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

Percentage Accurate: 94.2% → 99.4%

Time: 12.3s

Alternatives: 12

Speedup: 1.0×

• 32.1% 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: 94.2% 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.083333333333333 \cdot \frac{1}{x} + z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\right) + x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 9.8e+73)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))
     x))
   (+
    (+
     (* 0.083333333333333 (/ 1.0 x))
     (* z (* (+ 0.0007936500793651 y) (/ z x))))
    (* x (+ (log x) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.8e+73) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	} else {
		tmp = ((0.083333333333333 * (1.0 / x)) + (z * ((0.0007936500793651 + y) * (z / x)))) + (x * (log(x) + -1.0));
	}
	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 (x <= 9.8d+73) then
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((0.083333333333333d0 + (z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0))) / x)
    else
        tmp = ((0.083333333333333d0 * (1.0d0 / x)) + (z * ((0.0007936500793651d0 + y) * (z / x)))) + (x * (log(x) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.8e+73) {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	} else {
		tmp = ((0.083333333333333 * (1.0 / x)) + (z * ((0.0007936500793651 + y) * (z / x)))) + (x * (Math.log(x) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 9.8e+73:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x)
	else:
		tmp = ((0.083333333333333 * (1.0 / x)) + (z * ((0.0007936500793651 + y) * (z / x)))) + (x * (math.log(x) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 9.8e+73)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))) / x));
	else
		tmp = Float64(Float64(Float64(0.083333333333333 * Float64(1.0 / x)) + Float64(z * Float64(Float64(0.0007936500793651 + y) * Float64(z / x)))) + Float64(x * Float64(log(x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 9.8e+73)
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	else
		tmp = ((0.083333333333333 * (1.0 / x)) + (z * ((0.0007936500793651 + y) * (z / x)))) + (x * (log(x) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 9.8e+73], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $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[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.7999999999999998e73

   1. Initial program 99.7%

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

   if 9.7999999999999998e73 < x

   1. Initial program 83.6%

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

   3. Taylor expanded in z around 0 99.6%

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

   4. Taylor expanded in z around inf 88.8%

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

      1. unpow2 88.8%

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

      2. associate-*r/ 88.8%

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

      3. metadata-eval 88.8%

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

      4. associate-*l* 99.6%

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

      5. distribute-rgt-in 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-*r/ 99.6%

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

      8. associate-*l/ 96.6%

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

      9. associate-/l* 99.6%

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

      10. distribute-rgt-out 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 99.7%

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

      1. sub-neg 99.7%

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

      2. mul-1-neg 99.7%

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

      3. log-rec 99.7%

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

      4. remove-double-neg 99.7%

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

      5. metadata-eval 99.7%

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

      6. +-commutative 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.083333333333333 \cdot \frac{1}{x} + z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\right) + x \cdot \left(\log x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((z * ((0.0007936500793651 + y) / (x / z))) + (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(0.0007936500793651 + y) / Float64(x / z))) + 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 * ((0.0007936500793651 + y) / (x / z))) + (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[(0.0007936500793651 + y), $MachinePrecision] / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in z around 0 95.2%

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

4. Taylor expanded in z around inf 91.2%

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

   1. unpow2 91.2%

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

   2. associate-*r/ 91.2%

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

   3. metadata-eval 91.2%

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

   4. associate-*l* 95.2%

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

   5. distribute-rgt-in 92.4%

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

   6. associate-*l/ 92.4%

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

   7. associate-*r/ 92.4%

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

   8. associate-*l/ 95.1%

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

   9. associate-/l* 94.2%

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

   10. distribute-rgt-out 98.9%

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

6. Simplified 98.9%

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

   1. *-commutative 98.9%

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

   2. clear-num 98.8%

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

   3. un-div-inv 98.9%

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

8. Applied egg-rr 98.9%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((0.083333333333333d0 * (1.0d0 / x)) + (z * ((0.0007936500793651d0 + y) * (z / 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 * (1.0 / x)) + (z * ((0.0007936500793651 + y) * (z / x))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in z around 0 95.2%

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

4. Taylor expanded in z around inf 91.2%

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

   1. unpow2 91.2%

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

   2. associate-*r/ 91.2%

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

   3. metadata-eval 91.2%

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

   4. associate-*l* 95.2%

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

   5. distribute-rgt-in 92.4%

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

   6. associate-*l/ 92.4%

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

   7. associate-*r/ 92.4%

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

   8. associate-*l/ 95.1%

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

   9. associate-/l* 94.2%

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

   10. distribute-rgt-out 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 4: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\log x + -1\right)\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{+199}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ (log x) -1.0))))
   (if (<= x 2.6e+199)
     (+
      (/
       (+
        0.083333333333333
        (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))
       x)
      t_0)
     (+ 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 <= 2.6e+199) {
		tmp = ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x) + t_0;
	} 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 <= 2.6d+199) then
        tmp = ((0.083333333333333d0 + (z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0))) / x) + t_0
    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 <= 2.6e+199) {
		tmp = ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x) + t_0;
	} 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 <= 2.6e+199:
		tmp = ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x) + t_0
	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 <= 2.6e+199)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))) / x) + t_0);
	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 <= 2.6e+199)
		tmp = ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x) + t_0;
	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, 2.6e+199], N[(N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e199

