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

Percentage Accurate: 94.3% → 99.4%

Time: 16.4s

Alternatives: 19

Speedup: 1.0×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.3 \cdot 10^{+54}:\\ \;\;\;\;\left(\left(-0.5 \cdot \log x + x \cdot \left(\log x + -1\right)\right) + 0.91893853320467\right) + \frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.3e+54)
   (+
    (+ (+ (* -0.5 (log x)) (* x (+ (log x) -1.0))) 0.91893853320467)
    (/
     (+
      (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
      0.083333333333333)
     x))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (* z (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.3e+54) {
		tmp = (((-0.5 * log(x)) + (x * (log(x) + -1.0))) + 0.91893853320467) + (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x);
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 5.3d+54) then
        tmp = ((((-0.5d0) * log(x)) + (x * (log(x) + (-1.0d0)))) + 0.91893853320467d0) + (((z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)) + 0.083333333333333d0) / x)
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (z * (z * ((y / x) + (0.0007936500793651d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.3e+54) {
		tmp = (((-0.5 * Math.log(x)) + (x * (Math.log(x) + -1.0))) + 0.91893853320467) + (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x);
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 5.3e+54:
		tmp = (((-0.5 * math.log(x)) + (x * (math.log(x) + -1.0))) + 0.91893853320467) + (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x)
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.3e+54)
		tmp = Float64(Float64(Float64(Float64(-0.5 * log(x)) + Float64(x * Float64(log(x) + -1.0))) + 0.91893853320467) + Float64(Float64(Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(z * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5.3e+54)
		tmp = (((-0.5 * log(x)) + (x * (log(x) + -1.0))) + 0.91893853320467) + (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x);
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.3e+54], N[(N[(N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.30000000000000018e54

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.7%

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

   if 5.30000000000000018e54 < x

   1. Initial program 83.0%

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

   2. Taylor expanded in y around 0 73.5%

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

      1. +-commutative 73.5%

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

      2. +-commutative 73.5%

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

      3. associate-+l+ 73.5%

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

      4. associate-/l* 73.6%

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

      5. associate-/r/ 73.6%

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

      6. fma-neg 73.6%

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

      7. metadata-eval 73.6%

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

      8. +-commutative 73.6%

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

      9. associate-/l* 78.6%

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

      10. unpow2 78.6%

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

      11. associate-/r* 90.5%

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

      12. associate-*r/ 90.5%

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

      13. metadata-eval 90.5%

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

   4. Simplified 90.5%

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

   5. Taylor expanded in z around inf 88.0%

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

      1. *-commutative 88.0%

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

      2. unpow2 88.0%

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

      3. associate-*l* 99.7%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.3 \cdot 10^{+54}:\\ \;\;\;\;\left(\left(-0.5 \cdot \log x + x \cdot \left(\log x + -1\right)\right) + 0.91893853320467\right) + \frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\\ \mathbf{if}\;x \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))))
   (if (<= x 3e+22)
     (+
      (/
       (+
        (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
        0.083333333333333)
       x)
      t_0)
     (+ t_0 (* z (* z (+ (/ y x) (/ 0.0007936500793651 x))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	double tmp;
	if (x <= 3e+22) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + t_0;
	} else {
		tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651 / 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 = 0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)
    if (x <= 3d+22) then
        tmp = (((z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + t_0
    else
        tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x);
	double tmp;
	if (x <= 3e+22) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + t_0;
	} else {
		tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)
	tmp = 0
	if x <= 3e+22:
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + t_0
	else:
		tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651 / x))))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x))
	tmp = 0.0
	if (x <= 3e+22)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + t_0);
	else
		tmp = Float64(t_0 + Float64(z * Float64(z * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	tmp = 0.0;
	if (x <= 3e+22)
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + t_0;
	else
		tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3e+22], N[(N[(N[(N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 + N[(z * N[(z * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3e22

   1. Initial program 99.7%

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

   if 3e22 < x

   1. Initial program 84.7%

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

   2. Taylor expanded in y around 0 74.6%

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

      1. +-commutative 74.6%

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

      2. +-commutative 74.6%

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

      3. associate-+l+ 74.6%

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

      4. associate-/l* 74.7%

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

      5. associate-/r/ 74.7%

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

      6. fma-neg 74.7%

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

      7. metadata-eval 74.7%

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

      8. +-commutative 74.7%

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

      9. associate-/l* 79.2%

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

      10. unpow2 79.2%

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

      11. associate-/r* 89.8%

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

      12. associate-*r/ 89.8%

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

      13. metadata-eval 89.8%

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

   4. Simplified 89.8%

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

   5. Taylor expanded in z around inf 89.2%

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

      1. *-commutative 89.2%

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

      2. unpow2 89.2%

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

      3. associate-*l* 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(\log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5e+21)
   (+
    (/
     (+
      (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
      0.083333333333333)
     x)
    (- (* (log x) (+ x -0.5)) (+ x -0.91893853320467)))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (* z (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5e+21) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((log(x) * (x + -0.5)) - (x + -0.91893853320467));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 5d+21) then
        tmp = (((z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + ((log(x) * (x + (-0.5d0))) - (x + (-0.91893853320467d0)))
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (z * (z * ((y / x) + (0.0007936500793651d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 5e+21) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((Math.log(x) * (x + -0.5)) - (x + -0.91893853320467));
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 5e+21:
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((math.log(x) * (x + -0.5)) - (x + -0.91893853320467))
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5e+21)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(Float64(log(x) * Float64(x + -0.5)) - Float64(x + -0.91893853320467)));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(z * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5e+21)
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((log(x) * (x + -0.5)) - (x + -0.91893853320467));
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5e+21], N[(N[(N[(N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e21

