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

Percentage Accurate: 93.7% → 99.7%

Time: 13.7s

Alternatives: 16

Speedup: 1.0×

• 31.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;\log x \cdot \left(x - 0.5\right) - \left(\left(x - 0.91893853320467\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.083333333333333, \frac{-1}{x}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.3e-10)
   (-
    (* (log x) (- x 0.5))
    (-
     (- x 0.91893853320467)
     (/
      (fma
       (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
       z
       0.083333333333333)
      x)))
   (fma
    -0.083333333333333
    (/ -1.0 x)
    (fma
     (fma (/ y x) z (/ (fma 0.0007936500793651 z -0.0027777777777778) x))
     z
     (fma (log x) (- x 0.5) (- 0.91893853320467 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.3e-10) {
		tmp = (log(x) * (x - 0.5)) - ((x - 0.91893853320467) - (fma(fma(z, (0.0007936500793651 + y), -0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = fma(-0.083333333333333, (-1.0 / x), fma(fma((y / x), z, (fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, fma(log(x), (x - 0.5), (0.91893853320467 - x))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.3e-10)
		tmp = Float64(Float64(log(x) * Float64(x - 0.5)) - Float64(Float64(x - 0.91893853320467) - Float64(fma(fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = fma(-0.083333333333333, Float64(-1.0 / x), fma(fma(Float64(y / x), z, Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.3e-10], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.083333333333333 * N[(-1.0 / x), $MachinePrecision] + N[(N[(N[(y / x), $MachinePrecision] * z + N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * z + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999991e-10

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 1.29999999999999991e-10 < x

   1. Initial program 84.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) - x} \]

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \frac{1}{x \cdot 12.000000000000048}\right) + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right)\right) - x \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(-0.083333333333333, \color{blue}{\frac{1}{-x}}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;\log x \cdot \left(x - 0.5\right) - \left(\left(x - 0.91893853320467\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.083333333333333, \frac{-1}{x}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{z}{x}, y, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - x\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -1e+122)
     (fma (* z (/ z x)) y (/ 0.083333333333333 x))
     (if (<= t_0 4e+306)
       (+
        (fma (- x 0.5) (log x) (- 0.91893853320467 x))
        (/ 0.083333333333333 x))
       (* (* (+ (/ y x) (/ 0.0007936500793651 x)) z) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -1e+122) {
		tmp = fma((z * (z / x)), y, (0.083333333333333 / x));
	} else if (t_0 <= 4e+306) {
		tmp = fma((x - 0.5), log(x), (0.91893853320467 - x)) + (0.083333333333333 / x);
	} else {
		tmp = (((y / x) + (0.0007936500793651 / x)) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = 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))
	tmp = 0.0
	if (t_0 <= -1e+122)
		tmp = fma(Float64(z * Float64(z / x)), y, Float64(0.083333333333333 / x));
	elseif (t_0 <= 4e+306)
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(0.91893853320467 - x)) + Float64(0.083333333333333 / x));
	else
		tmp = Float64(Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -1e+122], N[(N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision] * y + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+306], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -1.00000000000000001e122

   1. Initial program 84.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} + \frac{{z}^{2}}{x}\right)\right)\right)\right) - \frac{x}{y}\right)} \]

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}, y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 8.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{{z}^{2}}{x}, y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.7%

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

   if -1.00000000000000001e122 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4.00000000000000007e306

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      11. lower--.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around 0

      \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 90.1%

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

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} \]

      3. associate--r- N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) - \left(x - \frac{91893853320467}{100000000000000}\right)\right) + \frac{\frac{83333333333333}{1000000000000000}}{x}} \]

   9. Applied rewrites 90.2%

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

   if 4.00000000000000007e306 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 87.9

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

   5. Applied rewrites 87.9%

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

Alternative 3: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{z}{x}, y, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -1e+122)
     (fma (* z (/ z x)) y (/ 0.083333333333333 x))
     (if (<= t_0 4e+306)
       (fma
        (- x 0.5)
        (log x)
        (- (+ (/ 0.083333333333333 x) 0.91893853320467) x))
       (* (* (+ (/ y x) (/ 0.0007936500793651 x)) z) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -1e+122) {
		tmp = fma((z * (z / x)), y, (0.083333333333333 / x));
	} else if (t_0 <= 4e+306) {
		tmp = fma((x - 0.5), log(x), (((0.083333333333333 / x) + 0.91893853320467) - x));
	} else {
		tmp = (((y / x) + (0.0007936500793651 / x)) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = 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))
	tmp = 0.0
	if (t_0 <= -1e+122)
		tmp = fma(Float64(z * Float64(z / x)), y, Float64(0.083333333333333 / x));
	elseif (t_0 <= 4e+306)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(0.083333333333333 / x) + 0.91893853320467) - x));
	else
		tmp = Float64(Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -1e+122], N[(N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision] * y + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+306], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -1.00000000000000001e122

