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

Percentage Accurate: 94.3% → 99.5%

Time: 16.8s

Alternatives: 13

Speedup: 1.1×

• 28.2% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -0.5\right) \cdot \log x - \left(x + -0.91893853320467\right)\\ \mathbf{if}\;x \leq 5.3 \cdot 10^{+35}:\\ \;\;\;\;t_0 + \frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ x -0.5) (log x)) (+ x -0.91893853320467))))
   (if (<= x 5.3e+35)
     (+
      t_0
      (/
       (+
        (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
        0.083333333333333)
       x))
     (+ t_0 (* z (* z (+ (/ y x) (/ 0.0007936500793651 x))))))))

C

double code(double x, double y, double z) {
	double t_0 = ((x + -0.5) * log(x)) - (x + -0.91893853320467);
	double tmp;
	if (x <= 5.3e+35) {
		tmp = t_0 + (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x);
	} 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 = ((x + (-0.5d0)) * log(x)) - (x + (-0.91893853320467d0))
    if (x <= 5.3d+35) then
        tmp = t_0 + (((z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)) + 0.083333333333333d0) / x)
    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 = ((x + -0.5) * Math.log(x)) - (x + -0.91893853320467);
	double tmp;
	if (x <= 5.3e+35) {
		tmp = t_0 + (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x);
	} else {
		tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((x + -0.5) * math.log(x)) - (x + -0.91893853320467)
	tmp = 0
	if x <= 5.3e+35:
		tmp = t_0 + (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x)
	else:
		tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651 / x))))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x + -0.5) * log(x)) - Float64(x + -0.91893853320467))
	tmp = 0.0
	if (x <= 5.3e+35)
		tmp = Float64(t_0 + Float64(Float64(Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x));
	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 = ((x + -0.5) * log(x)) - (x + -0.91893853320467);
	tmp = 0.0;
	if (x <= 5.3e+35)
		tmp = t_0 + (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x);
	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[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.3e+35], N[(t$95$0 + N[(N[(N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $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 < 5.30000000000000009e35

   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- 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   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 5.30000000000000009e35 < x

   1. Initial program 89.4%

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

      1. associate-+l- 89.4%

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

      2. sub-neg 89.4%

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

      3. metadata-eval 89.4%

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

      4. sub-neg 89.4%

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

      5. metadata-eval 89.4%

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

   3. Applied egg-rr 89.4%

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

   4. Taylor expanded in z around inf 89.4%

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

      1. +-commutative 89.4%

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

      2. associate-+r+ 89.4%

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

      3. +-commutative 89.4%

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

      4. fma-def 89.4%

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

      5. associate-*r/ 89.4%

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

      6. metadata-eval 89.4%

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

      7. associate-/l* 92.6%

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

      8. +-commutative 92.6%

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

      9. associate-/r/ 92.6%

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

      10. unpow2 92.6%

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

      11. +-commutative 92.6%

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

   6. Simplified 92.6%

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

   7. Taylor expanded in z around inf 92.6%

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

      1. *-commutative 92.6%

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

      2. unpow2 92.6%

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

      3. associate-*l* 99.6%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.3 \cdot 10^{+35}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x - \left(x + -0.91893853320467\right)\right) + \frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x - \left(x + -0.91893853320467\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 2: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{y}{\frac{\frac{x}{z}}{z}} - x\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))))
   (if (<= t_0 -1e+150)
     (- (/ y (/ (/ x z) z)) x)
     (if (<= t_0 5e+88)
       (+ (* x (+ (log x) -1.0)) (/ 0.083333333333333 x))
       (* z (* z (+ (/ y x) (/ 0.0007936500793651 x))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (((y + 0.0007936500793651) * z) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -1e+150) {
		tmp = (y / ((x / z) / z)) - x;
	} else if (t_0 <= 5e+88) {
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
	} else {
		tmp = 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 = z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)
    if (t_0 <= (-1d+150)) then
        tmp = (y / ((x / z) / z)) - x
    else if (t_0 <= 5d+88) then
        tmp = (x * (log(x) + (-1.0d0))) + (0.083333333333333d0 / x)
    else
        tmp = 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 = z * (((y + 0.0007936500793651) * z) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -1e+150) {
		tmp = (y / ((x / z) / z)) - x;
	} else if (t_0 <= 5e+88) {
		tmp = (x * (Math.log(x) + -1.0)) + (0.083333333333333 / x);
	} else {
		tmp = z * (z * ((y / x) + (0.0007936500793651 / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)
	tmp = 0
	if t_0 <= -1e+150:
		tmp = (y / ((x / z) / z)) - x
	elif t_0 <= 5e+88:
		tmp = (x * (math.log(x) + -1.0)) + (0.083333333333333 / x)
	else:
		tmp = z * (z * ((y / x) + (0.0007936500793651 / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -1e+150)
		tmp = Float64(Float64(y / Float64(Float64(x / z) / z)) - x);
	elseif (t_0 <= 5e+88)
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(0.083333333333333 / x));
	else
		tmp = 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 = z * (((y + 0.0007936500793651) * z) - 0.0027777777777778);
	tmp = 0.0;
	if (t_0 <= -1e+150)
		tmp = (y / ((x / z) / z)) - x;
	elseif (t_0 <= 5e+88)
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
	else
		tmp = z * (z * ((y / x) + (0.0007936500793651 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+150], N[(N[(y / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[t$95$0, 5e+88], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) < -9.99999999999999981e149

