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

Percentage Accurate: 94.4% → 98.2%

Time: 17.8s

Alternatives: 13

Speedup: 1.0×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((z * ((0.0007936500793651 + y) / (x / z))) + (0.083333333333333 * (1.0 / x)));
end

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around 0 93.2%

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

4. Taylor expanded in x around 0 92.4%

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

   1. associate-/l* 97.9%

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

   2. fma-neg 97.9%

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

   3. metadata-eval 97.9%

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

6. Simplified 97.9%

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

7. Taylor expanded in z around inf 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-/l* 99.0%

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

9. Simplified 99.0%

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

    1. clear-num 99.0%

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

    2. un-div-inv 99.1%

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

11. Applied egg-rr 99.1%

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

12. Final simplification 99.1%

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

Alternative 2: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-43} \lor \neg \left(z \leq 2000000\right):\\ \;\;\;\;x \cdot \left(\left(\log x + -1\right) + \frac{0.083333333333333}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8.5e-43) (not (<= z 2000000.0)))
   (* x (+ (+ (log x) -1.0) (/ 0.083333333333333 (* x x))))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (* 0.083333333333333 (/ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.5e-43) || !(z <= 2000000.0)) {
		tmp = x * ((log(x) + -1.0) + (0.083333333333333 / (x * x)));
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 * (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-8.5d-43)) .or. (.not. (z <= 2000000.0d0))) then
        tmp = x * ((log(x) + (-1.0d0)) + (0.083333333333333d0 / (x * x)))
    else
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (0.083333333333333d0 * (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.5e-43) || !(z <= 2000000.0)) {
		tmp = x * ((Math.log(x) + -1.0) + (0.083333333333333 / (x * x)));
	} else {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 * (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8.5e-43) or not (z <= 2000000.0):
		tmp = x * ((math.log(x) + -1.0) + (0.083333333333333 / (x * x)))
	else:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 * (1.0 / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8.5e-43) || !(z <= 2000000.0))
		tmp = Float64(x * Float64(Float64(log(x) + -1.0) + Float64(0.083333333333333 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(0.083333333333333 * Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8.5e-43) || ~((z <= 2000000.0)))
		tmp = x * ((log(x) + -1.0) + (0.083333333333333 / (x * x)));
	else
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 * (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8.5e-43], N[Not[LessEqual[z, 2000000.0]], $MachinePrecision]], N[(x * N[(N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision] + N[(0.083333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.50000000000000056e-43 or 2e6 < z

   1. Initial program 87.1%

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

   3. Taylor expanded in z around 0 22.5%

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

   4. Taylor expanded in x around inf 22.6%

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

      1. sub-neg 22.6%

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

      2. mul-1-neg 22.6%

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

      3. log-rec 22.6%

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

      4. remove-double-neg 22.6%

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

      5. metadata-eval 22.6%

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

      6. +-commutative 22.6%

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

   6. Simplified 22.6%

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

   7. Taylor expanded in x around inf 37.1%

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

      1. +-commutative 37.1%

         \[\leadsto x \cdot \left(\color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + -1 \cdot \log \left(\frac{1}{x}\right)\right)} - 1\right) \]

      2. mul-1-neg 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}\right) - 1\right) \]

      3. log-rec 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(-\color{blue}{\left(-\log x\right)}\right)\right) - 1\right) \]

      4. remove-double-neg 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\log x}\right) - 1\right) \]

      5. associate--l+ 37.1%

         \[\leadsto x \cdot \color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x - 1\right)\right)} \]

      6. sub-neg 37.1%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\log x + \left(-1\right)\right)}\right) \]

      7. metadata-eval 37.1%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x + \color{blue}{-1}\right)\right) \]

      8. associate-*r/ 37.1%

         \[\leadsto x \cdot \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{{x}^{2}}} + \left(\log x + -1\right)\right) \]

      9. metadata-eval 37.1%

         \[\leadsto x \cdot \left(\frac{\color{blue}{0.083333333333333}}{{x}^{2}} + \left(\log x + -1\right)\right) \]

      10. +-commutative 37.1%

          \[\leadsto x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \color{blue}{\left(-1 + \log x\right)}\right) \]