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf 96.9%

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

      1. sub-neg 97.7%

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

      2. mul-1-neg 97.7%

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

      3. log-rec 97.7%

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

      4. remove-double-neg 97.7%

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

      5. metadata-eval 97.7%

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

      6. +-commutative 97.7%

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

   5. Simplified 96.9%

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

   if 2.6000000000000001e199 < x

   1. Initial program 71.7%

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

   3. Taylor expanded in z around 0 87.8%

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

   4. Taylor expanded in x around inf 87.8%

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

      1. sub-neg 99.6%

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

      2. mul-1-neg 99.6%

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

      3. log-rec 99.6%

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

      4. remove-double-neg 99.6%

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

      5. metadata-eval 99.6%

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

      6. +-commutative 99.6%

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

   6. Simplified 87.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+199}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log x - x\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{-20}:\\ \;\;\;\;t\_0 + \frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{0.083333333333333 + z \cdot \left(z \cdot 0.0007936500793651 - 0.0027777777777778\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log x)) x)))
   (if (<= y -6.8e-20)
     (+ t_0 (/ (+ 0.083333333333333 (* z -0.0027777777777778)) x))
     (+
      t_0
      (/
       (+
        0.083333333333333
        (* z (- (* z 0.0007936500793651) 0.0027777777777778)))
       x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x * log(x)) - x;
	double tmp;
	if (y <= -6.8e-20) {
		tmp = t_0 + ((0.083333333333333 + (z * -0.0027777777777778)) / x);
	} else {
		tmp = t_0 + ((0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / 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)) - x
    if (y <= (-6.8d-20)) then
        tmp = t_0 + ((0.083333333333333d0 + (z * (-0.0027777777777778d0))) / x)
    else
        tmp = t_0 + ((0.083333333333333d0 + (z * ((z * 0.0007936500793651d0) - 0.0027777777777778d0))) / 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)) - x;
	double tmp;
	if (y <= -6.8e-20) {
		tmp = t_0 + ((0.083333333333333 + (z * -0.0027777777777778)) / x);
	} else {
		tmp = t_0 + ((0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.log(x)) - x
	tmp = 0
	if y <= -6.8e-20:
		tmp = t_0 + ((0.083333333333333 + (z * -0.0027777777777778)) / x)
	else:
		tmp = t_0 + ((0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * log(x)) - x)
	tmp = 0.0
	if (y <= -6.8e-20)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * -0.0027777777777778)) / x));
	else
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * 0.0007936500793651) - 0.0027777777777778))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * log(x)) - x;
	tmp = 0.0;
	if (y <= -6.8e-20)
		tmp = t_0 + ((0.083333333333333 + (z * -0.0027777777777778)) / x);
	else
		tmp = t_0 + ((0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[y, -6.8e-20], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * -0.0027777777777778), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(N[(z * 0.0007936500793651), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.7999999999999994e-20

   1. Initial program 92.6%

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

   3. Taylor expanded in x around 0 92.7%

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

      1. sub-neg 92.7%

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

      2. metadata-eval 92.7%

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

      3. distribute-rgt-in 92.6%

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

      4. *-commutative 92.6%

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

      5. neg-mul-1 92.6%

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

      6. associate-+l+ 92.6%

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

      7. distribute-rgt-out 92.6%

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

      8. +-commutative 92.6%

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

      9. *-commutative 92.6%

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

      10. fma-define 92.7%

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

      11. fma-define 92.6%

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

      12. associate-+l+ 92.6%

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

      13. sub-neg 92.6%

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

      14. +-commutative 92.6%

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

      15. *-commutative 92.6%

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

      16. fma-define 92.6%

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

   5. Simplified 92.6%

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

   6. Taylor expanded in x around inf 90.3%

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

      1. mul-1-neg 90.3%

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

      2. distribute-rgt-neg-in 90.3%

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

      3. log-rec 90.3%

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

      4. remove-double-neg 90.3%

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

   8. Simplified 90.3%

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

   9. Taylor expanded in z around 0 66.1%

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

       1. *-commutative 66.1%

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

   11. Simplified 66.1%

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

   if -6.7999999999999994e-20 < y

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0 94.3%

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

      1. sub-neg 94.3%

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

      2. metadata-eval 94.3%

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

      3. distribute-rgt-in 94.4%

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

      4. *-commutative 94.4%

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

      5. neg-mul-1 94.4%

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

      6. associate-+l+ 94.4%

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

      7. distribute-rgt-out 94.4%

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

      8. +-commutative 94.4%

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

      9. *-commutative 94.4%

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

      10. fma-define 94.3%

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

      11. fma-define 94.4%

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

      12. associate-+l+ 94.4%

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

      13. sub-neg 94.4%

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

      14. +-commutative 94.4%

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

      15. *-commutative 94.4%

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

      16. fma-define 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in x around inf 94.0%