   1. Initial program 85.7%

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

      1. add-sqr-sqrt 85.6%

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

      2. pow2 85.6%

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

      3. sub-neg 85.6%

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

      4. metadata-eval 85.6%

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

   3. Applied egg-rr 85.6%

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

   4. Taylor expanded in x around inf 76.7%

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

      1. neg-mul-1 76.7%

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

   6. Simplified 76.7%

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

   7. Taylor expanded in y around inf 76.6%

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

      1. associate-/l* 85.0%

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

      2. unpow2 85.0%

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

      3. associate-/r* 87.8%

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

   9. Simplified 87.8%

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

   if -9.99999999999999981e149 < (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) < 4.99999999999999997e88

   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 93.9%

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

   3. Taylor expanded in x around inf 91.0%

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

      1. *-commutative 91.0%

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

      2. sub-neg 91.0%

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

      3. mul-1-neg 91.0%

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

      4. log-rec 91.0%

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

      5. remove-double-neg 91.0%

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

      6. metadata-eval 91.0%

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

   5. Simplified 91.0%

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

   if 4.99999999999999997e88 < (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z)

   1. Initial program 92.9%

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

      1. add-sqr-sqrt 92.8%

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

      2. pow2 92.8%

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

      3. sub-neg 92.8%

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

      4. metadata-eval 92.8%

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

   3. Applied egg-rr 92.8%

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

   4. Taylor expanded in x around inf 79.7%

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

      1. neg-mul-1 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in z around inf 79.7%

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

      1. unpow2 79.7%

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

      2. +-commutative 79.7%

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

   9. Simplified 79.7%

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

   10. Taylor expanded in z around inf 80.2%

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

       1. unpow2 80.2%

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

       2. *-commutative 80.2%

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

       3. associate-*l* 82.7%

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

       4. associate-*r/ 82.7%

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

       5. metadata-eval 82.7%

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

   12. Simplified 82.7%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{y}{\frac{\frac{x}{z}}{z}} - x\\ \mathbf{elif}\;z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \leq 5 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 15500000:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x - \left(x + -0.91893853320467\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 15500000.0)
   (+
    (/
     (+
      (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
      0.083333333333333)
     x)
    (+ (- (* (log x) (- x 0.5)) x) 0.91893853320467))
   (+
    (- (* (+ x -0.5) (log x)) (+ x -0.91893853320467))
    (* z (* z (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 15500000.0) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + (((log(x) * (x - 0.5)) - x) + 0.91893853320467);
	} else {
		tmp = (((x + -0.5) * log(x)) - (x + -0.91893853320467)) + (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 <= 15500000.0d0) then
        tmp = (((z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + (((log(x) * (x - 0.5d0)) - x) + 0.91893853320467d0)
    else
        tmp = (((x + (-0.5d0)) * log(x)) - (x + (-0.91893853320467d0))) + (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 <= 15500000.0) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + (((Math.log(x) * (x - 0.5)) - x) + 0.91893853320467);
	} else {
		tmp = (((x + -0.5) * Math.log(x)) - (x + -0.91893853320467)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 15500000.0:
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + (((math.log(x) * (x - 0.5)) - x) + 0.91893853320467)
	else:
		tmp = (((x + -0.5) * math.log(x)) - (x + -0.91893853320467)) + (z * (z * ((y / x) + (0.0007936500793651 / x))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 15500000.0)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) + 0.91893853320467));
	else
		tmp = Float64(Float64(Float64(Float64(x + -0.5) * log(x)) - Float64(x + -0.91893853320467)) + 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 <= 15500000.0)
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) + (((log(x) * (x - 0.5)) - x) + 0.91893853320467);
	else
		tmp = (((x + -0.5) * log(x)) - (x + -0.91893853320467)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 15500000.0], 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[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $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 < 1.55e7

   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 1.55e7 < x

   1. Initial program 90.7%

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

      1. associate-+l- 90.7%

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

      2. sub-neg 90.7%

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

      3. metadata-eval 90.7%

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

      4. sub-neg 90.7%

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

      5. metadata-eval 90.7%

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

   3. Applied egg-rr 90.7%

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

   4. Taylor expanded in z around inf 90.7%

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

      1. +-commutative 90.7%

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

      2. associate-+r+ 90.7%

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

      3. +-commutative 90.7%

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

      4. fma-def 90.7%

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

      5. associate-*r/ 90.7%

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

      6. metadata-eval 90.7%

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

      7. associate-/l* 93.5%

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

      8. +-commutative 93.5%

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

      9. associate-/r/ 93.5%

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

      10. unpow2 93.5%

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

      11. +-commutative 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in z around inf 93.5%

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

      1. *-commutative 93.5%

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

      2. unpow2 93.5%

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

      3. associate-*l* 99.6%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 15500000:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x - \left(x + -0.91893853320467\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.035:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x - \left(x + -0.91893853320467\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.035)
   (-
    (/
     (+
      (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))
      0.083333333333333)
     x)
    x)
   (+
    (- (* (+ x -0.5) (log x)) (+ x -0.91893853320467))
    (* z (* z (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.035) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) - x;
	} else {
		tmp = (((x + -0.5) * log(x)) - (x + -0.91893853320467)) + (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 <= 0.035d0) then
        tmp = (((z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) - x
    else
        tmp = (((x + (-0.5d0)) * log(x)) - (x + (-0.91893853320467d0))) + (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 <= 0.035) {
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) - x;
	} else {
		tmp = (((x + -0.5) * Math.log(x)) - (x + -0.91893853320467)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.035:
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) - x
	else:
		tmp = (((x + -0.5) * math.log(x)) - (x + -0.91893853320467)) + (z * (z * ((y / x) + (0.0007936500793651 / x))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.035)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) - x);
	else
		tmp = Float64(Float64(Float64(Float64(x + -0.5) * log(x)) - Float64(x + -0.91893853320467)) + 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 <= 0.035)
		tmp = (((z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)) + 0.083333333333333) / x) - x;
	else
		tmp = (((x + -0.5) * log(x)) - (x + -0.91893853320467)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.035], N[(N[(N[(N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $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 < 0.035000000000000003

