp = ((0.0007936500793651 + y) / x) * pow(z, 2.0);
	} else if (z <= 5.2e-137) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = ((0.0007936500793651 + y) * pow(z, 2.0)) / 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) :: tmp
    if (z <= (-4.6d-13)) then
        tmp = ((0.0007936500793651d0 + y) / x) * (z ** 2.0d0)
    else if (z <= 5.2d-137) then
        tmp = 0.083333333333333d0 * (1.0d0 / x)
    else
        tmp = ((0.0007936500793651d0 + y) * (z ** 2.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.6e-13) {
		tmp = ((0.0007936500793651 + y) / x) * Math.pow(z, 2.0);
	} else if (z <= 5.2e-137) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = ((0.0007936500793651 + y) * Math.pow(z, 2.0)) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.6e-13:
		tmp = ((0.0007936500793651 + y) / x) * math.pow(z, 2.0)
	elif z <= 5.2e-137:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = ((0.0007936500793651 + y) * math.pow(z, 2.0)) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.6e-13)
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) / x) * (z ^ 2.0));
	elseif (z <= 5.2e-137)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) * (z ^ 2.0)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.6e-13)
		tmp = ((0.0007936500793651 + y) / x) * (z ^ 2.0);
	elseif (z <= 5.2e-137)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = ((0.0007936500793651 + y) * (z ^ 2.0)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.6e-13], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.2e-137], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.59999999999999958e-13

   1. Initial program 90.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 76.7%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 76.7%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 76.7%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 76.7%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 76.7%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 76.7%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 76.7%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 76.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 72.2%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 72.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 72.2%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 72.2%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 69.8%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 70.1%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   10. Step-by-step derivation

       1. associate-/l* 71.3%

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   11. Simplified 71.3%

       \[\leadsto \color{blue}{{z}^{2} \cdot \frac{0.0007936500793651 + y}{x}} \]

   if -4.59999999999999958e-13 < z < 5.1999999999999999e-137

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 93.7%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]
   4. Step-by-step derivation

      1. clear-num 93.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. inv-pow 93.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   5. Applied egg-rr 93.6%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 93.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. associate-/r/ 93.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   7. Simplified 93.7%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   8. Taylor expanded in x around 0 49.2%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{1}{x} \cdot 0.083333333333333 \]
   9. Step-by-step derivation

      1. +-commutative 54.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   10. Simplified 49.2%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{1}{x} \cdot 0.083333333333333 \]

   11. Taylor expanded in x around 0 49.1%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num 93.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

       2. inv-pow 93.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   13. Applied egg-rr 49.0%

       \[\leadsto \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   14. Step-by-step derivation

       1. unpow-1 93.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

       2. associate-/r/ 93.7%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   15. Simplified 49.1%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   if 5.1999999999999999e-137 < z

   1. Initial program 93.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 79.5%

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - 0.5 \cdot 0.5}{x + 0.5}} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. metadata-eval 79.5%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{0.25}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. metadata-eval 79.5%

         \[\leadsto \left(\left(\frac{x \cdot x - \color{blue}{-0.5 \cdot -0.5}}{x + 0.5} \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. associate-*l/ 79.4%

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - -0.5 \cdot -0.5\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. fma-neg 79.4%

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, --0.5 \cdot -0.5\right)} \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 79.4%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -\color{blue}{0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. metadata-eval 79.4%

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied egg-rr 79.4%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Taylor expanded in x around 0 67.1%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. +-commutative 67.1%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. *-commutative 67.1%

         \[\leadsto \left(\color{blue}{\log x \cdot -0.5} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   7. Simplified 67.1%

      \[\leadsto \color{blue}{\left(\log x \cdot -0.5 + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   8. Taylor expanded in z around inf 62.8%

      \[\leadsto \left(\log x \cdot -0.5 + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   9. Taylor expanded in x around 0 63.2%

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot {z}^{2}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-137}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0055:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - 0.916666666666667\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.0055)
   (* 0.083333333333333 (/ 1.0 x))
   (* x (- (log x) 0.916666666666667))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0055) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = x * (log(x) - 0.916666666666667);
	}
	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 (x <= 0.0055d0) then
        tmp = 0.083333333333333d0 * (1.0d0 / x)
    else
        tmp = x * (log(x) - 0.916666666666667d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0055) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = x * (Math.log(x) - 0.916666666666667);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.0055:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = x * (math.log(x) - 0.916666666666667)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.0055)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = Float64(x * Float64(log(x) - 0.916666666666667));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.0055)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = x * (log(x) - 0.916666666666667);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.0055], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - 0.916666666666667), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0054999999999999997

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 44.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]
   4. Step-by-step derivation