   9. Simplified 37.1%

      \[\leadsto \color{blue}{x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \left(-1 + \log x\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 37.1%

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

   11. Applied egg-rr 37.1%

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

   if -8.50000000000000056e-43 < z < 2e6

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0 93.1%

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

      1. div-inv 46.3%

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

      2. *-commutative 46.3%

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

   5. Applied egg-rr 93.2%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-43} \lor \neg \left(z \leq 2000000\right):\\ \;\;\;\;x \cdot \left(\left(\log x + -1\right) + \frac{0.083333333333333}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-43} \lor \neg \left(z \leq 1850000\right):\\ \;\;\;\;x \cdot \left(\left(\log x + -1\right) + \frac{0.083333333333333}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{x \cdot 12.000000000000048}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9e-43) (not (<= z 1850000.0)))
   (* x (+ (+ (log x) -1.0) (/ 0.083333333333333 (* x x))))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/ 1.0 (* x 12.000000000000048)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9e-43) || !(z <= 1850000.0)) {
		tmp = x * ((log(x) + -1.0) + (0.083333333333333 / (x * x)));
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (1.0 / (x * 12.000000000000048));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-9d-43)) .or. (.not. (z <= 1850000.0d0))) then
        tmp = x * ((log(x) + (-1.0d0)) + (0.083333333333333d0 / (x * x)))
    else
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (1.0d0 / (x * 12.000000000000048d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9e-43) || !(z <= 1850000.0)) {
		tmp = x * ((Math.log(x) + -1.0) + (0.083333333333333 / (x * x)));
	} else {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (1.0 / (x * 12.000000000000048));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9e-43) or not (z <= 1850000.0):
		tmp = x * ((math.log(x) + -1.0) + (0.083333333333333 / (x * x)))
	else:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (1.0 / (x * 12.000000000000048))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9e-43) || !(z <= 1850000.0))
		tmp = Float64(x * Float64(Float64(log(x) + -1.0) + Float64(0.083333333333333 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(1.0 / Float64(x * 12.000000000000048)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9e-43) || ~((z <= 1850000.0)))
		tmp = x * ((log(x) + -1.0) + (0.083333333333333 / (x * x)));
	else
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (1.0 / (x * 12.000000000000048));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9e-43], N[Not[LessEqual[z, 1850000.0]], $MachinePrecision]], N[(x * N[(N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision] + N[(0.083333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.0000000000000005e-43 or 1.85e6 < z

   1. Initial program 87.1%

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

   3. Taylor expanded in z around 0 22.5%

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

   4. Taylor expanded in x around inf 22.6%

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

      1. sub-neg 22.6%

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

      2. mul-1-neg 22.6%

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

      3. log-rec 22.6%

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

      4. remove-double-neg 22.6%

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

      5. metadata-eval 22.6%

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

      6. +-commutative 22.6%

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

   6. Simplified 22.6%

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

   7. Taylor expanded in x around inf 37.1%

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

      1. +-commutative 37.1%

         \[\leadsto x \cdot \left(\color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + -1 \cdot \log \left(\frac{1}{x}\right)\right)} - 1\right) \]

      2. mul-1-neg 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}\right) - 1\right) \]

      3. log-rec 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(-\color{blue}{\left(-\log x\right)}\right)\right) - 1\right) \]

      4. remove-double-neg 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\log x}\right) - 1\right) \]

      5. associate--l+ 37.1%

         \[\leadsto x \cdot \color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x - 1\right)\right)} \]

      6. sub-neg 37.1%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\log x + \left(-1\right)\right)}\right) \]

      7. metadata-eval 37.1%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x + \color{blue}{-1}\right)\right) \]

      8. associate-*r/ 37.1%

         \[\leadsto x \cdot \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{{x}^{2}}} + \left(\log x + -1\right)\right) \]

      9. metadata-eval 37.1%

         \[\leadsto x \cdot \left(\frac{\color{blue}{0.083333333333333}}{{x}^{2}} + \left(\log x + -1\right)\right) \]

      10. +-commutative 37.1%

          \[\leadsto x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \color{blue}{\left(-1 + \log x\right)}\right) \]

   9. Simplified 37.1%

      \[\leadsto \color{blue}{x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \left(-1 + \log x\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 37.1%