   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. add-sqr-sqrt 99.7%

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

      2. pow2 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around inf 97.5%

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

      1. neg-mul-1 97.5%

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

   6. Simplified 97.5%

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

   if 0.035000000000000003 < x

   1. Initial program 90.8%

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

      1. associate-+l- 90.8%

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

      2. sub-neg 90.8%

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

      3. metadata-eval 90.8%

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

      4. sub-neg 90.8%

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

      5. metadata-eval 90.8%

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

   3. Applied egg-rr 90.8%

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

   4. Taylor expanded in z around inf 90.8%

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

      1. +-commutative 90.8%

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

      2. associate-+r+ 90.8%

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

      3. +-commutative 90.8%

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

      4. fma-def 90.8%

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

      5. associate-*r/ 90.8%

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

      6. metadata-eval 90.8%

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

      7. associate-/l* 93.6%

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

      8. +-commutative 93.6%

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

      9. associate-/r/ 93.6%

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

      10. unpow2 93.6%

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

      11. +-commutative 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in z around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. unpow2 93.3%

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

      3. associate-*l* 99.4%

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

      4. associate-*r/ 99.4%

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

      5. metadata-eval 99.4%

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

   9. Simplified 99.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.035:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x - \left(x + -0.91893853320467\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\log x + -1\right)\\ t_1 := \left(y + 0.0007936500793651\right) \cdot z\\ \mathbf{if}\;x \leq 10^{+36}:\\ \;\;\;\;\frac{z \cdot \left(t_1 - 0.0027777777777778\right) + 0.083333333333333}{x} + t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{z}{x} + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ (log x) -1.0))) (t_1 (* (+ y 0.0007936500793651) z)))
   (if (<= x 1e+36)
     (+ (/ (+ (* z (- t_1 0.0027777777777778)) 0.083333333333333) x) t_0)
     (+ (* t_1 (/ z x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (log(x) + -1.0);
	double t_1 = (y + 0.0007936500793651) * z;
	double tmp;
	if (x <= 1e+36) {
		tmp = (((z * (t_1 - 0.0027777777777778)) + 0.083333333333333) / x) + t_0;
	} else {
		tmp = (t_1 * (z / x)) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (log(x) + (-1.0d0))
    t_1 = (y + 0.0007936500793651d0) * z
    if (x <= 1d+36) then
        tmp = (((z * (t_1 - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + t_0
    else
        tmp = (t_1 * (z / x)) + t_0
    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 t_1 = (y + 0.0007936500793651) * z;
	double tmp;
	if (x <= 1e+36) {
		tmp = (((z * (t_1 - 0.0027777777777778)) + 0.083333333333333) / x) + t_0;
	} else {
		tmp = (t_1 * (z / x)) + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (math.log(x) + -1.0)
	t_1 = (y + 0.0007936500793651) * z
	tmp = 0
	if x <= 1e+36:
		tmp = (((z * (t_1 - 0.0027777777777778)) + 0.083333333333333) / x) + t_0
	else:
		tmp = (t_1 * (z / x)) + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(log(x) + -1.0))
	t_1 = Float64(Float64(y + 0.0007936500793651) * z)
	tmp = 0.0
	if (x <= 1e+36)
		tmp = Float64(Float64(Float64(Float64(z * Float64(t_1 - 0.0027777777777778)) + 0.083333333333333) / x) + t_0);
	else
		tmp = Float64(Float64(t_1 * Float64(z / x)) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (log(x) + -1.0);
	t_1 = (y + 0.0007936500793651) * z;
	tmp = 0.0;
	if (x <= 1e+36)
		tmp = (((z * (t_1 - 0.0027777777777778)) + 0.083333333333333) / x) + t_0;
	else
		tmp = (t_1 * (z / x)) + t_0;
	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]}, Block[{t$95$1 = N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[x, 1e+36], N[(N[(N[(N[(z * N[(t$95$1 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(t$95$1 * N[(z / x), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e36

   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 inf 97.2%

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

      1. *-commutative 37.5%

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

      2. sub-neg 37.5%

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

      3. mul-1-neg 37.5%

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

      4. log-rec 37.5%

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

      5. remove-double-neg 37.5%

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

      6. metadata-eval 37.5%

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

   4. Simplified 97.2%

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

   if 1.00000000000000004e36 < x

   1. Initial program 89.3%

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

      1. associate-+l- 89.3%

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

      2. sub-neg 89.3%

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

      3. metadata-eval 89.3%

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

      4. sub-neg 89.3%

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

      5. metadata-eval 89.3%

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

   3. Applied egg-rr 89.3%

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

      1. *-un-lft-identity 89.3%

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

      2. add-sqr-sqrt 89.3%

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

      3. times-frac 89.4%

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

      4. *-commutative 89.4%

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

      5. fma-udef 89.4%

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

      6. fma-neg 89.4%

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

      7. metadata-eval 89.4%

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

   5. Applied egg-rr 89.4%

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

      1. associate-*l/ 89.4%

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

      2. *-lft-identity 89.4%

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

      3. fma-def 89.4%

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

      4. *-commutative 89.4%

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

      5. fma-def 89.4%

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

      6. +-commutative 89.4%

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

   7. Simplified 89.4%

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

   8. Taylor expanded in z around inf 89.4%

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

      1. associate-/l* 92.6%

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

      2. associate-/r/ 92.6%

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

      3. unpow2 92.6%

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

      4. associate-*l/ 99.6%

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

      5. associate-*l* 98.8%

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

      6. +-commutative 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in x around inf 99.0%