      1. clear-num 44.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. inv-pow 44.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   5. Applied egg-rr 44.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 44.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. associate-/r/ 44.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   7. Simplified 44.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   8. Taylor expanded in x around 0 44.1%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{1}{x} \cdot 0.083333333333333 \]
   9. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   10. Simplified 44.1%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{1}{x} \cdot 0.083333333333333 \]

   11. Taylor expanded in x around 0 43.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num 44.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

       2. inv-pow 44.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   13. Applied egg-rr 43.0%

       \[\leadsto \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   14. Step-by-step derivation

       1. unpow-1 44.4%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

       2. associate-/r/ 44.5%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   15. Simplified 43.1%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   if 0.0054999999999999997 < x

   1. Initial program 90.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 68.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]

   4. Taylor expanded in x around inf 67.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. sub-neg 89.0%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. mul-1-neg 89.0%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. log-rec 89.0%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. remove-double-neg 89.0%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 89.0%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. +-commutative 89.0%

         \[\leadsto x \cdot \color{blue}{\left(-1 + \log x\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   6. Simplified 67.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} + \frac{0.083333333333333}{x} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 67.0%

         \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{1 \cdot \frac{0.083333333333333}{x}} \]

      2. div-inv 67.0%

         \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \color{blue}{\left(0.083333333333333 \cdot \frac{1}{x}\right)} \]

      3. add-exp-log 67.0%

         \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot \color{blue}{e^{\log \left(\frac{1}{x}\right)}}\right) \]

      4. add-sqr-sqrt 0.0%

         \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log \left(\frac{1}{x}\right)} \cdot \sqrt{\log \left(\frac{1}{x}\right)}}}\right) \]

      5. sqrt-unprod 25.7%

         \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log \left(\frac{1}{x}\right) \cdot \log \left(\frac{1}{x}\right)}}}\right) \]

      6. log-rec 25.7%

         \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\color{blue}{\left(-\log x\right)} \cdot \log \left(\frac{1}{x}\right)}}\right) \]

      7. log-rec 25.7%

         \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\left(-\log x\right) \cdot \color{blue}{\left(-\log x\right)}}}\right) \]

      8. sqr-neg 25.7%

         \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\color{blue}{\log x \cdot \log x}}}\right) \]

      9. sqrt-unprod 25.7%

         \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}\right) \]

      10. add-sqr-sqrt 25.7%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\log x}}\right) \]

      11. add-exp-log 25.7%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot \color{blue}{x}\right) \]

   8. Applied egg-rr 25.7%

      \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{1 \cdot \left(0.083333333333333 \cdot x\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.7%

         \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{0.083333333333333 \cdot x} \]

      2. *-commutative 25.7%

         \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{x \cdot 0.083333333333333} \]

   10. Simplified 25.7%

       \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{x \cdot 0.083333333333333} \]

   11. Taylor expanded in x around 0 25.7%

       \[\leadsto \color{blue}{x \cdot \left(\log x - 0.916666666666667\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0055:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - 0.916666666666667\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.083333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.85e-8) (* 0.083333333333333 (/ 1.0 x)) (* x 0.083333333333333)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = x * 0.083333333333333;
	}
	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 (x <= 1.85d-8) then
        tmp = 0.083333333333333d0 * (1.0d0 / x)
    else
        tmp = x * 0.083333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = x * 0.083333333333333;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.85e-8:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = x * 0.083333333333333
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.85e-8)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = Float64(x * 0.083333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.85e-8)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = x * 0.083333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.85e-8], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x * 0.083333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85e-8

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 43.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]
   4. Step-by-step derivation

      1. clear-num 43.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. inv-pow 43.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   5. Applied egg-rr 43.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 43.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. associate-/r/ 43.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   7. Simplified 43.6%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   8. Taylor expanded in x around 0 43.6%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{1}{x} \cdot 0.083333333333333 \]
   9. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   10. Simplified 43.6%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{1}{x} \cdot 0.083333333333333 \]

   11. Taylor expanded in x around 0 43.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num 43.5%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

       2. inv-pow 43.5%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   13. Applied egg-rr 43.0%

       \[\leadsto \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   14. Step-by-step derivation

       1. unpow-1 43.5%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

       2. associate-/r/ 43.6%

          \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   15. Simplified 43.0%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   if 1.85e-8 < x

   1. Initial program 90.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 68.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]
   4. Step-by-step derivation

      1. clear-num 68.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. inv-pow 68.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   5. Applied egg-rr 68.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 68.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. associate-/r/ 68.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   7. Simplified 68.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   8. Taylor expanded in x around 0 2.4%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{1}{x} \cdot 0.083333333333333 \]
   9. Step-by-step derivation