   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. Step-by-step derivation

      1. associate-+l- 50.8%

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

      2. sub-neg 50.8%

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

      3. metadata-eval 50.8%

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

      4. sub-neg 50.8%

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

      5. metadata-eval 50.8%

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

   3. Applied egg-rr 99.7%

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

   if 5e21 < x

   1. Initial program 84.7%

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

   2. Taylor expanded in y around 0 74.6%

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

      1. +-commutative 74.6%

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

      2. +-commutative 74.6%

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

      3. associate-+l+ 74.6%

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

      4. associate-/l* 74.7%

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

      5. associate-/r/ 74.7%

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

      6. fma-neg 74.7%

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

      7. metadata-eval 74.7%

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

      8. +-commutative 74.7%

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

      9. associate-/l* 79.2%

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

      10. unpow2 79.2%

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

      11. associate-/r* 89.8%

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

      12. associate-*r/ 89.8%

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

      13. metadata-eval 89.8%

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

   4. Simplified 89.8%

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

   5. Taylor expanded in z around inf 89.2%

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

      1. *-commutative 89.2%

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

      2. unpow2 89.2%

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

      3. associate-*l* 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(\log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 4: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\\ \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(-0.5 \cdot \log x + 0.91893853320467\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+223}:\\ \;\;\;\;t_0 + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + z \cdot \frac{y}{\frac{x}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))))
   (if (<= x 33000000.0)
     (+
      (/
       (+
        (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
        0.083333333333333)
       x)
      (+ (* -0.5 (log x)) 0.91893853320467))
     (if (<= x 3.8e+223)
       (+ t_0 (/ (* z z) (/ x (+ y 0.0007936500793651))))
       (+ t_0 (* z (/ y (/ x z))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	double tmp;
	if (x <= 33000000.0) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * log(x)) + 0.91893853320467);
	} else if (x <= 3.8e+223) {
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)));
	} else {
		tmp = t_0 + (z * (y / (x / z)));
	}
	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 = 0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)
    if (x <= 33000000.0d0) then
        tmp = (((z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + (((-0.5d0) * log(x)) + 0.91893853320467d0)
    else if (x <= 3.8d+223) then
        tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651d0)))
    else
        tmp = t_0 + (z * (y / (x / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x);
	double tmp;
	if (x <= 33000000.0) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * Math.log(x)) + 0.91893853320467);
	} else if (x <= 3.8e+223) {
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)));
	} else {
		tmp = t_0 + (z * (y / (x / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)
	tmp = 0
	if x <= 33000000.0:
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * math.log(x)) + 0.91893853320467)
	elif x <= 3.8e+223:
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)))
	else:
		tmp = t_0 + (z * (y / (x / z)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x))
	tmp = 0.0
	if (x <= 33000000.0)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(Float64(-0.5 * log(x)) + 0.91893853320467));
	elseif (x <= 3.8e+223)
		tmp = Float64(t_0 + Float64(Float64(z * z) / Float64(x / Float64(y + 0.0007936500793651))));
	else
		tmp = Float64(t_0 + Float64(z * Float64(y / Float64(x / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	tmp = 0.0;
	if (x <= 33000000.0)
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * log(x)) + 0.91893853320467);
	elseif (x <= 3.8e+223)
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)));
	else
		tmp = t_0 + (z * (y / (x / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 33000000.0], N[(N[(N[(N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+223], N[(t$95$0 + N[(N[(z * z), $MachinePrecision] / N[(x / N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(z * N[(y / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.3e7

   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. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 99.2%

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

   if 3.3e7 < x < 3.8e223

   1. Initial program 88.8%

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

   2. Taylor expanded in z around inf 88.8%

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

      1. associate-/l* 30.6%

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

      2. unpow2 30.6%

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

   4. Simplified 92.5%

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

   if 3.8e223 < x

   1. Initial program 73.5%

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

   2. Taylor expanded in y around 0 73.2%

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

      1. +-commutative 73.2%

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

      2. +-commutative 73.2%

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

      3. associate-+l+ 73.2%

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

      4. associate-/l* 73.5%

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

      5. associate-/r/ 73.5%

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

      6. fma-neg 73.5%

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

      7. metadata-eval 73.5%

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

      8. +-commutative 73.5%

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

      9. associate-/l* 79.9%

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

      10. unpow2 79.9%

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

      11. associate-/r* 99.4%

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

      12. associate-*r/ 99.4%

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

      13. metadata-eval 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in y around inf 73.5%

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

      1. associate-*l/ 79.2%

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

      2. unpow2 79.2%

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

      3. associate-*r* 98.9%

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

      4. *-commutative 98.9%

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

      5. associate-*r* 98.9%

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

      6. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in z around 0 95.6%

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

      1. associate-/l* 98.9%

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

   10. Simplified 98.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(-0.5 \cdot \log x + 0.91893853320467\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+223}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{y}{\frac{x}{z}}\\ \end{array} \]

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.075:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(-0.5 \cdot \log x + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.075)
   (+
    (/
     (+
      (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
      0.083333333333333)
     x)
    (+ (* -0.5 (log x)) 0.91893853320467))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (* z (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.075) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * log(x)) + 0.91893853320467);
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.075d0) then
        tmp = (((z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + (((-0.5d0) * log(x)) + 0.91893853320467d0)
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (z * (z * ((y / x) + (0.0007936500793651d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.075) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * Math.log(x)) + 0.91893853320467);
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.075:
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * math.log(x)) + 0.91893853320467)
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.075)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(Float64(-0.5 * log(x)) + 0.91893853320467));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(z * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.075)
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * log(x)) + 0.91893853320467);
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.075], N[(N[(N[(N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0749999999999999972

   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. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 99.2%

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

   if 0.0749999999999999972 < x

   1. Initial program 86.2%

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

   2. Taylor expanded in y around 0 75.7%

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

      1. +-commutative 75.7%

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

      2. +-commutative 75.7%

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

      3. associate-+l+ 75.7%

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

      4. associate-/l* 75.8%

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

      5. associate-/r/ 75.8%

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

      6. fma-neg 75.8%

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

      7. metadata-eval 75.8%

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

      8. +-commutative 75.8%

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

      9. associate-/l* 79.8%

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

      10. unpow2 79.8%

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

      11. associate-/r* 89.3%

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

      12. associate-*r/ 89.3%

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

      13. metadata-eval 89.3%

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

   4. Simplified 89.3%

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

   5. Taylor expanded in z around inf 90.2%

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

      1. *-commutative 90.2%

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

      2. unpow2 90.2%

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

      3. associate-*l* 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.075:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(-0.5 \cdot \log x + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 6: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \log x + 0.91893853320467\\ \mathbf{if}\;x \leq 6.2 \cdot 10^{-50}:\\ \;\;\;\;t_0 + \frac{0.083333333333333 + z \cdot \left(y \cdot z\right)}{x}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;t_0 + \frac{0.083333333333333 + z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* -0.5 (log x)) 0.91893853320467)))
   (if (<= x 6.2e-50)
     (+ t_0 (/ (+ 0.083333333333333 (* z (* y z))) x))
     (if (<= x 33000000.0)
       (+
        t_0
        (/
         (+
          0.083333333333333
          (* z (- (* 0.0007936500793651 z) 0.0027777777777778)))
         x))
       (+ (* x (+ (log x) -1.0)) (* z (* z (/ y x))))))))

