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

Percentage Accurate: 93.7% → 97.9%

Time: 17.7s

Alternatives: 16

Speedup: 1.0×

• 32.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

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) / (x / z))));
}

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) / (x / z))))
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) / (x / z))));
}

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) / (x / z))))

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(x / z)))))
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) / (x / z))));
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[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in y around 0 82.8%

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

4. Taylor expanded in z around 0 94.8%

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

   1. associate-*r/ 94.8%

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

   2. metadata-eval 94.8%

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

   3. associate-*r/ 94.8%

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

   4. metadata-eval 94.8%

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

6. Simplified 94.8%

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

7. Taylor expanded in z around inf 90.6%

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

   1. unpow2 90.6%

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

   2. associate-*l* 94.0%

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

   3. distribute-rgt-in 90.5%

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

   4. associate-*r/ 90.5%

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

   5. metadata-eval 90.5%

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

   6. associate-*l/ 90.5%

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

   7. associate-*r/ 90.1%

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

   8. associate-*l/ 93.8%

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

   9. associate-/l* 91.9%

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

   10. distribute-rgt-out 98.1%

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

9. Simplified 98.1%

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

    1. *-commutative 98.1%

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

    2. clear-num 98.1%

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

    3. un-div-inv 98.1%

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

11. Applied egg-rr 98.1%

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

12. Final simplification 98.1%

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

Alternative 2: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) - \frac{z \cdot \left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(0.083333333333333 \cdot \frac{1}{x} + z \cdot \frac{z \cdot y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.55e+141)
   (-
    (* x (+ (log x) -1.0))
    (/
     (-
      (* z (- 0.0027777777777778 (* z (+ 0.0007936500793651 y))))
      0.083333333333333)
     x))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (+ (* 0.083333333333333 (/ 1.0 x)) (* z (/ (* z y) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.55e+141) {
		tmp = (x * (log(x) + -1.0)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.55d+141) then
        tmp = (x * (log(x) + (-1.0d0))) - (((z * (0.0027777777777778d0 - (z * (0.0007936500793651d0 + y)))) - 0.083333333333333d0) / x)
    else
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((0.083333333333333d0 * (1.0d0 / x)) + (z * ((z * y) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.55e+141) {
		tmp = (x * (Math.log(x) + -1.0)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
	} else {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.55e+141:
		tmp = (x * (math.log(x) + -1.0)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x)
	else:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.55e+141)
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) - Float64(Float64(Float64(z * Float64(0.0027777777777778 - Float64(z * Float64(0.0007936500793651 + y)))) - 0.083333333333333) / x));
	else
		tmp = 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(z * y) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.55e+141)
		tmp = (x * (log(x) + -1.0)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
	else
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.55e+141], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(z * N[(0.0027777777777778 - N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 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[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5499999999999999e141

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf 95.7%

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

      1. sub-neg 43.3%

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

      2. mul-1-neg 43.3%

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

      3. log-rec 43.3%

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

      4. remove-double-neg 43.3%

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

      5. metadata-eval 43.3%

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

      6. +-commutative 43.3%

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

   5. Simplified 95.7%

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

   if 2.5499999999999999e141 < x

   1. Initial program 90.0%

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

   3. Taylor expanded in y around 0 85.2%

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

   4. Taylor expanded in z around 0 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-*r/ 99.7%

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

      4. metadata-eval 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in z around inf 91.5%

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

      1. unpow2 91.5%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-in 99.7%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-*l/ 99.7%