      1. +-commutative 31.5%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   10. Simplified 2.4%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{1}{x} \cdot 0.083333333333333 \]

   11. Taylor expanded in x around 0 3.7%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 66.7%

          \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{1 \cdot \frac{0.083333333333333}{x}} \]

       2. div-inv 66.7%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \color{blue}{\left(0.083333333333333 \cdot \frac{1}{x}\right)} \]

       3. add-exp-log 66.7%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot \color{blue}{e^{\log \left(\frac{1}{x}\right)}}\right) \]

       4. add-sqr-sqrt 0.7%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log \left(\frac{1}{x}\right)} \cdot \sqrt{\log \left(\frac{1}{x}\right)}}}\right) \]

       5. sqrt-unprod 25.9%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log \left(\frac{1}{x}\right) \cdot \log \left(\frac{1}{x}\right)}}}\right) \]

       6. log-rec 25.9%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\color{blue}{\left(-\log x\right)} \cdot \log \left(\frac{1}{x}\right)}}\right) \]

       7. log-rec 25.9%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\left(-\log x\right) \cdot \color{blue}{\left(-\log x\right)}}}\right) \]

       8. sqr-neg 25.9%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\color{blue}{\log x \cdot \log x}}}\right) \]

       9. sqrt-unprod 25.3%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}\right) \]

       10. add-sqr-sqrt 25.3%

           \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\log x}}\right) \]

       11. add-exp-log 25.3%

           \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot \color{blue}{x}\right) \]

   13. Applied egg-rr 10.0%

       \[\leadsto \color{blue}{1 \cdot \left(0.083333333333333 \cdot x\right)} \]
   14. Step-by-step derivation

       1. *-lft-identity 25.3%

          \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{0.083333333333333 \cdot x} \]

       2. *-commutative 25.3%

          \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{x \cdot 0.083333333333333} \]

   15. Simplified 10.0%

       \[\leadsto \color{blue}{x \cdot 0.083333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.083333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.083333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.85e-8) (/ 0.083333333333333 x) (* x 0.083333333333333)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = x * 0.083333333333333;
	}
	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 (x <= 1.85d-8) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = x * 0.083333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = x * 0.083333333333333;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.85e-8:
		tmp = 0.083333333333333 / x
	else:
		tmp = x * 0.083333333333333
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.85e-8)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = Float64(x * 0.083333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.85e-8)
		tmp = 0.083333333333333 / x;
	else
		tmp = x * 0.083333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.85e-8], N[(0.083333333333333 / x), $MachinePrecision], N[(x * 0.083333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85e-8

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 43.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]
   4. Step-by-step derivation

      1. clear-num 43.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. inv-pow 43.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   5. Applied egg-rr 43.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 43.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. associate-/r/ 43.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   7. Simplified 43.6%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   8. Taylor expanded in x around 0 43.6%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{1}{x} \cdot 0.083333333333333 \]
   9. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   10. Simplified 43.6%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{1}{x} \cdot 0.083333333333333 \]

   11. Taylor expanded in x around 0 43.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

   if 1.85e-8 < x

   1. Initial program 90.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 68.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]
   4. Step-by-step derivation

      1. clear-num 68.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. inv-pow 68.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

   5. Applied egg-rr 68.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 68.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

      2. associate-/r/ 68.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   7. Simplified 68.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   8. Taylor expanded in x around 0 2.4%

      \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{1}{x} \cdot 0.083333333333333 \]
   9. Step-by-step derivation

      1. +-commutative 31.5%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   10. Simplified 2.4%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{1}{x} \cdot 0.083333333333333 \]

   11. Taylor expanded in x around 0 3.7%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 66.7%

          \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{1 \cdot \frac{0.083333333333333}{x}} \]

       2. div-inv 66.7%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \color{blue}{\left(0.083333333333333 \cdot \frac{1}{x}\right)} \]

       3. add-exp-log 66.7%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot \color{blue}{e^{\log \left(\frac{1}{x}\right)}}\right) \]

       4. add-sqr-sqrt 0.7%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log \left(\frac{1}{x}\right)} \cdot \sqrt{\log \left(\frac{1}{x}\right)}}}\right) \]

       5. sqrt-unprod 25.9%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log \left(\frac{1}{x}\right) \cdot \log \left(\frac{1}{x}\right)}}}\right) \]

       6. log-rec 25.9%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\color{blue}{\left(-\log x\right)} \cdot \log \left(\frac{1}{x}\right)}}\right) \]

       7. log-rec 25.9%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\left(-\log x\right) \cdot \color{blue}{\left(-\log x\right)}}}\right) \]