C

double code(double x, double y, double z) {
	double t_0 = (-0.5 * log(x)) + 0.91893853320467;
	double tmp;
	if (x <= 6.2e-50) {
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	} else if (x <= 33000000.0) {
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x);
	} else {
		tmp = (x * (log(x) + -1.0)) + (z * (z * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-0.5d0) * log(x)) + 0.91893853320467d0
    if (x <= 6.2d-50) then
        tmp = t_0 + ((0.083333333333333d0 + (z * (y * z))) / x)
    else if (x <= 33000000.0d0) then
        tmp = t_0 + ((0.083333333333333d0 + (z * ((0.0007936500793651d0 * z) - 0.0027777777777778d0))) / x)
    else
        tmp = (x * (log(x) + (-1.0d0))) + (z * (z * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-0.5 * Math.log(x)) + 0.91893853320467;
	double tmp;
	if (x <= 6.2e-50) {
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	} else if (x <= 33000000.0) {
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x);
	} else {
		tmp = (x * (Math.log(x) + -1.0)) + (z * (z * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-0.5 * math.log(x)) + 0.91893853320467
	tmp = 0
	if x <= 6.2e-50:
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x)
	elif x <= 33000000.0:
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x)
	else:
		tmp = (x * (math.log(x) + -1.0)) + (z * (z * (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-0.5 * log(x)) + 0.91893853320467)
	tmp = 0.0
	if (x <= 6.2e-50)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(y * z))) / x));
	elseif (x <= 33000000.0)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778))) / x));
	else
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(z * Float64(z * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-0.5 * log(x)) + 0.91893853320467;
	tmp = 0.0;
	if (x <= 6.2e-50)
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	elseif (x <= 33000000.0)
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x);
	else
		tmp = (x * (log(x) + -1.0)) + (z * (z * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[x, 6.2e-50], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 33000000.0], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.2000000000000004e-50

   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. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   4. Taylor expanded in y around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. unpow2 90.0%

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

      3. associate-*l* 88.2%

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

   6. Simplified 88.2%

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

   if 6.2000000000000004e-50 < x < 3.3e7

   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. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 97.5%

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

   4. Taylor expanded in y around 0 83.1%

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

      1. *-commutative 83.1%

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

   6. Simplified 83.1%

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

   if 3.3e7 < x

   1. Initial program 85.5%

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

   2. Taylor expanded in y around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. +-commutative 76.0%

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

      3. associate-+l+ 76.0%

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

      4. associate-/l* 76.0%

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

      5. associate-/r/ 76.0%

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

      6. fma-neg 76.0%

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

      7. metadata-eval 76.0%

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

      8. +-commutative 76.0%

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

      9. associate-/l* 80.2%

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

      10. unpow2 80.2%

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

      11. associate-/r* 90.3%

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

      12. associate-*r/ 90.3%

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

      13. metadata-eval 90.3%

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

   4. Simplified 90.3%

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

   5. Taylor expanded in y around inf 79.3%

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

      1. associate-*l/ 82.5%

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

      2. unpow2 82.5%

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

      3. associate-*r* 87.7%

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

      4. *-commutative 87.7%

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

      5. associate-*r* 87.7%

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

      6. *-commutative 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in x around inf 87.5%

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

      1. *-commutative 68.5%

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

      2. sub-neg 68.5%

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

      3. mul-1-neg 68.5%

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

      4. log-rec 68.5%

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

      5. remove-double-neg 68.5%

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

      6. metadata-eval 68.5%

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

   10. Simplified 87.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-50}:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(y \cdot z\right)}{x}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 7: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \log x + 0.91893853320467\\ \mathbf{if}\;x \leq 5 \cdot 10^{-51}:\\ \;\;\;\;t_0 + \frac{0.083333333333333 + z \cdot \left(y \cdot z\right)}{x}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;t_0 + \frac{0.083333333333333 + z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* -0.5 (log x)) 0.91893853320467)))
   (if (<= x 5e-51)
     (+ t_0 (/ (+ 0.083333333333333 (* z (* y z))) x))
     (if (<= x 33000000.0)
       (+
        t_0
        (/
         (+
          0.083333333333333
          (* z (- (* 0.0007936500793651 z) 0.0027777777777778)))
         x))
       (+
        (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
        (* z (* z (/ y x))))))))

C

double code(double x, double y, double z) {
	double t_0 = (-0.5 * log(x)) + 0.91893853320467;
	double tmp;
	if (x <= 5e-51) {
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	} else if (x <= 33000000.0) {
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x);
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-0.5d0) * log(x)) + 0.91893853320467d0
    if (x <= 5d-51) then
        tmp = t_0 + ((0.083333333333333d0 + (z * (y * z))) / x)
    else if (x <= 33000000.0d0) then
        tmp = t_0 + ((0.083333333333333d0 + (z * ((0.0007936500793651d0 * z) - 0.0027777777777778d0))) / x)
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (z * (z * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-0.5 * Math.log(x)) + 0.91893853320467;
	double tmp;
	if (x <= 5e-51) {
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	} else if (x <= 33000000.0) {
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x);
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (z * (z * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-0.5 * math.log(x)) + 0.91893853320467
	tmp = 0
	if x <= 5e-51:
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x)
	elif x <= 33000000.0:
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x)
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (z * (z * (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-0.5 * log(x)) + 0.91893853320467)
	tmp = 0.0
	if (x <= 5e-51)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(y * z))) / x));
	elseif (x <= 33000000.0)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778))) / x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(z * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-0.5 * log(x)) + 0.91893853320467;
	tmp = 0.0;
	if (x <= 5e-51)
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	elseif (x <= 33000000.0)
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x);
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[x, 5e-51], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 33000000.0], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.00000000000000004e-51

   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. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   4. Taylor expanded in y around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. unpow2 90.0%

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

      3. associate-*l* 88.2%

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

   6. Simplified 88.2%

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

   if 5.00000000000000004e-51 < x < 3.3e7

   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. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 97.5%