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

      7. associate-*r/ 99.7%

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

      8. associate-*l/ 98.2%

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

      9. associate-/l* 99.7%

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

      10. distribute-rgt-out 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in y around inf 96.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) - \frac{z \cdot \left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(0.083333333333333 \cdot \frac{1}{x} + z \cdot \frac{z \cdot y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;x \leq 5 \cdot 10^{+137}:\\ \;\;\;\;t\_0 - \frac{z \cdot \left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(0.083333333333333 \cdot \frac{1}{x} + z \cdot \frac{z \cdot y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
   (if (<= x 5e+137)
     (-
      t_0
      (/
       (-
        (* z (- 0.0027777777777778 (* z (+ 0.0007936500793651 y))))
        0.083333333333333)
       x))
     (+ t_0 (+ (* 0.083333333333333 (/ 1.0 x)) (* z (/ (* z y) x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double tmp;
	if (x <= 5e+137) {
		tmp = t_0 - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
	} else {
		tmp = t_0 + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0
    if (x <= 5d+137) then
        tmp = t_0 - (((z * (0.0027777777777778d0 - (z * (0.0007936500793651d0 + y)))) - 0.083333333333333d0) / x)
    else
        tmp = t_0 + ((0.083333333333333d0 * (1.0d0 / x)) + (z * ((z * y) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * Math.log(x)) - x) + 0.91893853320467;
	double tmp;
	if (x <= 5e+137) {
		tmp = t_0 - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
	} else {
		tmp = t_0 + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x - 0.5) * math.log(x)) - x) + 0.91893853320467
	tmp = 0
	if x <= 5e+137:
		tmp = t_0 - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x)
	else:
		tmp = t_0 + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)
	tmp = 0.0
	if (x <= 5e+137)
		tmp = Float64(t_0 - Float64(Float64(Float64(z * Float64(0.0027777777777778 - Float64(z * Float64(0.0007936500793651 + y)))) - 0.083333333333333) / x));
	else
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 * Float64(1.0 / x)) + Float64(z * Float64(Float64(z * y) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	tmp = 0.0;
	if (x <= 5e+137)
		tmp = t_0 - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
	else
		tmp = t_0 + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[x, 5e+137], N[(t$95$0 - N[(N[(N[(z * N[(0.0027777777777778 - N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e137

   1. Initial program 97.2%

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

   if 5.0000000000000002e137 < x

   1. Initial program 90.5%

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

   3. Taylor expanded in y around 0 86.0%

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

   4. Taylor expanded in z around 0 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. associate-*r/ 99.6%

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

      4. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in z around inf 92.0%

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

      1. unpow2 92.0%

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

      2. associate-*l* 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-*r/ 99.6%

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

      8. associate-*l/ 98.2%

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

      9. associate-/l* 99.6%

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

      10. distribute-rgt-out 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in y around inf 96.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{z \cdot \left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(0.083333333333333 \cdot \frac{1}{x} + z \cdot \frac{z \cdot y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+137}:\\ \;\;\;\;\left(\log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right) - \frac{z \cdot \left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(0.083333333333333 \cdot \frac{1}{x} + z \cdot \frac{z \cdot y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.2e+137)
   (-
    (- (* (log x) (+ x -0.5)) (+ x -0.91893853320467))
    (/
     (-
      (* z (- 0.0027777777777778 (* z (+ 0.0007936500793651 y))))
      0.083333333333333)
     x))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (+ (* 0.083333333333333 (/ 1.0 x)) (* z (/ (* z y) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.2e+137) {
		tmp = ((log(x) * (x + -0.5)) - (x + -0.91893853320467)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 3.2d+137) then
        tmp = ((log(x) * (x + (-0.5d0))) - (x + (-0.91893853320467d0))) - (((z * (0.0027777777777778d0 - (z * (0.0007936500793651d0 + y)))) - 0.083333333333333d0) / x)
    else
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((0.083333333333333d0 * (1.0d0 / x)) + (z * ((z * y) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.2e+137) {
		tmp = ((Math.log(x) * (x + -0.5)) - (x + -0.91893853320467)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
	} else {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3.2e+137:
		tmp = ((math.log(x) * (x + -0.5)) - (x + -0.91893853320467)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x)
	else:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.2e+137)
		tmp = Float64(Float64(Float64(log(x) * Float64(x + -0.5)) - Float64(x + -0.91893853320467)) - Float64(Float64(Float64(z * Float64(0.0027777777777778 - Float64(z * Float64(0.0007936500793651 + y)))) - 0.083333333333333) / x));
	else
		tmp = 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(z * y) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3.2e+137)
		tmp = ((log(x) * (x + -0.5)) - (x + -0.91893853320467)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
	else
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 * (1.0 / x)) + (z * ((z * y) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.2e+137], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(z * N[(0.0027777777777778 - N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 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[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.20000000000000019e137