       8. sqr-neg 25.9%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\color{blue}{\log x \cdot \log x}}}\right) \]

       9. sqrt-unprod 25.3%

          \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}\right) \]

       10. add-sqr-sqrt 25.3%

           \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\log x}}\right) \]

       11. add-exp-log 25.3%

           \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot \color{blue}{x}\right) \]

   13. Applied egg-rr 10.0%

       \[\leadsto \color{blue}{1 \cdot \left(0.083333333333333 \cdot x\right)} \]
   14. Step-by-step derivation

       1. *-lft-identity 25.3%

          \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{0.083333333333333 \cdot x} \]

       2. *-commutative 25.3%

          \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{x \cdot 0.083333333333333} \]

   15. Simplified 10.0%

       \[\leadsto \color{blue}{x \cdot 0.083333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 6.6% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.083333333333333 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x 0.083333333333333))

C

double code(double x, double y, double z) {
	return x * 0.083333333333333;
}

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 * 0.083333333333333d0
end function

Java

public static double code(double x, double y, double z) {
	return x * 0.083333333333333;
}

Python

def code(x, y, z):
	return x * 0.083333333333333

Julia

function code(x, y, z)
	return Float64(x * 0.083333333333333)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * 0.083333333333333;
end

Wolfram

code[x_, y_, z_] := N[(x * 0.083333333333333), $MachinePrecision]

Derivation

1. Initial program 94.8%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around 0 56.9%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.083333333333333}{x}} \]
4. Step-by-step derivation

   1. clear-num 56.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

   2. inv-pow 56.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]

5. Applied egg-rr 56.9%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 56.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} \]

   2. associate-/r/ 56.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

7. Simplified 56.9%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

8. Taylor expanded in x around 0 21.9%

   \[\leadsto \color{blue}{\left(0.91893853320467 + -0.5 \cdot \log x\right)} + \frac{1}{x} \cdot 0.083333333333333 \]
9. Step-by-step derivation

   1. +-commutative 63.7%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

10. Simplified 21.9%

    \[\leadsto \color{blue}{\left(-0.5 \cdot \log x + 0.91893853320467\right)} + \frac{1}{x} \cdot 0.083333333333333 \]

11. Taylor expanded in x around 0 22.3%

    \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
12. Step-by-step derivation

    1. *-un-lft-identity 55.5%

       \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{1 \cdot \frac{0.083333333333333}{x}} \]

    2. div-inv 55.5%

       \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \color{blue}{\left(0.083333333333333 \cdot \frac{1}{x}\right)} \]

    3. add-exp-log 54.0%

       \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot \color{blue}{e^{\log \left(\frac{1}{x}\right)}}\right) \]

    4. add-sqr-sqrt 18.9%

       \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log \left(\frac{1}{x}\right)} \cdot \sqrt{\log \left(\frac{1}{x}\right)}}}\right) \]

    5. sqrt-unprod 32.5%

       \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log \left(\frac{1}{x}\right) \cdot \log \left(\frac{1}{x}\right)}}}\right) \]

    6. log-rec 32.5%

       \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\color{blue}{\left(-\log x\right)} \cdot \log \left(\frac{1}{x}\right)}}\right) \]

    7. log-rec 32.5%

       \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\left(-\log x\right) \cdot \color{blue}{\left(-\log x\right)}}}\right) \]

    8. sqr-neg 32.5%

       \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\sqrt{\color{blue}{\log x \cdot \log x}}}\right) \]

    9. sqrt-unprod 13.3%

       \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}\right) \]

    10. add-sqr-sqrt 14.1%

        \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot e^{\color{blue}{\log x}}\right) \]

    11. add-exp-log 14.1%

        \[\leadsto x \cdot \left(-1 + \log x\right) + 1 \cdot \left(0.083333333333333 \cdot \color{blue}{x}\right) \]

13. Applied egg-rr 6.6%

    \[\leadsto \color{blue}{1 \cdot \left(0.083333333333333 \cdot x\right)} \]
14. Step-by-step derivation

    1. *-lft-identity 14.1%

       \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{0.083333333333333 \cdot x} \]

    2. *-commutative 14.1%

       \[\leadsto x \cdot \left(-1 + \log x\right) + \color{blue}{x \cdot 0.083333333333333} \]

15. Simplified 6.6%

    \[\leadsto \color{blue}{x \cdot 0.083333333333333} \]
16. Add Preprocessing

Developer target: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x))
  (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

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 - 0.5d0) * log(x)) + (0.91893853320467d0 - x)) + (0.083333333333333d0 / x)) + ((z / x) * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) + Float64(0.91893853320467 - x)) + Float64(0.083333333333333 / x)) + Float64(Float64(z / x) * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024107 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))