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

   4. Taylor expanded in y around 0 83.1%

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

      1. *-commutative 83.1%

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

   6. Simplified 83.1%

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

   if 3.3e7 < x

   1. Initial program 85.5%

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

   2. Taylor expanded in y around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. +-commutative 76.0%

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

      3. associate-+l+ 76.0%

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

      4. associate-/l* 76.0%

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

      5. associate-/r/ 76.0%

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

      6. fma-neg 76.0%

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

      7. metadata-eval 76.0%

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

      8. +-commutative 76.0%

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

      9. associate-/l* 80.2%

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

      10. unpow2 80.2%

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

      11. associate-/r* 90.3%

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

      12. associate-*r/ 90.3%

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

      13. metadata-eval 90.3%

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

   4. Simplified 90.3%

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

   5. Taylor expanded in y around inf 79.3%

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

      1. associate-*l/ 82.5%

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

      2. unpow2 82.5%

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

      3. associate-*r* 87.7%

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

      4. *-commutative 87.7%

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

      5. associate-*r* 87.7%

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

      6. *-commutative 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-51}:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(y \cdot z\right)}{x}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 8: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \log x + 0.91893853320467\\ \mathbf{if}\;x \leq 3 \cdot 10^{-51}:\\ \;\;\;\;t_0 + \frac{0.083333333333333 + z \cdot \left(y \cdot z\right)}{x}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;t_0 + \frac{0.083333333333333 + z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{y}{\frac{\frac{x}{z}}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* -0.5 (log x)) 0.91893853320467)))
   (if (<= x 3e-51)
     (+ t_0 (/ (+ 0.083333333333333 (* z (* y z))) x))
     (if (<= x 33000000.0)
       (+
        t_0
        (/
         (+
          0.083333333333333
          (* z (- (* 0.0007936500793651 z) 0.0027777777777778)))
         x))
       (+
        (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
        (/ y (/ (/ x z) z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (-0.5 * log(x)) + 0.91893853320467;
	double tmp;
	if (x <= 3e-51) {
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	} else if (x <= 33000000.0) {
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x);
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (y / ((x / z) / z));
	}
	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 = ((-0.5d0) * log(x)) + 0.91893853320467d0
    if (x <= 3d-51) then
        tmp = t_0 + ((0.083333333333333d0 + (z * (y * z))) / x)
    else if (x <= 33000000.0d0) then
        tmp = t_0 + ((0.083333333333333d0 + (z * ((0.0007936500793651d0 * z) - 0.0027777777777778d0))) / x)
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (y / ((x / z) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-0.5 * Math.log(x)) + 0.91893853320467;
	double tmp;
	if (x <= 3e-51) {
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	} else if (x <= 33000000.0) {
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x);
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (y / ((x / z) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-0.5 * math.log(x)) + 0.91893853320467
	tmp = 0
	if x <= 3e-51:
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x)
	elif x <= 33000000.0:
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x)
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (y / ((x / z) / z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-0.5 * log(x)) + 0.91893853320467)
	tmp = 0.0
	if (x <= 3e-51)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(y * z))) / x));
	elseif (x <= 33000000.0)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778))) / x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(y / Float64(Float64(x / z) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-0.5 * log(x)) + 0.91893853320467;
	tmp = 0.0;
	if (x <= 3e-51)
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	elseif (x <= 33000000.0)
		tmp = t_0 + ((0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x);
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (y / ((x / z) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[x, 3e-51], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 33000000.0], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(y / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.00000000000000002e-51

   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. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   4. Taylor expanded in y around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. unpow2 90.0%

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

      3. associate-*l* 88.2%

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

   6. Simplified 88.2%

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

   if 3.00000000000000002e-51 < x < 3.3e7

   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. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 97.5%

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

   4. Taylor expanded in y around 0 83.1%

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

      1. *-commutative 83.1%

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

   6. Simplified 83.1%

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

   if 3.3e7 < x

   1. Initial program 85.5%

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

   2. Taylor expanded in y around inf 79.3%

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

      1. associate-/l* 83.5%

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

      2. unpow2 83.5%

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

      3. associate-/r* 88.4%

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

   4. Simplified 88.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-51}:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(y \cdot z\right)}{x}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{y}{\frac{\frac{x}{z}}{z}}\\ \end{array} \]

Alternative 9: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{+223}:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{y}{\frac{x}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.3e+223)
   (+
    (/
     (+
      (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
      0.083333333333333)
     x)
    (* x (+ (log x) -1.0)))
   (+ (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)) (* z (/ y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.3e+223) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + (x * (log(x) + -1.0));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (y / (x / z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 3.3d+223) then
        tmp = (((z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + (x * (log(x) + (-1.0d0)))
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (z * (y / (x / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.3e+223) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + (x * (Math.log(x) + -1.0));
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (z * (y / (x / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3.3e+223:
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + (x * (math.log(x) + -1.0))
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (z * (y / (x / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.3e+223)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(x * Float64(log(x) + -1.0)));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(y / Float64(x / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3.3e+223)
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + (x * (log(x) + -1.0));
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (y / (x / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.3e+223], N[(N[(N[(N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(y / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e223

   1. Initial program 94.8%

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

   2. Taylor expanded in x around inf 94.0%

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

      1. *-commutative 56.0%

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

      2. sub-neg 56.0%

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

      3. mul-1-neg 56.0%

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

      4. log-rec 56.0%

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

      5. remove-double-neg 56.0%

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

      6. metadata-eval 56.0%

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

   4. Simplified 94.0%

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

   if 3.3e223 < x

   1. Initial program 73.5%

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

   2. Taylor expanded in y around 0 73.2%

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

      1. +-commutative 73.2%

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

      2. +-commutative 73.2%

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

      3. associate-+l+ 73.2%

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

      4. associate-/l* 73.5%

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

      5. associate-/r/ 73.5%

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

      6. fma-neg 73.5%

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

      7. metadata-eval 73.5%

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

      8. +-commutative 73.5%

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

      9. associate-/l* 79.9%

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

      10. unpow2 79.9%

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

      11. associate-/r* 99.4%

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

      12. associate-*r/ 99.4%

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

      13. metadata-eval 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in y around inf 73.5%

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

      1. associate-*l/ 79.2%

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

      2. unpow2 79.2%

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

      3. associate-*r* 98.9%

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

      4. *-commutative 98.9%

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

      5. associate-*r* 98.9%

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

      6. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in z around 0 95.6%

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

      1. associate-/l* 98.9%

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

   10. Simplified 98.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{+223}:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{y}{\frac{x}{z}}\\ \end{array} \]

Alternative 10: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(-0.5 \cdot \log x + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 33000000.0)
   (+
    (/
     (+
      (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
      0.083333333333333)
     x)
    (+ (* -0.5 (log x)) 0.91893853320467))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (/ (+ y 0.0007936500793651) (/ (/ x z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * log(x)) + 0.91893853320467);
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((y + 0.0007936500793651) / ((x / z) / z));
	}
	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 <= 33000000.0d0) then
        tmp = (((z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + (((-0.5d0) * log(x)) + 0.91893853320467d0)
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + ((y + 0.0007936500793651d0) / ((x / z) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * Math.log(x)) + 0.91893853320467);
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + ((y + 0.0007936500793651) / ((x / z) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 33000000.0:
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * math.log(x)) + 0.91893853320467)
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + ((y + 0.0007936500793651) / ((x / z) / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 33000000.0)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(Float64(-0.5 * log(x)) + 0.91893853320467));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(y + 0.0007936500793651) / Float64(Float64(x / z) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 33000000.0)
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + ((-0.5 * log(x)) + 0.91893853320467);
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((y + 0.0007936500793651) / ((x / z) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 33000000.0], N[(N[(N[(N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(y + 0.0007936500793651), $MachinePrecision] / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e7

   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. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 99.2%

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

   if 3.3e7 < x

   1. Initial program 85.5%

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

      1. *-un-lft-identity 85.5%

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

      2. add-cube-cbrt 85.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1 \cdot \left(\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\right)}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}} \]

      3. times-frac 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{\sqrt[3]{x}}} \]

      4. pow2 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}} \cdot \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{\sqrt[3]{x}} \]

      5. *-commutative 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\color{blue}{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)} + 0.083333333333333}{\sqrt[3]{x}} \]