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

   11. Applied egg-rr 37.1%

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

   if -9.0000000000000005e-43 < z < 1.85e6

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0 93.1%

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

      1. div-inv 46.3%

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

      2. *-commutative 46.3%

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

   5. Applied egg-rr 93.2%

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

      1. associate-/r/ 93.1%

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

      2. div-inv 93.2%

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

      3. metadata-eval 93.2%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

   7. Applied egg-rr 93.2%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-43} \lor \neg \left(z \leq 1850000\right):\\ \;\;\;\;x \cdot \left(\left(\log x + -1\right) + \frac{0.083333333333333}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{x \cdot 12.000000000000048}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 * (1.0 / x)) + (z * ((0.0007936500793651 + y) * (z / x))));
end

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around 0 93.2%

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

4. Taylor expanded in x around 0 92.4%

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

   1. associate-/l* 97.9%

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

   2. fma-neg 97.9%

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

   3. metadata-eval 97.9%

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

6. Simplified 97.9%

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

7. Taylor expanded in z around inf 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-/l* 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 5: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-43} \lor \neg \left(z \leq 1500000\right):\\ \;\;\;\;x \cdot \left(\left(\log x + -1\right) + \frac{0.083333333333333}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8e-43) (not (<= z 1500000.0)))
   (* x (+ (+ (log x) -1.0) (/ 0.083333333333333 (* x x))))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-43) || !(z <= 1500000.0)) {
		tmp = x * ((log(x) + -1.0) + (0.083333333333333 / (x * x)));
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-8d-43)) .or. (.not. (z <= 1500000.0d0))) then
        tmp = x * ((log(x) + (-1.0d0)) + (0.083333333333333d0 / (x * x)))
    else
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-43) || !(z <= 1500000.0)) {
		tmp = x * ((Math.log(x) + -1.0) + (0.083333333333333 / (x * x)));
	} else {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8e-43) or not (z <= 1500000.0):
		tmp = x * ((math.log(x) + -1.0) + (0.083333333333333 / (x * x)))
	else:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8e-43) || !(z <= 1500000.0))
		tmp = Float64(x * Float64(Float64(log(x) + -1.0) + Float64(0.083333333333333 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8e-43) || ~((z <= 1500000.0)))
		tmp = x * ((log(x) + -1.0) + (0.083333333333333 / (x * x)));
	else
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8e-43], N[Not[LessEqual[z, 1500000.0]], $MachinePrecision]], N[(x * N[(N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision] + N[(0.083333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.00000000000000062e-43 or 1.5e6 < z

   1. Initial program 87.1%

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

   3. Taylor expanded in z around 0 22.5%

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

   4. Taylor expanded in x around inf 22.6%

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

      1. sub-neg 22.6%

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

      2. mul-1-neg 22.6%

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

      3. log-rec 22.6%

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

      4. remove-double-neg 22.6%

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

      5. metadata-eval 22.6%

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

      6. +-commutative 22.6%

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

   6. Simplified 22.6%

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

   7. Taylor expanded in x around inf 37.1%

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

      1. +-commutative 37.1%

         \[\leadsto x \cdot \left(\color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + -1 \cdot \log \left(\frac{1}{x}\right)\right)} - 1\right) \]

      2. mul-1-neg 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}\right) - 1\right) \]

      3. log-rec 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(-\color{blue}{\left(-\log x\right)}\right)\right) - 1\right) \]

      4. remove-double-neg 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\log x}\right) - 1\right) \]

      5. associate--l+ 37.1%

         \[\leadsto x \cdot \color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x - 1\right)\right)} \]

      6. sub-neg 37.1%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\log x + \left(-1\right)\right)}\right) \]

      7. metadata-eval 37.1%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x + \color{blue}{-1}\right)\right) \]

      8. associate-*r/ 37.1%

         \[\leadsto x \cdot \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{{x}^{2}}} + \left(\log x + -1\right)\right) \]

      9. metadata-eval 37.1%

         \[\leadsto x \cdot \left(\frac{\color{blue}{0.083333333333333}}{{x}^{2}} + \left(\log x + -1\right)\right) \]

