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

Percentage Accurate: 93.7% → 97.9%

Time: 13.4s

Alternatives: 7

Speedup: 1.0×

• 32.5% 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: 93.7% 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: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) + 0.083333333333333 \cdot \frac{1}{x}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (+
   (* z (* (/ z x) (+ 0.0007936500793651 y)))
   (* 0.083333333333333 (/ 1.0 x)))))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((z * ((z / x) * (0.0007936500793651 + y))) + (0.083333333333333 * (1.0 / 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) + ((z * ((z / x) * (0.0007936500793651d0 + y))) + (0.083333333333333d0 * (1.0d0 / x)))
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((z * ((z / x) * (0.0007936500793651 + y))) + (0.083333333333333 * (1.0 / x)));
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((z * ((z / x) * (0.0007936500793651 + y))) + (0.083333333333333 * (1.0 / x)))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(z * Float64(Float64(z / x) * Float64(0.0007936500793651 + y))) + Float64(0.083333333333333 * Float64(1.0 / x))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((z * ((z / x) * (0.0007936500793651 + y))) + (0.083333333333333 * (1.0 / 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[(z * N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Taylor expanded in z around 0 95.9%

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

4. Taylor expanded in z around inf 90.3%

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

   1. unpow2 90.3%

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

   2. associate-*r/ 90.3%

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

   3. metadata-eval 90.3%

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

   4. associate-*l* 95.8%

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

   5. distribute-rgt-in 90.7%

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

   6. associate-*l/ 90.7%

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

   7. associate-*r/ 90.7%

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

   8. associate-*l/ 91.9%

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

   9. associate-/l* 92.1%

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

   10. distribute-rgt-out 99.1%

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

6. Simplified 99.1%

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

Alternative 2: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\log x + -1\right)\\ \mathbf{if}\;x \leq 4.2 \cdot 10^{+191}:\\ \;\;\;\;t\_0 + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ (log x) -1.0))))
   (if (<= x 4.2e+191)
     (+
      t_0
      (/
       (+
        0.083333333333333
        (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))
       x))
     (+ t_0 (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x * (log(x) + -1.0);
	double tmp;
	if (x <= 4.2e+191) {
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = x * (log(x) + (-1.0d0))
    if (x <= 4.2d+191) then
        tmp = t_0 + ((0.083333333333333d0 + (z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0))) / x)
    else
        tmp = t_0 + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (Math.log(x) + -1.0);
	double tmp;
	if (x <= 4.2e+191) {
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	} else {
		tmp = t_0 + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (math.log(x) + -1.0)
	tmp = 0
	if x <= 4.2e+191:
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x)
	else:
		tmp = t_0 + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(log(x) + -1.0))
	tmp = 0.0
	if (x <= 4.2e+191)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))) / x));
	else
		tmp = Float64(t_0 + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (log(x) + -1.0);
	tmp = 0.0;
	if (x <= 4.2e+191)
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	else
		tmp = t_0 + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.2e+191], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2000000000000001e191

   1. Initial program 96.6%

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

   3. Taylor expanded in x around inf 95.8%

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

      1. sub-neg 98.3%

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

      2. mul-1-neg 98.3%

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

      3. log-rec 98.3%

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

      4. remove-double-neg 98.3%

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

      5. metadata-eval 98.3%

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

      6. +-commutative 98.3%

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

   5. Simplified 95.8%

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

   if 4.2000000000000001e191 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in z around 0 86.6%

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

   4. Taylor expanded in x around inf 86.6%

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

      1. sub-neg 99.5%

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

      2. mul-1-neg 99.5%

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

      3. log-rec 99.5%

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

      4. remove-double-neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

   6. Simplified 86.6%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+191}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) + 0.083333333333333 \cdot \frac{1}{x}\right) + x \cdot \left(\log x + -1\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+
   (* z (* (/ z x) (+ 0.0007936500793651 y)))
   (* 0.083333333333333 (/ 1.0 x)))
  (* x (+ (log x) -1.0))))

C

double code(double x, double y, double z) {
	return ((z * ((z / x) * (0.0007936500793651 + y))) + (0.083333333333333 * (1.0 / x))) + (x * (log(x) + -1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((z * ((z / x) * (0.0007936500793651d0 + y))) + (0.083333333333333d0 * (1.0d0 / x))) + (x * (log(x) + (-1.0d0)))
end function

Java

public static double code(double x, double y, double z) {
	return ((z * ((z / x) * (0.0007936500793651 + y))) + (0.083333333333333 * (1.0 / x))) + (x * (Math.log(x) + -1.0));
}