      6. fma-udef 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\color{blue}{\mathsf{fma}\left(z, \left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, 0.083333333333333\right)}}{\sqrt[3]{x}} \]

      7. fma-neg 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)}, 0.083333333333333\right)}{\sqrt[3]{x}} \]

      8. metadata-eval 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, \color{blue}{-0.0027777777777778}\right), 0.083333333333333\right)}{\sqrt[3]{x}} \]

   3. Applied egg-rr 85.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt[3]{x}}} \]
   4. Step-by-step derivation

      1. associate-*l/ 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt[3]{x}}}{{\left(\sqrt[3]{x}\right)}^{2}}} \]

      2. *-lft-identity 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt[3]{x}}}}{{\left(\sqrt[3]{x}\right)}^{2}} \]

      3. fma-def 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778}, 0.083333333333333\right)}{\sqrt[3]{x}}}{{\left(\sqrt[3]{x}\right)}^{2}} \]

      4. *-commutative 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(y + 0.0007936500793651\right)} + -0.0027777777777778, 0.083333333333333\right)}{\sqrt[3]{x}}}{{\left(\sqrt[3]{x}\right)}^{2}} \]

      5. fma-def 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right)}, 0.083333333333333\right)}{\sqrt[3]{x}}}{{\left(\sqrt[3]{x}\right)}^{2}} \]

      6. +-commutative 85.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{0.0007936500793651 + y}, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt[3]{x}}}{{\left(\sqrt[3]{x}\right)}^{2}} \]

   5. Simplified 85.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{\sqrt[3]{x}}}{{\left(\sqrt[3]{x}\right)}^{2}}} \]

   6. Taylor expanded in z around inf 85.5%

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

      1. *-commutative 85.5%

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

      2. associate-/l* 89.7%

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

      3. +-commutative 89.7%

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

      4. unpow2 89.7%

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

      5. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(-0.5 \cdot \log x + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}}\\ \end{array} \]

Alternative 11: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+14} \lor \neg \left(z \leq 3.2 \cdot 10^{+21}\right):\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.4e+14) (not (<= z 3.2e+21)))
   (+
    (+ (* -0.5 (log x)) 0.91893853320467)
    (* (+ y 0.0007936500793651) (/ (* z z) x)))
   (+
    (- (* (log x) (+ x -0.5)) (+ x -0.91893853320467))
    (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.4e+14) || !(z <= 3.2e+21)) {
		tmp = ((-0.5 * log(x)) + 0.91893853320467) + ((y + 0.0007936500793651) * ((z * z) / x));
	} else {
		tmp = ((log(x) * (x + -0.5)) - (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 <= (-4.4d+14)) .or. (.not. (z <= 3.2d+21))) then
        tmp = (((-0.5d0) * log(x)) + 0.91893853320467d0) + ((y + 0.0007936500793651d0) * ((z * z) / x))
    else
        tmp = ((log(x) * (x + (-0.5d0))) - (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 <= -4.4e+14) || !(z <= 3.2e+21)) {
		tmp = ((-0.5 * Math.log(x)) + 0.91893853320467) + ((y + 0.0007936500793651) * ((z * z) / x));
	} else {
		tmp = ((Math.log(x) * (x + -0.5)) - (x + -0.91893853320467)) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.4e+14) or not (z <= 3.2e+21):
		tmp = ((-0.5 * math.log(x)) + 0.91893853320467) + ((y + 0.0007936500793651) * ((z * z) / x))
	else:
		tmp = ((math.log(x) * (x + -0.5)) - (x + -0.91893853320467)) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.4e+14) || !(z <= 3.2e+21))
		tmp = Float64(Float64(Float64(-0.5 * log(x)) + 0.91893853320467) + Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x)));
	else
		tmp = Float64(Float64(Float64(log(x) * Float64(x + -0.5)) - Float64(x + -0.91893853320467)) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.4e+14) || ~((z <= 3.2e+21)))
		tmp = ((-0.5 * log(x)) + 0.91893853320467) + ((y + 0.0007936500793651) * ((z * z) / x));
	else
		tmp = ((log(x) * (x + -0.5)) - (x + -0.91893853320467)) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.4e+14], N[Not[LessEqual[z, 3.2e+21]], $MachinePrecision]], N[(N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4e14 or 3.2e21 < z

   1. Initial program 83.6%

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

   2. Taylor expanded in x around 0 83.6%

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

   3. Taylor expanded in x around 0 71.6%

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

   4. Taylor expanded in z around inf 71.5%

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

      1. associate-/l* 74.7%

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

      2. +-commutative 74.7%

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

      3. associate-/r/ 73.9%

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

      4. unpow2 73.9%

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

      5. +-commutative 73.9%

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

   6. Simplified 73.9%

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

   if -4.4e14 < z < 3.2e21

   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. Taylor expanded in z around 0 90.3%

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

      1. associate-+l- 90.3%

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

      2. sub-neg 90.3%

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

      3. metadata-eval 90.3%

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

      4. sub-neg 90.3%

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

      5. metadata-eval 90.3%

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

   4. Applied egg-rr 90.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+14} \lor \neg \left(z \leq 3.2 \cdot 10^{+21}\right):\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 12: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+14} \lor \neg \left(z \leq 1.05 \cdot 10^{+23}\right):\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.1e+14) (not (<= z 1.05e+23)))
   (+
    (+ (* -0.5 (log x)) 0.91893853320467)
    (/ (* z z) (/ x (+ y 0.0007936500793651))))
   (+
    (- (* (log x) (+ x -0.5)) (+ x -0.91893853320467))
    (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.1e+14) || !(z <= 1.05e+23)) {
		tmp = ((-0.5 * log(x)) + 0.91893853320467) + ((z * z) / (x / (y + 0.0007936500793651)));
	} else {
		tmp = ((log(x) * (x + -0.5)) - (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 <= (-5.1d+14)) .or. (.not. (z <= 1.05d+23))) then
        tmp = (((-0.5d0) * log(x)) + 0.91893853320467d0) + ((z * z) / (x / (y + 0.0007936500793651d0)))
    else
        tmp = ((log(x) * (x + (-0.5d0))) - (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 <= -5.1e+14) || !(z <= 1.05e+23)) {
		tmp = ((-0.5 * Math.log(x)) + 0.91893853320467) + ((z * z) / (x / (y + 0.0007936500793651)));
	} else {
		tmp = ((Math.log(x) * (x + -0.5)) - (x + -0.91893853320467)) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.1e+14) or not (z <= 1.05e+23):
		tmp = ((-0.5 * math.log(x)) + 0.91893853320467) + ((z * z) / (x / (y + 0.0007936500793651)))
	else:
		tmp = ((math.log(x) * (x + -0.5)) - (x + -0.91893853320467)) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.1e+14) || !(z <= 1.05e+23))
		tmp = Float64(Float64(Float64(-0.5 * log(x)) + 0.91893853320467) + Float64(Float64(z * z) / Float64(x / Float64(y + 0.0007936500793651))));
	else
		tmp = Float64(Float64(Float64(log(x) * Float64(x + -0.5)) - Float64(x + -0.91893853320467)) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.1e+14) || ~((z <= 1.05e+23)))
		tmp = ((-0.5 * log(x)) + 0.91893853320467) + ((z * z) / (x / (y + 0.0007936500793651)));
	else
		tmp = ((log(x) * (x + -0.5)) - (x + -0.91893853320467)) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.1e+14], N[Not[LessEqual[z, 1.05e+23]], $MachinePrecision]], N[(N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(x / N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1e14 or 1.0500000000000001e23 < z