      10. +-commutative 37.1%

          \[\leadsto x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \color{blue}{\left(-1 + \log x\right)}\right) \]

   9. Simplified 37.1%

      \[\leadsto \color{blue}{x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \left(-1 + \log x\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 37.1%

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

   11. Applied egg-rr 37.1%

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

   if -8.00000000000000062e-43 < z < 1.5e6

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0 93.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-43} \lor \neg \left(z \leq 1500000\right):\\ \;\;\;\;x \cdot \left(\left(\log x + -1\right) + \frac{0.083333333333333}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-43} \lor \neg \left(z \leq 1820000\right):\\ \;\;\;\;x \cdot \left(\left(\log x + -1\right) + \frac{0.083333333333333}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.91893853320467 + \log x \cdot \left(x + -0.5\right)\right) - x\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9e-43) (not (<= z 1820000.0)))
   (* x (+ (+ (log x) -1.0) (/ 0.083333333333333 (* x x))))
   (+
    (- (+ 0.91893853320467 (* (log x) (+ x -0.5))) x)
    (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9e-43) || !(z <= 1820000.0)) {
		tmp = x * ((log(x) + -1.0) + (0.083333333333333 / (x * x)));
	} else {
		tmp = ((0.91893853320467 + (log(x) * (x + -0.5))) - x) + (0.083333333333333 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-9d-43)) .or. (.not. (z <= 1820000.0d0))) then
        tmp = x * ((log(x) + (-1.0d0)) + (0.083333333333333d0 / (x * x)))
    else
        tmp = ((0.91893853320467d0 + (log(x) * (x + (-0.5d0)))) - x) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9e-43) || !(z <= 1820000.0)) {
		tmp = x * ((Math.log(x) + -1.0) + (0.083333333333333 / (x * x)));
	} else {
		tmp = ((0.91893853320467 + (Math.log(x) * (x + -0.5))) - x) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9e-43) or not (z <= 1820000.0):
		tmp = x * ((math.log(x) + -1.0) + (0.083333333333333 / (x * x)))
	else:
		tmp = ((0.91893853320467 + (math.log(x) * (x + -0.5))) - x) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9e-43) || !(z <= 1820000.0))
		tmp = Float64(x * Float64(Float64(log(x) + -1.0) + Float64(0.083333333333333 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(0.91893853320467 + Float64(log(x) * Float64(x + -0.5))) - x) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9e-43) || ~((z <= 1820000.0)))
		tmp = x * ((log(x) + -1.0) + (0.083333333333333 / (x * x)));
	else
		tmp = ((0.91893853320467 + (log(x) * (x + -0.5))) - x) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9e-43], N[Not[LessEqual[z, 1820000.0]], $MachinePrecision]], N[(x * N[(N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision] + N[(0.083333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.0000000000000005e-43 or 1.82e6 < z

   1. Initial program 87.1%

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

   3. Taylor expanded in z around 0 22.5%

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

   4. Taylor expanded in x around inf 22.6%

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

      1. sub-neg 22.6%

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

      2. mul-1-neg 22.6%

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

      3. log-rec 22.6%

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

      4. remove-double-neg 22.6%

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

      5. metadata-eval 22.6%

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

      6. +-commutative 22.6%

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

   6. Simplified 22.6%

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

   7. Taylor expanded in x around inf 37.1%

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

      1. +-commutative 37.1%

         \[\leadsto x \cdot \left(\color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + -1 \cdot \log \left(\frac{1}{x}\right)\right)} - 1\right) \]

      2. mul-1-neg 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}\right) - 1\right) \]

      3. log-rec 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(-\color{blue}{\left(-\log x\right)}\right)\right) - 1\right) \]

      4. remove-double-neg 37.1%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\log x}\right) - 1\right) \]

      5. associate--l+ 37.1%

         \[\leadsto x \cdot \color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x - 1\right)\right)} \]

      6. sub-neg 37.1%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\log x + \left(-1\right)\right)}\right) \]

      7. metadata-eval 37.1%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x + \color{blue}{-1}\right)\right) \]

      8. associate-*r/ 37.1%

         \[\leadsto x \cdot \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{{x}^{2}}} + \left(\log x + -1\right)\right) \]