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

       1. *-commutative 99.0%

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

       2. sub-neg 99.0%

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

       3. mul-1-neg 99.0%

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

       4. log-rec 99.0%

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

       5. remove-double-neg 99.0%

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

       6. metadata-eval 99.0%

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

   13. Simplified 99.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+36}:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + 0.0007936500793651\right) \cdot z\right) \cdot \frac{z}{x} + x \cdot \left(\log x + -1\right)\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + 0.0007936500793651\right) \cdot z\\ \mathbf{if}\;x \leq 0.031:\\ \;\;\;\;\frac{z \cdot \left(t_0 - 0.0027777777777778\right) + 0.083333333333333}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x - \left(x + -0.91893853320467\right)\right) + t_0 \cdot \frac{z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y 0.0007936500793651) z)))
   (if (<= x 0.031)
     (- (/ (+ (* z (- t_0 0.0027777777777778)) 0.083333333333333) x) x)
     (+ (- (* (+ x -0.5) (log x)) (+ x -0.91893853320467)) (* t_0 (/ z x))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + 0.0007936500793651) * z;
	double tmp;
	if (x <= 0.031) {
		tmp = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) - x;
	} else {
		tmp = (((x + -0.5) * log(x)) - (x + -0.91893853320467)) + (t_0 * (z / 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 = (y + 0.0007936500793651d0) * z
    if (x <= 0.031d0) then
        tmp = (((z * (t_0 - 0.0027777777777778d0)) + 0.083333333333333d0) / x) - x
    else
        tmp = (((x + (-0.5d0)) * log(x)) - (x + (-0.91893853320467d0))) + (t_0 * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + 0.0007936500793651) * z;
	double tmp;
	if (x <= 0.031) {
		tmp = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) - x;
	} else {
		tmp = (((x + -0.5) * Math.log(x)) - (x + -0.91893853320467)) + (t_0 * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + 0.0007936500793651) * z
	tmp = 0
	if x <= 0.031:
		tmp = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) - x
	else:
		tmp = (((x + -0.5) * math.log(x)) - (x + -0.91893853320467)) + (t_0 * (z / x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + 0.0007936500793651) * z)
	tmp = 0.0
	if (x <= 0.031)
		tmp = Float64(Float64(Float64(Float64(z * Float64(t_0 - 0.0027777777777778)) + 0.083333333333333) / x) - x);
	else
		tmp = Float64(Float64(Float64(Float64(x + -0.5) * log(x)) - Float64(x + -0.91893853320467)) + Float64(t_0 * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + 0.0007936500793651) * z;
	tmp = 0.0;
	if (x <= 0.031)
		tmp = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) - x;
	else
		tmp = (((x + -0.5) * log(x)) - (x + -0.91893853320467)) + (t_0 * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[x, 0.031], N[(N[(N[(N[(z * N[(t$95$0 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.031

   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. add-sqr-sqrt 99.7%

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

      2. pow2 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around inf 97.5%

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

      1. neg-mul-1 97.5%

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

   6. Simplified 97.5%

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

   if 0.031 < x

   1. Initial program 90.8%

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

      1. associate-+l- 90.8%

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

      2. sub-neg 90.8%

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

      3. metadata-eval 90.8%

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

      4. sub-neg 90.8%

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

      5. metadata-eval 90.8%

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

   3. Applied egg-rr 90.8%

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

      1. *-un-lft-identity 90.8%

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

      2. add-sqr-sqrt 90.7%

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

      3. times-frac 90.8%

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

      4. *-commutative 90.8%

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

      5. fma-udef 90.8%

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

      6. fma-neg 90.8%

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

      7. metadata-eval 90.8%

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

   5. Applied egg-rr 90.8%

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

      1. associate-*l/ 90.8%

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

      2. *-lft-identity 90.8%

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

      3. fma-def 90.8%

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

      4. *-commutative 90.8%

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

      5. fma-def 90.8%

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

      6. +-commutative 90.8%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in z around inf 90.6%

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

      1. associate-/l* 93.3%

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

      2. associate-/r/ 93.3%

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

      3. unpow2 93.3%

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

      4. associate-*l/ 99.4%

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

      5. associate-*l* 98.7%

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

      6. +-commutative 98.7%

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

   10. Simplified 98.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.031:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x - \left(x + -0.91893853320467\right)\right) + \left(\left(y + 0.0007936500793651\right) \cdot z\right) \cdot \frac{z}{x}\\ \end{array} \]