   1. Initial program 83.6%

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

   2. Taylor expanded in x around 0 83.6%

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

   3. Taylor expanded in x around 0 71.6%

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

   4. Taylor expanded in z around inf 71.5%

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

      1. associate-/l* 74.7%

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

      2. unpow2 74.7%

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

   6. Simplified 74.7%

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

   if -5.1e14 < z < 1.0500000000000001e23

   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. Taylor expanded in z around 0 90.3%

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

      1. associate-+l- 90.3%

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

      2. sub-neg 90.3%

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

      3. metadata-eval 90.3%

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

      4. sub-neg 90.3%

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

      5. metadata-eval 90.3%

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

   4. Applied egg-rr 90.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+14} \lor \neg \left(z \leq 1.05 \cdot 10^{+23}\right):\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 13: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \log x + 0.91893853320467\\ \mathbf{if}\;x \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;t_0 + \frac{0.083333333333333 + z \cdot \left(y \cdot z\right)}{x}\\ \mathbf{elif}\;x \leq 52000000:\\ \;\;\;\;t_0 + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* -0.5 (log x)) 0.91893853320467)))
   (if (<= x 3.8e-8)
     (+ t_0 (/ (+ 0.083333333333333 (* z (* y z))) x))
     (if (<= x 52000000.0)
       (+ t_0 (/ (* z z) (/ x (+ y 0.0007936500793651))))
       (+ (* x (+ (log x) -1.0)) (* z (* z (/ y x))))))))

C

double code(double x, double y, double z) {
	double t_0 = (-0.5 * log(x)) + 0.91893853320467;
	double tmp;
	if (x <= 3.8e-8) {
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	} else if (x <= 52000000.0) {
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)));
	} else {
		tmp = (x * (log(x) + -1.0)) + (z * (z * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-0.5d0) * log(x)) + 0.91893853320467d0
    if (x <= 3.8d-8) then
        tmp = t_0 + ((0.083333333333333d0 + (z * (y * z))) / x)
    else if (x <= 52000000.0d0) then
        tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651d0)))
    else
        tmp = (x * (log(x) + (-1.0d0))) + (z * (z * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-0.5 * Math.log(x)) + 0.91893853320467;
	double tmp;
	if (x <= 3.8e-8) {
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	} else if (x <= 52000000.0) {
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)));
	} else {
		tmp = (x * (Math.log(x) + -1.0)) + (z * (z * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-0.5 * math.log(x)) + 0.91893853320467
	tmp = 0
	if x <= 3.8e-8:
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x)
	elif x <= 52000000.0:
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)))
	else:
		tmp = (x * (math.log(x) + -1.0)) + (z * (z * (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-0.5 * log(x)) + 0.91893853320467)
	tmp = 0.0
	if (x <= 3.8e-8)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(y * z))) / x));
	elseif (x <= 52000000.0)
		tmp = Float64(t_0 + Float64(Float64(z * z) / Float64(x / Float64(y + 0.0007936500793651))));
	else
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(z * Float64(z * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-0.5 * log(x)) + 0.91893853320467;
	tmp = 0.0;
	if (x <= 3.8e-8)
		tmp = t_0 + ((0.083333333333333 + (z * (y * z))) / x);
	elseif (x <= 52000000.0)
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)));
	else
		tmp = (x * (log(x) + -1.0)) + (z * (z * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[x, 3.8e-8], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 52000000.0], N[(t$95$0 + N[(N[(z * z), $MachinePrecision] / N[(x / N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.80000000000000028e-8

   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. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   4. Taylor expanded in y around inf 86.6%

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

      1. *-commutative 86.6%

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

      2. unpow2 86.6%

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

      3. associate-*l* 85.0%

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

   6. Simplified 85.0%

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

   if 3.80000000000000028e-8 < x < 5.2e7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   3. Taylor expanded in x around 0 93.5%

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

   4. Taylor expanded in z around inf 90.0%

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

      1. associate-/l* 89.9%

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

      2. unpow2 89.9%

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

   6. Simplified 89.9%

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

   if 5.2e7 < x

   1. Initial program 85.5%

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

   2. Taylor expanded in y around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. +-commutative 76.0%

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

      3. associate-+l+ 76.0%

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

      4. associate-/l* 76.0%

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

      5. associate-/r/ 76.0%

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

      6. fma-neg 76.0%

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

      7. metadata-eval 76.0%

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

      8. +-commutative 76.0%

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

      9. associate-/l* 80.2%

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

      10. unpow2 80.2%

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

      11. associate-/r* 90.3%

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

      12. associate-*r/ 90.3%

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

      13. metadata-eval 90.3%

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

   4. Simplified 90.3%

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

   5. Taylor expanded in y around inf 79.3%

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

      1. associate-*l/ 82.5%

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

      2. unpow2 82.5%

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

      3. associate-*r* 87.7%

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

      4. *-commutative 87.7%

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

      5. associate-*r* 87.7%

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

      6. *-commutative 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in x around inf 87.5%