      9. metadata-eval 37.1%

         \[\leadsto x \cdot \left(\frac{\color{blue}{0.083333333333333}}{{x}^{2}} + \left(\log x + -1\right)\right) \]

      10. +-commutative 37.1%

          \[\leadsto x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \color{blue}{\left(-1 + \log x\right)}\right) \]

   9. Simplified 37.1%

      \[\leadsto \color{blue}{x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \left(-1 + \log x\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 37.1%

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

   11. Applied egg-rr 37.1%

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

   if -9.0000000000000005e-43 < z < 1.82e6

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0 93.1%

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

      1. sub-neg 93.1%

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

      2. associate-+l+ 93.1%

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

      3. +-commutative 93.1%

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

      4. sub-neg 93.1%

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

      5. associate-+r- 93.1%

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

      6. sub-neg 93.1%

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

      7. metadata-eval 93.1%

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

      8. *-commutative 93.1%

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

   5. Applied egg-rr 93.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-43} \lor \neg \left(z \leq 1820000\right):\\ \;\;\;\;x \cdot \left(\left(\log x + -1\right) + \frac{0.083333333333333}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.91893853320467 + \log x \cdot \left(x + -0.5\right)\right) - x\right) + \frac{0.083333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log x + -1\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{-43} \lor \neg \left(z \leq 6.6 \cdot 10^{+40}\right):\\ \;\;\;\;x \cdot \left(t\_0 + \frac{0.083333333333333}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0 + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (log x) -1.0)))
   (if (or (<= z -8.6e-43) (not (<= z 6.6e+40)))
     (* x (+ t_0 (/ 0.083333333333333 (* x x))))
     (+ (* x t_0) (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double t_0 = log(x) + -1.0;
	double tmp;
	if ((z <= -8.6e-43) || !(z <= 6.6e+40)) {
		tmp = x * (t_0 + (0.083333333333333 / (x * x)));
	} else {
		tmp = (x * t_0) + (0.083333333333333 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) + (-1.0d0)
    if ((z <= (-8.6d-43)) .or. (.not. (z <= 6.6d+40))) then
        tmp = x * (t_0 + (0.083333333333333d0 / (x * x)))
    else
        tmp = (x * t_0) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(x) + -1.0;
	double tmp;
	if ((z <= -8.6e-43) || !(z <= 6.6e+40)) {
		tmp = x * (t_0 + (0.083333333333333 / (x * x)));
	} else {
		tmp = (x * t_0) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(x) + -1.0
	tmp = 0
	if (z <= -8.6e-43) or not (z <= 6.6e+40):
		tmp = x * (t_0 + (0.083333333333333 / (x * x)))
	else:
		tmp = (x * t_0) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(x) + -1.0)
	tmp = 0.0
	if ((z <= -8.6e-43) || !(z <= 6.6e+40))
		tmp = Float64(x * Float64(t_0 + Float64(0.083333333333333 / Float64(x * x))));
	else
		tmp = Float64(Float64(x * t_0) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(x) + -1.0;
	tmp = 0.0;
	if ((z <= -8.6e-43) || ~((z <= 6.6e+40)))
		tmp = x * (t_0 + (0.083333333333333 / (x * x)));
	else
		tmp = (x * t_0) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[z, -8.6e-43], N[Not[LessEqual[z, 6.6e+40]], $MachinePrecision]], N[(x * N[(t$95$0 + N[(0.083333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.59999999999999927e-43 or 6.5999999999999997e40 < z

   1. Initial program 86.4%

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

   3. Taylor expanded in z around 0 19.3%

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

   4. Taylor expanded in x around inf 19.4%

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

      1. sub-neg 19.4%

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

      2. mul-1-neg 19.4%

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

      3. log-rec 19.4%

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

      4. remove-double-neg 19.4%

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

      5. metadata-eval 19.4%

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

      6. +-commutative 19.4%

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

   6. Simplified 19.4%

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

   7. Taylor expanded in x around inf 34.7%

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

      1. +-commutative 34.7%

         \[\leadsto x \cdot \left(\color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + -1 \cdot \log \left(\frac{1}{x}\right)\right)} - 1\right) \]

      2. mul-1-neg 34.7%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}\right) - 1\right) \]