Alternative 7: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + 0.0007936500793651\right) \cdot z\\ \mathbf{if}\;x \leq 1.12:\\ \;\;\;\;\frac{z \cdot \left(t_0 - 0.0027777777777778\right) + 0.083333333333333}{x} - x\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{z}{x} + x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y 0.0007936500793651) z)))
   (if (<= x 1.12)
     (- (/ (+ (* z (- t_0 0.0027777777777778)) 0.083333333333333) x) x)
     (+ (* t_0 (/ z x)) (* x (+ (log x) -1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + 0.0007936500793651) * z;
	double tmp;
	if (x <= 1.12) {
		tmp = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) - x;
	} else {
		tmp = (t_0 * (z / x)) + (x * (log(x) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + 0.0007936500793651d0) * z
    if (x <= 1.12d0) then
        tmp = (((z * (t_0 - 0.0027777777777778d0)) + 0.083333333333333d0) / x) - x
    else
        tmp = (t_0 * (z / x)) + (x * (log(x) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + 0.0007936500793651) * z;
	double tmp;
	if (x <= 1.12) {
		tmp = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) - x;
	} else {
		tmp = (t_0 * (z / x)) + (x * (Math.log(x) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + 0.0007936500793651) * z
	tmp = 0
	if x <= 1.12:
		tmp = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) - x
	else:
		tmp = (t_0 * (z / x)) + (x * (math.log(x) + -1.0))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + 0.0007936500793651) * z)
	tmp = 0.0
	if (x <= 1.12)
		tmp = Float64(Float64(Float64(Float64(z * Float64(t_0 - 0.0027777777777778)) + 0.083333333333333) / x) - x);
	else
		tmp = Float64(Float64(t_0 * Float64(z / x)) + Float64(x * Float64(log(x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + 0.0007936500793651) * z;
	tmp = 0.0;
	if (x <= 1.12)
		tmp = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) - x;
	else
		tmp = (t_0 * (z / x)) + (x * (log(x) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[x, 1.12], N[(N[(N[(N[(z * N[(t$95$0 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(t$95$0 * N[(z / x), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1200000000000001

   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. add-sqr-sqrt 99.7%

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

      2. pow2 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around inf 97.5%

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

      1. neg-mul-1 97.5%

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

   6. Simplified 97.5%

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

   if 1.1200000000000001 < x

   1. Initial program 90.8%

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

      1. associate-+l- 90.8%

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

      2. sub-neg 90.8%

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

      3. metadata-eval 90.8%

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

      4. sub-neg 90.8%

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

      5. metadata-eval 90.8%

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

   3. Applied egg-rr 90.8%

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

      1. *-un-lft-identity 90.8%

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

      2. add-sqr-sqrt 90.7%

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

      3. times-frac 90.8%

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

      4. *-commutative 90.8%

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

      5. fma-udef 90.8%

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

      6. fma-neg 90.8%

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

      7. metadata-eval 90.8%

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

   5. Applied egg-rr 90.8%

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

      1. associate-*l/ 90.8%

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

      2. *-lft-identity 90.8%

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

      3. fma-def 90.8%

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

      4. *-commutative 90.8%

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

      5. fma-def 90.8%

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

      6. +-commutative 90.8%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in z around inf 90.6%

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

      1. associate-/l* 93.3%

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

      2. associate-/r/ 93.3%

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

      3. unpow2 93.3%

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

      4. associate-*l/ 99.4%

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

      5. associate-*l* 98.7%

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

      6. +-commutative 98.7%

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

   10. Simplified 98.7%

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

   11. Taylor expanded in x around inf 98.4%

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

       1. *-commutative 98.4%

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

       2. sub-neg 98.4%

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

       3. mul-1-neg 98.4%

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

       4. log-rec 98.4%

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

       5. remove-double-neg 98.4%

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

       6. metadata-eval 98.4%

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

   13. Simplified 98.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) + 0.083333333333333}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + 0.0007936500793651\right) \cdot z\right) \cdot \frac{z}{x} + x \cdot \left(\log x + -1\right)\\ \end{array} \]

Alternative 8: 89.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+39}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \left(y + 0.0007936500793651\right) \cdot z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{y}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.2e+39)
   (+ x (/ (fma z (* (+ y 0.0007936500793651) z) 0.083333333333333) x))
   (+ (* x (+ (log x) -1.0)) (* (/ y x) (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.2e+39) {
		tmp = x + (fma(z, ((y + 0.0007936500793651) * z), 0.083333333333333) / x);
	} else {
		tmp = (x * (log(x) + -1.0)) + ((y / x) * (z * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.2e+39)
		tmp = Float64(x + Float64(fma(z, Float64(Float64(y + 0.0007936500793651) * z), 0.083333333333333) / x));
	else
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(Float64(y / x) * Float64(z * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.2e+39], N[(x + N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e39

   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. add-sqr-sqrt 99.7%

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

      2. pow2 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around inf 93.3%

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

      1. neg-mul-1 93.3%

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

   6. Simplified 93.3%

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

   7. Taylor expanded in z around inf 92.7%

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

      1. unpow2 92.7%

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

      2. +-commutative 92.7%

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

   9. Simplified 92.7%

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

       1. expm1-log1p-u 72.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x\right) + \frac{\left(z \cdot z\right) \cdot \left(y + 0.0007936500793651\right) + 0.083333333333333}{x}\right)\right)} \]

       2. expm1-udef 72.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x\right) + \frac{\left(z \cdot z\right) \cdot \left(y + 0.0007936500793651\right) + 0.083333333333333}{x}\right)} - 1} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + \frac{\left(z \cdot z\right) \cdot \left(y + 0.0007936500793651\right) + 0.083333333333333}{x}\right)} - 1 \]

       4. sqrt-unprod 73.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + \frac{\left(z \cdot z\right) \cdot \left(y + 0.0007936500793651\right) + 0.083333333333333}{x}\right)} - 1 \]