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

      1. *-commutative 68.5%

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

      2. sub-neg 68.5%

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

      3. mul-1-neg 68.5%

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

      4. log-rec 68.5%

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

      5. remove-double-neg 68.5%

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

      6. metadata-eval 68.5%

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

   10. Simplified 87.5%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(y \cdot z\right)}{x}\\ \mathbf{elif}\;x \leq 52000000:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 14: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\log x + -1\right)\\ \mathbf{if}\;z \leq -8 \cdot 10^{-17} \lor \neg \left(z \leq 1.95 \cdot 10^{-21}\right):\\ \;\;\;\;t_0 + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ (log x) -1.0))))
   (if (or (<= z -8e-17) (not (<= z 1.95e-21)))
     (+ t_0 (* z (* z (/ y x))))
     (+ t_0 (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x * (log(x) + -1.0);
	double tmp;
	if ((z <= -8e-17) || !(z <= 1.95e-21)) {
		tmp = t_0 + (z * (z * (y / x)));
	} else {
		tmp = t_0 + (0.083333333333333 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (log(x) + (-1.0d0))
    if ((z <= (-8d-17)) .or. (.not. (z <= 1.95d-21))) then
        tmp = t_0 + (z * (z * (y / x)))
    else
        tmp = t_0 + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (Math.log(x) + -1.0);
	double tmp;
	if ((z <= -8e-17) || !(z <= 1.95e-21)) {
		tmp = t_0 + (z * (z * (y / x)));
	} else {
		tmp = t_0 + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (math.log(x) + -1.0)
	tmp = 0
	if (z <= -8e-17) or not (z <= 1.95e-21):
		tmp = t_0 + (z * (z * (y / x)))
	else:
		tmp = t_0 + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(log(x) + -1.0))
	tmp = 0.0
	if ((z <= -8e-17) || !(z <= 1.95e-21))
		tmp = Float64(t_0 + Float64(z * Float64(z * Float64(y / x))));
	else
		tmp = Float64(t_0 + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (log(x) + -1.0);
	tmp = 0.0;
	if ((z <= -8e-17) || ~((z <= 1.95e-21)))
		tmp = t_0 + (z * (z * (y / x)));
	else
		tmp = t_0 + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[z, -8e-17], N[Not[LessEqual[z, 1.95e-21]], $MachinePrecision]], N[(t$95$0 + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.00000000000000057e-17 or 1.95e-21 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in y around 0 61.3%

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

      1. +-commutative 61.3%

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

      2. +-commutative 61.3%

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

      3. associate-+l+ 61.3%

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

      4. associate-/l* 61.3%

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

      5. associate-/r/ 61.3%

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

      6. fma-neg 61.3%

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

      7. metadata-eval 61.3%

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

      8. +-commutative 61.3%

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

      9. associate-/l* 62.4%

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

      10. unpow2 62.4%

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

      11. associate-/r* 72.6%

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

      12. associate-*r/ 72.6%

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

      13. metadata-eval 72.6%

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

   4. Simplified 72.6%

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

   5. Taylor expanded in y around inf 62.9%

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

      1. associate-*l/ 66.2%

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

      2. unpow2 66.2%

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

      3. associate-*r* 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-*r* 70.0%

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

      6. *-commutative 70.0%

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

   7. Simplified 70.0%

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

   8. Taylor expanded in x around inf 69.9%

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

      1. *-commutative 27.1%

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

      2. sub-neg 27.1%

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

      3. mul-1-neg 27.1%

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

      4. log-rec 27.1%

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

      5. remove-double-neg 27.1%

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

      6. metadata-eval 27.1%

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

   10. Simplified 69.9%

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

   if -8.00000000000000057e-17 < z < 1.95e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0 92.8%

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

   3. Taylor expanded in x around inf 91.2%

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

      1. *-commutative 91.2%

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

      2. sub-neg 91.2%

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

      3. mul-1-neg 91.2%

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

      4. log-rec 91.2%

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

      5. remove-double-neg 91.2%

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

      6. metadata-eval 91.2%

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

   5. Simplified 91.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-17} \lor \neg \left(z \leq 1.95 \cdot 10^{-21}\right):\\ \;\;\;\;x \cdot \left(\log x + -1\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 15: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-59} \lor \neg \left(z \leq 4.6 \cdot 10^{-22}\right):\\ \;\;\;\;x \cdot \left(\log x + -1\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5e-59) (not (<= z 4.6e-22)))
   (+ (* x (+ (log x) -1.0)) (* z (* z (/ y 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 <= -5.5e-59) || !(z <= 4.6e-22)) {
		tmp = (x * (log(x) + -1.0)) + (z * (z * (y / 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 <= (-5.5d-59)) .or. (.not. (z <= 4.6d-22))) then
        tmp = (x * (log(x) + (-1.0d0))) + (z * (z * (y / 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 <= -5.5e-59) || !(z <= 4.6e-22)) {
		tmp = (x * (Math.log(x) + -1.0)) + (z * (z * (y / 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 <= -5.5e-59) or not (z <= 4.6e-22):
		tmp = (x * (math.log(x) + -1.0)) + (z * (z * (y / 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 <= -5.5e-59) || !(z <= 4.6e-22))
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(z * Float64(z * Float64(y / x))));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(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 <= -5.5e-59) || ~((z <= 4.6e-22)))
		tmp = (x * (log(x) + -1.0)) + (z * (z * (y / 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, -5.5e-59], N[Not[LessEqual[z, 4.6e-22]], $MachinePrecision]], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.50000000000000014e-59 or 4.5999999999999996e-22 < z

   1. Initial program 86.2%

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

   2. Taylor expanded in y around 0 62.9%

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

      1. +-commutative 62.9%

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

      2. +-commutative 62.9%

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

      3. associate-+l+ 62.9%

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

      4. associate-/l* 63.0%

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

      5. associate-/r/ 63.0%

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

      6. fma-neg 63.0%

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

      7. metadata-eval 63.0%

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

      8. +-commutative 63.0%

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

      9. associate-/l* 64.1%

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

      10. unpow2 64.1%

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

      11. associate-/r* 73.8%

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

      12. associate-*r/ 73.8%

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

      13. metadata-eval 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in y around inf 63.1%

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

      1. associate-*l/ 66.2%

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

      2. unpow2 66.2%

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

      3. associate-*r* 69.9%

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

      4. *-commutative 69.9%

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

      5. associate-*r* 69.9%

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

      6. *-commutative 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in x around inf 69.8%

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

      1. *-commutative 28.9%

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

      2. sub-neg 28.9%

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

      3. mul-1-neg 28.9%

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

      4. log-rec 28.9%

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

      5. remove-double-neg 28.9%

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

      6. metadata-eval 28.9%

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

   10. Simplified 69.8%

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

   if -5.50000000000000014e-59 < z < 4.5999999999999996e-22

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0 94.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-59} \lor \neg \left(z \leq 4.6 \cdot 10^{-22}\right):\\ \;\;\;\;x \cdot \left(\log x + -1\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 16: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-57} \lor \neg \left(z \leq 3.6 \cdot 10^{-21}\right):\\ \;\;\;\;x \cdot \left(\log x + -1\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7.8e-57) (not (<= z 3.6e-21)))
   (+ (* x (+ (log x) -1.0)) (* z (* z (/ y x))))
   (+
    (- (* (log x) (+ x -0.5)) (+ x -0.91893853320467))
    (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.8e-57) || !(z <= 3.6e-21)) {
		tmp = (x * (log(x) + -1.0)) + (z * (z * (y / x)));
	} else {
		tmp = ((log(x) * (x + -0.5)) - (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 <= (-7.8d-57)) .or. (.not. (z <= 3.6d-21))) then
        tmp = (x * (log(x) + (-1.0d0))) + (z * (z * (y / x)))
    else
        tmp = ((log(x) * (x + (-0.5d0))) - (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 <= -7.8e-57) || !(z <= 3.6e-21)) {
		tmp = (x * (Math.log(x) + -1.0)) + (z * (z * (y / x)));
	} else {
		tmp = ((Math.log(x) * (x + -0.5)) - (x + -0.91893853320467)) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7.8e-57) or not (z <= 3.6e-21):
		tmp = (x * (math.log(x) + -1.0)) + (z * (z * (y / x)))
	else:
		tmp = ((math.log(x) * (x + -0.5)) - (x + -0.91893853320467)) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7.8e-57) || !(z <= 3.6e-21))
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(z * Float64(z * Float64(y / x))));
	else
		tmp = Float64(Float64(Float64(log(x) * Float64(x + -0.5)) - Float64(x + -0.91893853320467)) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -7.8e-57) || ~((z <= 3.6e-21)))
		tmp = (x * (log(x) + -1.0)) + (z * (z * (y / x)));
	else
		tmp = ((log(x) * (x + -0.5)) - (x + -0.91893853320467)) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7.8e-57], N[Not[LessEqual[z, 3.6e-21]], $MachinePrecision]], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.80000000000000013e-57 or 3.59999999999999989e-21 < z