   1. Initial program 97.2%

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

      1. associate-+l- 97.2%

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

      2. sub-neg 97.2%

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

      3. metadata-eval 97.2%

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

      4. *-commutative 97.2%

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

      5. sub-neg 97.2%

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

      6. metadata-eval 97.2%

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

   4. Applied egg-rr 97.2%

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

   if 3.20000000000000019e137 < x

   1. Initial program 90.5%

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

   3. Taylor expanded in y around 0 86.0%

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

   4. Taylor expanded in z around 0 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. associate-*r/ 99.6%

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

      4. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in z around inf 92.0%

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

      1. unpow2 92.0%

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

      2. associate-*l* 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-*r/ 99.6%

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

      8. associate-*l/ 98.2%

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

      9. associate-/l* 99.6%

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

      10. distribute-rgt-out 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in y around inf 96.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+137}:\\ \;\;\;\;\left(\log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right) - \frac{z \cdot \left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(0.083333333333333 \cdot \frac{1}{x} + z \cdot \frac{z \cdot y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

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 95.4%

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

3. Taylor expanded in y around 0 82.8%

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

4. Taylor expanded in z around 0 94.8%

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

   1. associate-*r/ 94.8%

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

   2. metadata-eval 94.8%

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

   3. associate-*r/ 94.8%

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

   4. metadata-eval 94.8%

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

6. Simplified 94.8%

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

7. Taylor expanded in z around inf 90.6%

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

   1. unpow2 90.6%

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

   2. associate-*l* 94.0%

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

   3. distribute-rgt-in 90.5%

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

   4. associate-*r/ 90.5%

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

   5. metadata-eval 90.5%

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

   6. associate-*l/ 90.5%

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

   7. associate-*r/ 90.1%

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

   8. associate-*l/ 93.8%

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

   9. associate-/l* 91.9%

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

   10. distribute-rgt-out 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 6: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.6 \cdot 10^{-35}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + \left(0.083333333333333 \cdot \frac{1}{x} + \frac{z}{x} \cdot -0.0027777777777778\right)\\ \mathbf{else}:\\ \;\;\;\;\log x \cdot \left(x + -0.5\right) + \left(\left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 9.6e-35)
   (+
    (+ 0.91893853320467 (* (log x) -0.5))
    (+ (* 0.083333333333333 (/ 1.0 x)) (* (/ z x) -0.0027777777777778)))
   (+
    (* (log x) (+ x -0.5))
    (- (+ 0.91893853320467 (/ 0.083333333333333 x)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.6e-35) {
		tmp = (0.91893853320467 + (log(x) * -0.5)) + ((0.083333333333333 * (1.0 / x)) + ((z / x) * -0.0027777777777778));
	} else {
		tmp = (log(x) * (x + -0.5)) + ((0.91893853320467 + (0.083333333333333 / x)) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 9.6d-35) then
        tmp = (0.91893853320467d0 + (log(x) * (-0.5d0))) + ((0.083333333333333d0 * (1.0d0 / x)) + ((z / x) * (-0.0027777777777778d0)))
    else
        tmp = (log(x) * (x + (-0.5d0))) + ((0.91893853320467d0 + (0.083333333333333d0 / x)) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.6e-35) {
		tmp = (0.91893853320467 + (Math.log(x) * -0.5)) + ((0.083333333333333 * (1.0 / x)) + ((z / x) * -0.0027777777777778));
	} else {
		tmp = (Math.log(x) * (x + -0.5)) + ((0.91893853320467 + (0.083333333333333 / x)) - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 9.6e-35:
		tmp = (0.91893853320467 + (math.log(x) * -0.5)) + ((0.083333333333333 * (1.0 / x)) + ((z / x) * -0.0027777777777778))
	else:
		tmp = (math.log(x) * (x + -0.5)) + ((0.91893853320467 + (0.083333333333333 / x)) - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 9.6e-35)
		tmp = Float64(Float64(0.91893853320467 + Float64(log(x) * -0.5)) + Float64(Float64(0.083333333333333 * Float64(1.0 / x)) + Float64(Float64(z / x) * -0.0027777777777778)));
	else
		tmp = Float64(Float64(log(x) * Float64(x + -0.5)) + Float64(Float64(0.91893853320467 + Float64(0.083333333333333 / x)) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 9.6e-35)
		tmp = (0.91893853320467 + (log(x) * -0.5)) + ((0.083333333333333 * (1.0 / x)) + ((z / x) * -0.0027777777777778));
	else
		tmp = (log(x) * (x + -0.5)) + ((0.91893853320467 + (0.083333333333333 / x)) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 9.6e-35], N[(N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * -0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.6000000000000005e-35