       5. sqr-neg 73.6%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x \cdot x}} + \frac{\left(z \cdot z\right) \cdot \left(y + 0.0007936500793651\right) + 0.083333333333333}{x}\right)} - 1 \]

       6. sqrt-unprod 73.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \frac{\left(z \cdot z\right) \cdot \left(y + 0.0007936500793651\right) + 0.083333333333333}{x}\right)} - 1 \]

       7. add-sqr-sqrt 73.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x} + \frac{\left(z \cdot z\right) \cdot \left(y + 0.0007936500793651\right) + 0.083333333333333}{x}\right)} - 1 \]

       8. associate-*l* 73.6%

          \[\leadsto e^{\mathsf{log1p}\left(x + \frac{\color{blue}{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)} + 0.083333333333333}{x}\right)} - 1 \]

       9. fma-def 73.6%

          \[\leadsto e^{\mathsf{log1p}\left(x + \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot \left(y + 0.0007936500793651\right), 0.083333333333333\right)}}{x}\right)} - 1 \]

   11. Applied egg-rr 73.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\mathsf{fma}\left(z, z \cdot \left(y + 0.0007936500793651\right), 0.083333333333333\right)}{x}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 73.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\mathsf{fma}\left(z, z \cdot \left(y + 0.0007936500793651\right), 0.083333333333333\right)}{x}\right)\right)} \]

       2. expm1-log1p 93.3%

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

   13. Simplified 93.3%

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

   if 1.2e39 < x

   1. Initial program 89.2%

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

      1. add-cbrt-cube 83.3%

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

      2. pow3 83.3%

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

      3. *-commutative 83.3%

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

      4. fma-udef 83.3%

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

      5. fma-neg 83.3%

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

      6. metadata-eval 83.3%

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

   3. Applied egg-rr 83.3%

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

   4. Taylor expanded in y around inf 81.2%

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

      1. associate-/l* 84.5%

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

      2. associate-/r/ 82.7%

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

      3. unpow2 82.7%

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

   6. Simplified 82.7%

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

   7. Taylor expanded in x around inf 82.7%

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

      1. *-commutative 72.0%

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

      2. sub-neg 72.0%

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

      3. mul-1-neg 72.0%

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

      4. log-rec 72.0%

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

      5. remove-double-neg 72.0%

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

      6. metadata-eval 72.0%

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

   9. Simplified 82.7%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+39}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \left(y + 0.0007936500793651\right) \cdot z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{y}{x} \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 9: 66.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)\\ \mathbf{if}\;t_0 \leq -20000:\\ \;\;\;\;\frac{y}{\frac{\frac{x}{z}}{z}} - x\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+40}:\\ \;\;\;\;x + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778))))
   (if (<= t_0 -20000.0)
     (- (/ y (/ (/ x z) z)) x)
     (if (<= t_0 2e+40)
       (+ x (/ 0.083333333333333 x))
       (* z (* z (+ (/ y x) (/ 0.0007936500793651 x))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (((y + 0.0007936500793651) * z) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -20000.0) {
		tmp = (y / ((x / z) / z)) - x;
	} else if (t_0 <= 2e+40) {
		tmp = x + (0.083333333333333 / x);
	} else {
		tmp = 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 = z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0)
    if (t_0 <= (-20000.0d0)) then
        tmp = (y / ((x / z) / z)) - x
    else if (t_0 <= 2d+40) then
        tmp = x + (0.083333333333333d0 / x)
    else
        tmp = 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 = z * (((y + 0.0007936500793651) * z) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -20000.0) {
		tmp = (y / ((x / z) / z)) - x;
	} else if (t_0 <= 2e+40) {
		tmp = x + (0.083333333333333 / x);
	} else {
		tmp = z * (z * ((y / x) + (0.0007936500793651 / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (((y + 0.0007936500793651) * z) - 0.0027777777777778)
	tmp = 0
	if t_0 <= -20000.0:
		tmp = (y / ((x / z) / z)) - x
	elif t_0 <= 2e+40:
		tmp = x + (0.083333333333333 / x)
	else:
		tmp = z * (z * ((y / x) + (0.0007936500793651 / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -20000.0)
		tmp = Float64(Float64(y / Float64(Float64(x / z) / z)) - x);
	elseif (t_0 <= 2e+40)
		tmp = Float64(x + Float64(0.083333333333333 / x));
	else
		tmp = 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 = z * (((y + 0.0007936500793651) * z) - 0.0027777777777778);
	tmp = 0.0;
	if (t_0 <= -20000.0)
		tmp = (y / ((x / z) / z)) - x;
	elseif (t_0 <= 2e+40)
		tmp = x + (0.083333333333333 / x);
	else
		tmp = z * (z * ((y / x) + (0.0007936500793651 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20000.0], N[(N[(y / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[t$95$0, 2e+40], N[(x + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) < -2e4

   1. Initial program 87.2%

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

      1. add-sqr-sqrt 87.1%

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

      2. pow2 87.1%

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

      3. sub-neg 87.1%

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

      4. metadata-eval 87.1%

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

   3. Applied egg-rr 87.1%

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

   4. Taylor expanded in x around inf 73.8%

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

      1. neg-mul-1 73.8%

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

   6. Simplified 73.8%

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

   7. Taylor expanded in y around inf 72.3%

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

      1. associate-/l* 79.8%

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

      2. unpow2 79.8%

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

      3. associate-/r* 82.3%

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

   9. Simplified 82.3%

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

   if -2e4 < (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) < 2.00000000000000006e40