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

      1. mul-1-neg 94.0%

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

      2. distribute-rgt-neg-in 94.0%

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

      3. log-rec 94.0%

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

      4. remove-double-neg 94.0%

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

   8. Simplified 94.0%

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

   9. Taylor expanded in y around 0 89.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-20}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot 0.0007936500793651 - 0.0027777777777778\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in z around 0 95.2%

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

4. Taylor expanded in z around inf 91.2%

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

   1. unpow2 91.2%

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

   2. associate-*r/ 91.2%

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

   3. metadata-eval 91.2%

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

   4. associate-*l* 95.2%

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

   5. distribute-rgt-in 92.4%

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

   6. associate-*l/ 92.4%

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

   7. associate-*r/ 92.4%

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

   8. associate-*l/ 95.1%

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

   9. associate-/l* 94.2%

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

   10. distribute-rgt-out 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in x around inf 98.0%

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

   1. sub-neg 98.0%

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

   2. mul-1-neg 98.0%

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

   3. log-rec 98.0%

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

   4. remove-double-neg 98.0%

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

   5. metadata-eval 98.0%

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

   6. +-commutative 98.0%

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

9. Simplified 98.0%

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

10. Final simplification 98.0%

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

Alternative 7: 61.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in x around 0 93.9%

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

   1. sub-neg 93.9%

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

   2. metadata-eval 93.9%

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

   3. distribute-rgt-in 93.9%

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

   4. *-commutative 93.9%

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

   5. neg-mul-1 93.9%

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

   6. associate-+l+ 93.9%

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

   7. distribute-rgt-out 93.9%

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

   8. +-commutative 93.9%

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

   9. *-commutative 93.9%

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

   10. fma-define 93.9%

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

   11. fma-define 93.9%

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

   12. associate-+l+ 93.9%

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

   13. sub-neg 93.9%

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

   14. +-commutative 93.9%

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

   15. *-commutative 93.9%

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

   16. fma-define 93.9%

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

5. Simplified 93.9%

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

6. Taylor expanded in x around inf 93.0%

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

   1. mul-1-neg 93.0%

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

   2. distribute-rgt-neg-in 93.0%

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

   3. log-rec 93.0%

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

   4. remove-double-neg 93.0%

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

8. Simplified 93.0%

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

9. Taylor expanded in z around 0 63.5%

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

    1. *-commutative 63.5%

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

11. Simplified 63.5%

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

12. Final simplification 63.5%

    \[\leadsto \left(x \cdot \log x - x\right) + \frac{0.083333333333333 + z \cdot -0.0027777777777778}{x} \]
13. Add Preprocessing

Alternative 8: 56.7% 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 93.9%

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

3. Taylor expanded in z around 0 58.7%

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

Alternative 9: 55.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in z around 0 58.7%

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

4. Taylor expanded in x around inf 57.7%

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

   1. sub-neg 98.0%

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

   2. mul-1-neg 98.0%

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

   3. log-rec 98.0%

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

   4. remove-double-neg 98.0%

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

   5. metadata-eval 98.0%

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

   6. +-commutative 98.0%

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

6. Simplified 57.7%

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

   1. div-inv 57.8%

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

   2. *-commutative 57.8%

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

8. Applied egg-rr 57.8%

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

9. Final simplification 57.8%

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

Alternative 10: 55.7% 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 93.9%

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

3. Taylor expanded in z around 0 58.7%

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

4. Taylor expanded in x around inf 57.7%

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

   1. sub-neg 98.0%

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

   2. mul-1-neg 98.0%

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

   3. log-rec 98.0%

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

   4. remove-double-neg 98.0%

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

   5. metadata-eval 98.0%

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

   6. +-commutative 98.0%

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

6. Simplified 57.7%

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

7. Final simplification 57.7%

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

Alternative 11: 23.0% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ 0.083333333333333 \cdot \frac{1}{x} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 * (1.0 / x);
}

Python

def code(x, y, z):
	return 0.083333333333333 * (1.0 / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in z around 0 58.7%

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

4. Taylor expanded in x around inf 57.7%

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

   1. sub-neg 98.0%

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

   2. mul-1-neg 98.0%

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

   3. log-rec 98.0%

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

   4. remove-double-neg 98.0%

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

   5. metadata-eval 98.0%

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

   6. +-commutative 98.0%

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

6. Simplified 57.7%

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

7. Taylor expanded in x around 0 23.9%

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

   1. div-inv 57.8%

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

   2. *-commutative 57.8%

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

9. Applied egg-rr 24.0%

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

10. Final simplification 24.0%

    \[\leadsto 0.083333333333333 \cdot \frac{1}{x} \]
11. Add Preprocessing

Alternative 12: 23.0% 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 93.9%

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

3. Taylor expanded in z around 0 58.7%

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

4. Taylor expanded in x around inf 57.7%

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

   1. sub-neg 98.0%

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

   2. mul-1-neg 98.0%

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

   3. log-rec 98.0%

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

   4. remove-double-neg 98.0%

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

   5. metadata-eval 98.0%

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

   6. +-commutative 98.0%

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

6. Simplified 57.7%

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

7. Taylor expanded in x around 0 23.9%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
8. 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 2024108 
(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)))