   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

   3. Taylor expanded in y around 0 82.0%

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

   4. Taylor expanded in z around 0 88.2%

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

      1. associate-*r/ 88.2%

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

      2. metadata-eval 88.2%

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

      3. associate-*r/ 88.2%

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

      4. metadata-eval 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in x around 0 88.2%

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

   8. Taylor expanded in z around 0 52.7%

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

      1. *-commutative 52.7%

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

   10. Simplified 52.7%

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

   if 9.6000000000000005e-35 < x

   1. Initial program 92.2%

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

      1. associate-+l+ 92.3%

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

      2. fma-neg 92.2%

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

      3. sub-neg 92.2%

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

      4. metadata-eval 92.2%

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

      5. fma-define 92.2%

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

      6. fma-neg 92.2%

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

      7. metadata-eval 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in z around 0 66.6%

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

      1. fma-neg 66.7%

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

      2. *-commutative 66.7%

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

   7. Applied egg-rr 66.7%

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

      1. associate-+l- 66.7%

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

   9. Applied egg-rr 66.7%

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

       1. +-commutative 66.7%

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

   11. Simplified 66.7%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.6 \cdot 10^{-35}:\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + \left(0.083333333333333 \cdot \frac{1}{x} + \frac{z}{x} \cdot -0.0027777777777778\right)\\ \mathbf{else}:\\ \;\;\;\;\log x \cdot \left(x + -0.5\right) + \left(\left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\log x + -1\right) - \frac{z \cdot \left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) - 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (-
  (* x (+ (log x) -1.0))
  (/
   (-
    (* z (- 0.0027777777777778 (* z (+ 0.0007936500793651 y))))
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return (x * (log(x) + -1.0)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 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))) - (((z * (0.0027777777777778d0 - (z * (0.0007936500793651d0 + y)))) - 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.log(x) + -1.0)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return (x * (math.log(x) + -1.0)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(log(x) + -1.0)) - Float64(Float64(Float64(z * Float64(0.0027777777777778 - Float64(z * Float64(0.0007936500793651 + y)))) - 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (log(x) + -1.0)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x);
end

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in x around inf 94.2%

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

   1. sub-neg 54.3%

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

   2. mul-1-neg 54.3%

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

   3. log-rec 54.3%

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

   4. remove-double-neg 54.3%

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

   5. metadata-eval 54.3%

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

   6. +-commutative 54.3%

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

5. Simplified 94.2%

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

6. Final simplification 94.2%

   \[\leadsto x \cdot \left(\log x + -1\right) - \frac{z \cdot \left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) - 0.083333333333333}{x} \]
7. Add Preprocessing

Alternative 8: 92.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in x around 0 95.3%

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

   1. sub-neg 95.3%

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

   2. metadata-eval 95.3%

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

   3. distribute-rgt-in 95.4%

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

   4. *-commutative 95.4%

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

   5. neg-mul-1 95.4%

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

   6. associate-+l+ 95.4%

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

   7. +-commutative 95.4%

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

   8. distribute-rgt-in 95.4%

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

   9. sub-neg 95.4%

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

   10. associate--l+ 95.4%

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

   11. +-commutative 95.4%

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

   12. fma-define 95.4%

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

5. Simplified 95.4%

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

6. Taylor expanded in x around inf 94.2%

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

   1. mul-1-neg 54.3%

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

   2. distribute-rgt-neg-in 54.3%

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

   3. log-rec 54.3%

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

   4. remove-double-neg 54.3%

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

8. Simplified 94.2%

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

9. Final simplification 94.2%

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

Alternative 9: 56.4% 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 95.4%

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

3. Taylor expanded in z around 0 55.4%

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

4. Final simplification 55.4%

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

Alternative 10: 56.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. associate-+l+ 95.4%

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

   2. fma-neg 95.4%

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

   3. sub-neg 95.4%

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

   4. metadata-eval 95.4%

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

   5. fma-define 95.3%

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

   6. fma-neg 95.3%

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

   7. metadata-eval 95.3%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 55.4%