   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. Step-by-step derivation

      1. add-sqr-sqrt 99.1%

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

      2. pow2 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in x around inf 43.3%

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

      1. neg-mul-1 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in z around 0 42.3%

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

      1. expm1-log1p-u 38.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x\right) + \frac{0.083333333333333}{x}\right)\right)} \]

      2. expm1-udef 38.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x\right) + \frac{0.083333333333333}{x}\right)} - 1} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      4. sqrt-unprod 45.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      5. sqr-neg 45.0%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x \cdot x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      6. sqrt-unprod 46.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      7. add-sqr-sqrt 46.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x} + \frac{0.083333333333333}{x}\right)} - 1 \]

   9. Applied egg-rr 46.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{0.083333333333333}{x}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 46.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{0.083333333333333}{x}\right)\right)} \]

       2. expm1-log1p 49.6%

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

   11. Simplified 49.6%

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

   if 2.00000000000000006e40 < (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z)

   1. Initial program 93.4%

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

      1. add-sqr-sqrt 93.3%

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

      2. pow2 93.3%

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

      3. sub-neg 93.3%

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

      4. metadata-eval 93.3%

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

   3. Applied egg-rr 93.3%

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

   4. Taylor expanded in x around inf 76.4%

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

      1. neg-mul-1 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in z around inf 76.3%

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

      1. unpow2 76.3%

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

      2. +-commutative 76.3%

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

   9. Simplified 76.3%

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

   10. Taylor expanded in z around inf 77.0%

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

       1. unpow2 77.0%

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

       2. *-commutative 77.0%

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

       3. associate-*l* 79.3%

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

       4. associate-*r/ 79.3%

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

       5. metadata-eval 79.3%

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

   12. Simplified 79.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \leq -20000:\\ \;\;\;\;\frac{y}{\frac{\frac{x}{z}}{z}} - x\\ \mathbf{elif}\;z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \leq 2 \cdot 10^{+40}:\\ \;\;\;\;x + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 10: 51.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-46} \lor \neg \left(z \leq 2.2 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}} - x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.2e-46) (not (<= z 2.2e-54)))
   (- (/ y (/ x (* z z))) x)
   (+ x (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.2e-46) || !(z <= 2.2e-54)) {
		tmp = (y / (x / (z * z))) - x;
	} else {
		tmp = 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 <= (-6.2d-46)) .or. (.not. (z <= 2.2d-54))) then
        tmp = (y / (x / (z * z))) - x
    else
        tmp = 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 <= -6.2e-46) || !(z <= 2.2e-54)) {
		tmp = (y / (x / (z * z))) - x;
	} else {
		tmp = x + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.2e-46) or not (z <= 2.2e-54):
		tmp = (y / (x / (z * z))) - x
	else:
		tmp = x + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.2e-46) || !(z <= 2.2e-54))
		tmp = Float64(Float64(y / Float64(x / Float64(z * z))) - x);
	else
		tmp = Float64(x + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.2e-46) || ~((z <= 2.2e-54)))
		tmp = (y / (x / (z * z))) - x;
	else
		tmp = x + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.2e-46], N[Not[LessEqual[z, 2.2e-54]], $MachinePrecision]], N[(N[(y / N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(x + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2000000000000002e-46 or 2.2e-54 < z

   1. Initial program 92.3%

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

      1. add-sqr-sqrt 92.2%

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

      2. pow2 92.2%

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

      3. sub-neg 92.2%

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

      4. metadata-eval 92.2%

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

   3. Applied egg-rr 92.2%

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

   4. Taylor expanded in x around inf 74.0%

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

      1. neg-mul-1 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in y around inf 45.7%

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

      1. associate-/l* 48.7%

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

      2. unpow2 48.7%

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

   9. Simplified 48.7%

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

   if -6.2000000000000002e-46 < z < 2.2e-54

   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. Step-by-step derivation

      1. add-sqr-sqrt 99.1%

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

      2. pow2 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in x around inf 43.2%

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

      1. neg-mul-1 43.2%

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

   6. Simplified 43.2%

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

   7. Taylor expanded in z around 0 42.2%

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

      1. expm1-log1p-u 38.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x\right) + \frac{0.083333333333333}{x}\right)\right)} \]

      2. expm1-udef 38.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x\right) + \frac{0.083333333333333}{x}\right)} - 1} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      4. sqrt-unprod 44.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      5. sqr-neg 44.1%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x \cdot x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      6. sqrt-unprod 45.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      7. add-sqr-sqrt 45.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x} + \frac{0.083333333333333}{x}\right)} - 1 \]

   9. Applied egg-rr 45.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{0.083333333333333}{x}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 45.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{0.083333333333333}{x}\right)\right)} \]