      3. log-rec 34.7%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(-\color{blue}{\left(-\log x\right)}\right)\right) - 1\right) \]

      4. remove-double-neg 34.7%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\log x}\right) - 1\right) \]

      5. associate--l+ 34.7%

         \[\leadsto x \cdot \color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x - 1\right)\right)} \]

      6. sub-neg 34.7%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\log x + \left(-1\right)\right)}\right) \]

      7. metadata-eval 34.7%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x + \color{blue}{-1}\right)\right) \]

      8. associate-*r/ 34.7%

         \[\leadsto x \cdot \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{{x}^{2}}} + \left(\log x + -1\right)\right) \]

      9. metadata-eval 34.7%

         \[\leadsto x \cdot \left(\frac{\color{blue}{0.083333333333333}}{{x}^{2}} + \left(\log x + -1\right)\right) \]

      10. +-commutative 34.7%

          \[\leadsto x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \color{blue}{\left(-1 + \log x\right)}\right) \]

   9. Simplified 34.7%

      \[\leadsto \color{blue}{x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \left(-1 + \log x\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 34.7%

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

   11. Applied egg-rr 34.7%

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

   if -8.59999999999999927e-43 < z < 6.5999999999999997e40

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0 92.7%

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

   4. Taylor expanded in x around inf 91.5%

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

      1. sub-neg 91.5%

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

      2. mul-1-neg 91.5%

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

      3. log-rec 91.5%

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

      4. remove-double-neg 91.5%

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

      5. metadata-eval 91.5%

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

      6. +-commutative 91.5%

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

   6. Simplified 91.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{-43} \lor \neg \left(z \leq 6.6 \cdot 10^{+40}\right):\\ \;\;\;\;x \cdot \left(\left(\log x + -1\right) + \frac{0.083333333333333}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around 0 93.2%

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

4. Taylor expanded in x around 0 92.4%

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

   1. associate-/l* 97.9%

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

   2. fma-neg 97.9%

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

   3. metadata-eval 97.9%

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

6. Simplified 97.9%

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

7. Taylor expanded in z around inf 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-/l* 99.0%

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

9. Simplified 99.0%

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

10. Taylor expanded in x around inf 98.6%

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

    1. sub-neg 52.4%

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

    2. mul-1-neg 52.4%

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

    3. log-rec 52.4%

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

    4. remove-double-neg 52.4%

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

    5. metadata-eval 52.4%

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

    6. +-commutative 52.4%

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

12. Simplified 98.6%

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

13. Final simplification 98.6%

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

Alternative 9: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (log(x) + (-1.0d0))) + ((0.083333333333333d0 + (z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0))) / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in x around inf 91.9%

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

   1. sub-neg 52.4%

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

   2. mul-1-neg 52.4%

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

   3. log-rec 52.4%

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

   4. remove-double-neg 52.4%

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

   5. metadata-eval 52.4%

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

   6. +-commutative 52.4%

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

5. Simplified 91.9%

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

6. Final simplification 91.9%

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

Alternative 10: 56.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.8) (* 0.083333333333333 (/ 1.0 x)) (* x (+ (log x) -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.8) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = x * (log(x) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.8d0) then
        tmp = 0.083333333333333d0 * (1.0d0 / x)
    else
        tmp = x * (log(x) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.8) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = x * (Math.log(x) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.8:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = x * (math.log(x) + -1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.8)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = Float64(x * Float64(log(x) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.8)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = x * (log(x) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.8], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999998

   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 z around 0 44.6%

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

   4. Taylor expanded in x around inf 44.2%

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

      1. sub-neg 44.2%

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

      2. mul-1-neg 44.2%

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

      3. log-rec 44.2%

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

      4. remove-double-neg 44.2%

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

      5. metadata-eval 44.2%

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

      6. +-commutative 44.2%

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

   6. Simplified 44.2%

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

   7. Taylor expanded in x around 0 44.2%

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

      1. div-inv 44.2%

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

      2. *-commutative 44.2%

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

   9. Applied egg-rr 44.2%

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

   if 2.7999999999999998 < x

   1. Initial program 86.0%

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

   3. Taylor expanded in z around 0 60.1%

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

   4. Taylor expanded in x around inf 59.6%

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

      1. sub-neg 59.6%

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

      2. mul-1-neg 59.6%

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

      3. log-rec 59.6%

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

      4. remove-double-neg 59.6%

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

      5. metadata-eval 59.6%

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

      6. +-commutative 59.6%

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

   6. Simplified 59.6%

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

   7. Taylor expanded in x around inf 59.6%

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

      1. +-commutative 59.6%

         \[\leadsto x \cdot \left(\color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + -1 \cdot \log \left(\frac{1}{x}\right)\right)} - 1\right) \]