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

   1. fma-neg 55.5%

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

   2. *-commutative 55.5%

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

7. Applied egg-rr 55.5%

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

8. Final simplification 55.5%

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

Alternative 11: 56.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. associate-+l+ 95.4%

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

   2. fma-neg 95.4%

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

   3. sub-neg 95.4%

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

   4. metadata-eval 95.4%

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

   5. fma-define 95.3%

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

   6. fma-neg 95.3%

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

   7. metadata-eval 95.3%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 55.4%

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

   1. fma-neg 55.5%

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

   2. *-commutative 55.5%

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

7. Applied egg-rr 55.5%

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

   1. associate-+l- 55.5%

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

9. Applied egg-rr 55.5%

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

    1. +-commutative 55.5%

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

11. Simplified 55.5%

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

12. Final simplification 55.5%

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

Alternative 12: 55.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in z around 0 55.4%

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

4. Taylor expanded in x around inf 54.3%

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

   1. mul-1-neg 54.3%

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

   2. distribute-rgt-neg-in 54.3%

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

   3. log-rec 54.3%

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

   4. remove-double-neg 54.3%

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

6. Simplified 54.3%

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

7. Final simplification 54.3%

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

Alternative 13: 55.6% 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 95.4%

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

3. Taylor expanded in z around 0 55.4%

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

4. Taylor expanded in x around inf 54.3%

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

   1. sub-neg 54.3%

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

   2. mul-1-neg 54.3%

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

   3. log-rec 54.3%

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

   4. remove-double-neg 54.3%

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

   5. metadata-eval 54.3%

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

   6. +-commutative 54.3%

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

6. Simplified 54.3%

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

7. Final simplification 54.3%

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

Alternative 14: 55.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in x around 0 95.3%

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

   1. sub-neg 95.3%

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

   2. metadata-eval 95.3%

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

   3. distribute-rgt-in 95.4%

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

   4. *-commutative 95.4%

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

   5. neg-mul-1 95.4%

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

   6. associate-+l+ 95.4%

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

   7. +-commutative 95.4%

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

   8. distribute-rgt-in 95.4%

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

   9. sub-neg 95.4%

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

   10. associate--l+ 95.4%

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

   11. +-commutative 95.4%

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

   12. fma-define 95.4%

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

5. Simplified 95.4%

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

6. Taylor expanded in z around 0 55.5%

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

7. Taylor expanded in x around inf 54.3%

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

   1. mul-1-neg 54.3%

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

   2. distribute-rgt-neg-in 54.3%

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

   3. log-rec 54.3%

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

   4. remove-double-neg 54.3%

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

9. Simplified 54.3%

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

10. Final simplification 54.3%

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

Alternative 15: 23.2% 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 95.4%

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

   1. associate-+l+ 95.4%

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

   2. fma-neg 95.4%

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

   3. sub-neg 95.4%

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

   4. metadata-eval 95.4%

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

   5. fma-define 95.3%

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

   6. fma-neg 95.3%

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

   7. metadata-eval 95.3%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 55.4%

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

6. Taylor expanded in x around 0 20.7%

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

7. Final simplification 20.7%

   \[\leadsto \frac{0.083333333333333}{x} \]
8. Add Preprocessing

Alternative 16: 1.3% accurate, 61.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -x
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(-x)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := (-x)

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in x around 0 95.3%

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

   1. sub-neg 95.3%

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

   2. metadata-eval 95.3%

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

   3. distribute-rgt-in 95.4%

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

   4. *-commutative 95.4%

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

   5. neg-mul-1 95.4%

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

   6. associate-+l+ 95.4%

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

   7. +-commutative 95.4%

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

   8. distribute-rgt-in 95.4%

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

   9. sub-neg 95.4%

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

   10. associate--l+ 95.4%

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

   11. +-commutative 95.4%

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

   12. fma-define 95.4%

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

5. Simplified 95.4%

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

6. Taylor expanded in z around 0 55.5%

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

7. Taylor expanded in x around 0 20.1%

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

8. Taylor expanded in x around inf 1.3%

   \[\leadsto \color{blue}{-1 \cdot x} \]
9. Step-by-step derivation

   1. neg-mul-1 1.3%

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

10. Simplified 1.3%

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

11. Final simplification 1.3%

    \[\leadsto -x \]
12. 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 2024050 
(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)))