   1. Initial program 86.2%

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

   2. Taylor expanded in y around 0 62.9%

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

      1. +-commutative 62.9%

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

      2. +-commutative 62.9%

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

      3. associate-+l+ 62.9%

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

      4. associate-/l* 63.0%

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

      5. associate-/r/ 63.0%

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

      6. fma-neg 63.0%

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

      7. metadata-eval 63.0%

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

      8. +-commutative 63.0%

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

      9. associate-/l* 64.1%

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

      10. unpow2 64.1%

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

      11. associate-/r* 73.8%

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

      12. associate-*r/ 73.8%

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

      13. metadata-eval 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in y around inf 63.1%

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

      1. associate-*l/ 66.2%

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

      2. unpow2 66.2%

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

      3. associate-*r* 69.9%

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

      4. *-commutative 69.9%

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

      5. associate-*r* 69.9%

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

      6. *-commutative 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in x around inf 69.8%

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

      1. *-commutative 28.9%

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

      2. sub-neg 28.9%

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

      3. mul-1-neg 28.9%

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

      4. log-rec 28.9%

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

      5. remove-double-neg 28.9%

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

      6. metadata-eval 28.9%

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

   10. Simplified 69.8%

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

   if -7.80000000000000013e-57 < z < 3.59999999999999989e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0 94.0%

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

      1. associate-+l- 94.0%

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

      2. sub-neg 94.0%

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

      3. metadata-eval 94.0%

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

      4. sub-neg 94.0%

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

      5. metadata-eval 94.0%

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

   4. Applied egg-rr 94.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-57} \lor \neg \left(z \leq 3.6 \cdot 10^{-21}\right):\\ \;\;\;\;x \cdot \left(\log x + -1\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 17: 61.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\log x + -1\right)\\ \mathbf{if}\;z \leq -124000000:\\ \;\;\;\;t_0 + z \cdot \frac{-0.0027777777777778}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ (log x) -1.0))))
   (if (<= z -124000000.0)
     (+ t_0 (* z (/ -0.0027777777777778 x)))
     (+ t_0 (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x * (log(x) + -1.0);
	double tmp;
	if (z <= -124000000.0) {
		tmp = t_0 + (z * (-0.0027777777777778 / x));
	} else {
		tmp = t_0 + (0.083333333333333 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (log(x) + (-1.0d0))
    if (z <= (-124000000.0d0)) then
        tmp = t_0 + (z * ((-0.0027777777777778d0) / x))
    else
        tmp = t_0 + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (Math.log(x) + -1.0);
	double tmp;
	if (z <= -124000000.0) {
		tmp = t_0 + (z * (-0.0027777777777778 / x));
	} else {
		tmp = t_0 + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (math.log(x) + -1.0)
	tmp = 0
	if z <= -124000000.0:
		tmp = t_0 + (z * (-0.0027777777777778 / x))
	else:
		tmp = t_0 + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(log(x) + -1.0))
	tmp = 0.0
	if (z <= -124000000.0)
		tmp = Float64(t_0 + Float64(z * Float64(-0.0027777777777778 / x)));
	else
		tmp = Float64(t_0 + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (log(x) + -1.0);
	tmp = 0.0;
	if (z <= -124000000.0)
		tmp = t_0 + (z * (-0.0027777777777778 / x));
	else
		tmp = t_0 + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -124000000.0], N[(t$95$0 + N[(z * N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.24e8

   1. Initial program 87.6%

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

   2. Taylor expanded in z around inf 77.5%

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

      1. +-commutative 77.5%

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

      2. associate-/l* 82.1%

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

      3. unpow2 82.1%

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

   4. Simplified 82.1%

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

   5. Taylor expanded in z around 0 33.2%

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

      1. metadata-eval 33.2%

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

      2. times-frac 33.2%

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

      3. *-commutative 33.2%

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

      4. times-frac 33.2%

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

      5. /-rgt-identity 33.2%

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

   7. Simplified 33.2%

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

   8. Taylor expanded in x around inf 33.2%

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

      1. *-commutative 24.9%

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

      2. sub-neg 24.9%

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

      3. mul-1-neg 24.9%

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

      4. log-rec 24.9%

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

      5. remove-double-neg 24.9%

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

      6. metadata-eval 24.9%

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

   10. Simplified 33.2%

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

   if -1.24e8 < z

   1. Initial program 94.0%

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

   2. Taylor expanded in z around 0 70.3%

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

   3. Taylor expanded in x around inf 69.3%

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

      1. *-commutative 69.3%

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

      2. sub-neg 69.3%

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

      3. mul-1-neg 69.3%

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

      4. log-rec 69.3%

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

      5. remove-double-neg 69.3%

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

      6. metadata-eval 69.3%

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

   5. Simplified 69.3%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -124000000:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + z \cdot \frac{-0.0027777777777778}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 18: 57.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.5%

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

2. Taylor expanded in z around 0 59.7%

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

3. Taylor expanded in x around inf 58.9%

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

   1. *-commutative 58.9%

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

   2. sub-neg 58.9%

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

   3. mul-1-neg 58.9%

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

   4. log-rec 58.9%

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

   5. remove-double-neg 58.9%

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

   6. metadata-eval 58.9%

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

5. Simplified 58.9%

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

6. Final simplification 58.9%

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

Alternative 19: 23.6% 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 92.5%

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

2. Taylor expanded in z around 0 59.7%

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

3. Taylor expanded in x around inf 58.9%

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

   1. *-commutative 58.9%

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

   2. sub-neg 58.9%

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

   3. mul-1-neg 58.9%

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

   4. log-rec 58.9%

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

   5. remove-double-neg 58.9%

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

   6. metadata-eval 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around 0 25.7%

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

7. Final simplification 25.7%

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

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 2023238 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- 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)))