      2. mul-1-neg 59.6%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}\right) - 1\right) \]

      3. log-rec 59.6%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(-\color{blue}{\left(-\log x\right)}\right)\right) - 1\right) \]

      4. remove-double-neg 59.6%

         \[\leadsto x \cdot \left(\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\log x}\right) - 1\right) \]

      5. associate--l+ 59.6%

         \[\leadsto x \cdot \color{blue}{\left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x - 1\right)\right)} \]

      6. sub-neg 59.6%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\log x + \left(-1\right)\right)}\right) \]

      7. metadata-eval 59.6%

         \[\leadsto x \cdot \left(0.083333333333333 \cdot \frac{1}{{x}^{2}} + \left(\log x + \color{blue}{-1}\right)\right) \]

      8. associate-*r/ 59.6%

         \[\leadsto x \cdot \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{{x}^{2}}} + \left(\log x + -1\right)\right) \]

      9. metadata-eval 59.6%

         \[\leadsto x \cdot \left(\frac{\color{blue}{0.083333333333333}}{{x}^{2}} + \left(\log x + -1\right)\right) \]

      10. +-commutative 59.6%

          \[\leadsto x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \color{blue}{\left(-1 + \log x\right)}\right) \]

   9. Simplified 59.6%

      \[\leadsto \color{blue}{x \cdot \left(\frac{0.083333333333333}{{x}^{2}} + \left(-1 + \log x\right)\right)} \]

   10. Taylor expanded in x around inf 59.6%

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

       1. sub-neg 59.6%

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

       2. metadata-eval 59.6%

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

       3. mul-1-neg 59.6%

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

       4. log-rec 59.6%

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

       5. remove-double-neg 59.6%

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

       6. *-lft-identity 59.6%

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

       7. *-lft-identity 59.6%

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

       8. +-commutative 59.6%

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

   12. Simplified 59.6%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.2% 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 92.4%

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

3. Taylor expanded in z around 0 52.8%

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

4. Taylor expanded in x around inf 52.4%

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

   1. sub-neg 52.4%

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

   2. mul-1-neg 52.4%

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

   3. log-rec 52.4%

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

   4. remove-double-neg 52.4%

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

   5. metadata-eval 52.4%

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

   6. +-commutative 52.4%

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

6. Simplified 52.4%

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

7. Final simplification 52.4%

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

Alternative 12: 23.3% accurate, 24.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around 0 52.8%

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

4. Taylor expanded in x around inf 52.4%

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

   1. sub-neg 52.4%

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

   2. mul-1-neg 52.4%

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

   3. log-rec 52.4%

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

   4. remove-double-neg 52.4%

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

   5. metadata-eval 52.4%

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

   6. +-commutative 52.4%

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

6. Simplified 52.4%

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

7. Taylor expanded in x around 0 22.2%

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

   1. div-inv 22.2%

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

   2. *-commutative 22.2%

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

9. Applied egg-rr 22.2%

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

10. Final simplification 22.2%

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

Alternative 13: 23.3% 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 92.4%

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

3. Taylor expanded in z around 0 52.8%

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

4. Taylor expanded in x around inf 52.4%

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

   1. sub-neg 52.4%

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

   2. mul-1-neg 52.4%

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

   3. log-rec 52.4%

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

   4. remove-double-neg 52.4%

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

   5. metadata-eval 52.4%

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

   6. +-commutative 52.4%

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

6. Simplified 52.4%

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

7. Taylor expanded in x around 0 22.2%

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

8. Final simplification 22.2%

   \[\leadsto \frac{0.083333333333333}{x} \]
9. 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 2024096 
(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)))