       2. expm1-log1p 49.4%

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

   11. Simplified 49.4%

       \[\leadsto \color{blue}{x + \frac{0.083333333333333}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-46} \lor \neg \left(z \leq 2.2 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}} - x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 11: 51.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}} - x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-55}:\\ \;\;\;\;x + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{x}{z}}{z}} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e-42)
   (- (/ y (/ x (* z z))) x)
   (if (<= z 5e-55) (+ x (/ 0.083333333333333 x)) (- (/ y (/ (/ x z) z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-42) {
		tmp = (y / (x / (z * z))) - x;
	} else if (z <= 5e-55) {
		tmp = x + (0.083333333333333 / x);
	} else {
		tmp = (y / ((x / z) / z)) - 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 <= (-2d-42)) then
        tmp = (y / (x / (z * z))) - x
    else if (z <= 5d-55) then
        tmp = x + (0.083333333333333d0 / x)
    else
        tmp = (y / ((x / z) / z)) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-42) {
		tmp = (y / (x / (z * z))) - x;
	} else if (z <= 5e-55) {
		tmp = x + (0.083333333333333 / x);
	} else {
		tmp = (y / ((x / z) / z)) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2e-42:
		tmp = (y / (x / (z * z))) - x
	elif z <= 5e-55:
		tmp = x + (0.083333333333333 / x)
	else:
		tmp = (y / ((x / z) / z)) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e-42)
		tmp = Float64(Float64(y / Float64(x / Float64(z * z))) - x);
	elseif (z <= 5e-55)
		tmp = Float64(x + Float64(0.083333333333333 / x));
	else
		tmp = Float64(Float64(y / Float64(Float64(x / z) / z)) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2e-42)
		tmp = (y / (x / (z * z))) - x;
	elseif (z <= 5e-55)
		tmp = x + (0.083333333333333 / x);
	else
		tmp = (y / ((x / z) / z)) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e-42], N[(N[(y / N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[z, 5e-55], N[(x + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.00000000000000008e-42

   1. Initial program 93.1%

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

      1. add-sqr-sqrt 93.0%

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

      2. pow2 93.0%

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

      3. sub-neg 93.0%

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

      4. metadata-eval 93.0%

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

   3. Applied egg-rr 93.0%

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

   4. Taylor expanded in x around inf 73.5%

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

      1. neg-mul-1 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in y around inf 46.4%

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

      1. associate-/l* 50.8%

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

      2. unpow2 50.8%

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

   9. Simplified 50.8%

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

   if -2.00000000000000008e-42 < z < 5.0000000000000002e-55

   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. Step-by-step derivation

      1. add-sqr-sqrt 99.1%

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

      2. pow2 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in x around inf 43.2%

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

      1. neg-mul-1 43.2%

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

   6. Simplified 43.2%

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

   7. Taylor expanded in z around 0 42.2%

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

      1. expm1-log1p-u 38.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x\right) + \frac{0.083333333333333}{x}\right)\right)} \]

      2. expm1-udef 38.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x\right) + \frac{0.083333333333333}{x}\right)} - 1} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      4. sqrt-unprod 44.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      5. sqr-neg 44.1%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x \cdot x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      6. sqrt-unprod 45.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

      7. add-sqr-sqrt 45.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x} + \frac{0.083333333333333}{x}\right)} - 1 \]

   9. Applied egg-rr 45.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{0.083333333333333}{x}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 45.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{0.083333333333333}{x}\right)\right)} \]

       2. expm1-log1p 49.4%

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

   11. Simplified 49.4%

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

   if 5.0000000000000002e-55 < z

   1. Initial program 91.5%

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

      1. add-sqr-sqrt 91.4%

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

      2. pow2 91.4%

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

      3. sub-neg 91.4%

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

      4. metadata-eval 91.4%

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

   3. Applied egg-rr 91.4%

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

   4. Taylor expanded in x around inf 74.5%

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

      1. neg-mul-1 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in y around inf 45.0%

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

      1. associate-/l* 46.3%

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

      2. unpow2 46.3%

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

      3. associate-/r* 47.1%

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

   9. Simplified 47.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}} - x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-55}:\\ \;\;\;\;x + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{x}{z}}{z}} - x\\ \end{array} \]

Alternative 12: 28.2% accurate, 24.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. add-sqr-sqrt 94.8%

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

   2. pow2 94.8%

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

   3. sub-neg 94.8%

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

   4. metadata-eval 94.8%

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

3. Applied egg-rr 94.8%

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

4. Taylor expanded in x around inf 62.4%

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

   1. neg-mul-1 62.4%

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

6. Simplified 62.4%

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

7. Taylor expanded in z around 0 18.9%

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

   1. expm1-log1p-u 16.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x\right) + \frac{0.083333333333333}{x}\right)\right)} \]

   2. expm1-udef 16.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x\right) + \frac{0.083333333333333}{x}\right)} - 1} \]

   3. add-sqr-sqrt 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

   4. sqrt-unprod 23.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + \frac{0.083333333333333}{x}\right)} - 1 \]

   5. sqr-neg 23.0%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x \cdot x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

   6. sqrt-unprod 22.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \frac{0.083333333333333}{x}\right)} - 1 \]

   7. add-sqr-sqrt 22.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x} + \frac{0.083333333333333}{x}\right)} - 1 \]

9. Applied egg-rr 22.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{0.083333333333333}{x}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 22.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{0.083333333333333}{x}\right)\right)} \]

    2. expm1-log1p 23.9%

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

11. Simplified 23.9%

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

12. Final simplification 23.9%

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

Alternative 13: 1.3% accurate, 61.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. add-sqr-sqrt 94.8%

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

   2. pow2 94.8%

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

   3. sub-neg 94.8%

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

   4. metadata-eval 94.8%

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

3. Applied egg-rr 94.8%

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

4. Taylor expanded in x around inf 62.4%

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

   1. neg-mul-1 62.4%

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

6. Simplified 62.4%

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

7. Taylor expanded in z around 0 18.9%

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

8. Taylor expanded in x around inf 1.3%

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

   1. mul-1-neg 1.3%

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

10. Simplified 1.3%

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

11. Final simplification 1.3%

    \[\leadsto -x \]

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 2023224 
(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)))