Python

def code(x, y, z):
	return ((z * ((z / x) * (0.0007936500793651 + y))) + (0.083333333333333 * (1.0 / x))) + (x * (math.log(x) + -1.0))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(z * Float64(Float64(z / x) * Float64(0.0007936500793651 + y))) + Float64(0.083333333333333 * Float64(1.0 / x))) + Float64(x * Float64(log(x) + -1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((z * ((z / x) * (0.0007936500793651 + y))) + (0.083333333333333 * (1.0 / x))) + (x * (log(x) + -1.0));
end

Wolfram

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

Derivation

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. Taylor expanded in z around 0 95.9%

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

4. Taylor expanded in z around inf 90.3%

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

   1. unpow2 90.3%

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

   2. associate-*r/ 90.3%

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

   3. metadata-eval 90.3%

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

   4. associate-*l* 95.8%

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

   5. distribute-rgt-in 90.7%

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

   6. associate-*l/ 90.7%

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

   7. associate-*r/ 90.7%

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

   8. associate-*l/ 91.9%

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

   9. associate-/l* 92.1%

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

   10. distribute-rgt-out 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in x around inf 98.5%

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

   1. sub-neg 98.5%

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

   2. mul-1-neg 98.5%

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

   3. log-rec 98.5%

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

   4. remove-double-neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

    \[\leadsto \left(z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) + 0.083333333333333 \cdot \frac{1}{x}\right) + x \cdot \left(\log x + -1\right) \]
11. Add Preprocessing

Alternative 4: 56.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (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) + (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) + (0.083333333333333 / x);
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (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[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]

Derivation

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. 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. Add Preprocessing

Alternative 5: 55.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (* x (+ (log x) -1.0)) (/ 0.083333333333333 x)))

C

double code(double x, double y, double z) {
	return (x * (log(x) + -1.0)) + (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 * (log(x) + (-1.0d0))) + (0.083333333333333d0 / x)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.log(x) + -1.0)) + (0.083333333333333 / x);
}

Python

def code(x, y, z):
	return (x * (math.log(x) + -1.0)) + (0.083333333333333 / x)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(0.083333333333333 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
end

Wolfram

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

Derivation

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. 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. Taylor expanded in x around inf 56.3%

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

   1. sub-neg 98.5%

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

   2. mul-1-neg 98.5%

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

   3. log-rec 98.5%

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

   4. remove-double-neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

6. Simplified 56.3%

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

7. Final simplification 56.3%

   \[\leadsto x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x} \]
8. Add Preprocessing

Alternative 6: 23.4% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ 0.91893853320467 + \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 0.91893853320467 (/ 0.083333333333333 x)))

C

double code(double x, double y, double z) {
	return 0.91893853320467 + (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 = 0.91893853320467d0 + (0.083333333333333d0 / x)
end function

Java

public static double code(double x, double y, double z) {
	return 0.91893853320467 + (0.083333333333333 / x);
}

Python

def code(x, y, z):
	return 0.91893853320467 + (0.083333333333333 / x)

Julia

function code(x, y, z)
	return Float64(0.91893853320467 + Float64(0.083333333333333 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.91893853320467 + (0.083333333333333 / x);
end

Wolfram

code[x_, y_, z_] := N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]

Derivation

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. 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. Taylor expanded in x around inf 56.2%

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

   1. mul-1-neg 56.2%

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

   2. distribute-rgt-neg-in 56.2%

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

   3. log-rec 56.2%

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

   4. remove-double-neg 56.2%

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

6. Simplified 56.2%

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

7. Taylor expanded in x around 0 25.1%

   \[\leadsto \color{blue}{0.91893853320467} + \frac{0.083333333333333}{x} \]
8. Add Preprocessing

Alternative 7: 22.8% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 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 = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

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. 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. sub-neg 56.9%

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

   2. associate-+l+ 56.9%

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

   3. +-commutative 56.9%

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

   4. sub-neg 56.9%

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

   5. sub-neg 56.9%

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

   6. metadata-eval 56.9%

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

   7. fma-undefine 56.9%

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

   8. add-cube-cbrt 56.3%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right)}} + \frac{0.083333333333333}{x} \]

   9. pow3 56.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right)}\right)}^{3}} + \frac{0.083333333333333}{x} \]

5. Applied egg-rr 56.3%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right)}\right)}^{3}} + \frac{0.083333333333333}{x} \]

6. Taylor expanded in x around 0 24.0%

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

   1. *-commutative 24.0%

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

8. Simplified 24.0%

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

9. Taylor expanded in x around 0 24.5%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
10. Add Preprocessing

Developer target: 98.7% 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 2024106 
(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)))