   1. Initial program 84.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} + \frac{{z}^{2}}{x}\right)\right)\right)\right) - \frac{x}{y}\right)} \]

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}, y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 8.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{{z}^{2}}{x}, y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.7%

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

   if -1.00000000000000001e122 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4.00000000000000007e306

   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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)} - x \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)\right)} - x \]

      3. associate--l+ N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right) + \left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} + \left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x - \frac{1}{2}}, \log x, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      7. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \color{blue}{\log x}, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{91893853320467}{100000000000000}\right)} - x\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{91893853320467}{100000000000000}\right)} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}} + \frac{91893853320467}{100000000000000}\right) - x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) \]

      13. lower-/.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if 4.00000000000000007e306 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 87.9

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

   5. Applied rewrites 87.9%

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

Alternative 4: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{z}{x}, y, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x} + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -1e+122)
     (fma (* z (/ z x)) y (/ 0.083333333333333 x))
     (if (<= t_0 4e+306)
       (-
        (fma (log x) (- x 0.5) (+ (/ 0.083333333333333 x) 0.91893853320467))
        x)
       (* (* (+ (/ y x) (/ 0.0007936500793651 x)) z) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -1e+122) {
		tmp = fma((z * (z / x)), y, (0.083333333333333 / x));
	} else if (t_0 <= 4e+306) {
		tmp = fma(log(x), (x - 0.5), ((0.083333333333333 / x) + 0.91893853320467)) - x;
	} else {
		tmp = (((y / x) + (0.0007936500793651 / x)) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = 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))
	tmp = 0.0
	if (t_0 <= -1e+122)
		tmp = fma(Float64(z * Float64(z / x)), y, Float64(0.083333333333333 / x));
	elseif (t_0 <= 4e+306)
		tmp = Float64(fma(log(x), Float64(x - 0.5), Float64(Float64(0.083333333333333 / x) + 0.91893853320467)) - x);
	else
		tmp = Float64(Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -1e+122], N[(N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision] * y + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+306], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -1.00000000000000001e122

   1. Initial program 84.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} + \frac{{z}^{2}}{x}\right)\right)\right)\right) - \frac{x}{y}\right)} \]

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}, y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 8.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{{z}^{2}}{x}, y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.7%

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

   if -1.00000000000000001e122 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4.00000000000000007e306

   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. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{y \cdot \left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \frac{{z}^{2}}{x}\right)\right)} \]

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)} - x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)\right)} - x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} - x \]

      5. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x - \frac{1}{2}}, \frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log x, x - \frac{1}{2}, \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{91893853320467}{100000000000000}}\right) - x \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log x, x - \frac{1}{2}, \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{91893853320467}{100000000000000}}\right) - x \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\log x, x - \frac{1}{2}, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}} + \frac{91893853320467}{100000000000000}\right) - x \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} + \frac{91893853320467}{100000000000000}\right) - x \]

      11. lower-/.f64 90.1

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

   8. Applied rewrites 90.1%

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

   if 4.00000000000000007e306 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 87.9

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

   5. Applied rewrites 87.9%

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

Alternative 5: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{z}{x}, y, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\log x \cdot x - \left(\left(x - 0.91893853320467\right) - \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -1e+122)
     (fma (* z (/ z x)) y (/ 0.083333333333333 x))
     (if (<= t_0 4e+306)
       (- (* (log x) x) (- (- x 0.91893853320467) (/ 0.083333333333333 x)))
       (* (* (+ (/ y x) (/ 0.0007936500793651 x)) z) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -1e+122) {
		tmp = fma((z * (z / x)), y, (0.083333333333333 / x));
	} else if (t_0 <= 4e+306) {
		tmp = (log(x) * x) - ((x - 0.91893853320467) - (0.083333333333333 / x));
	} else {
		tmp = (((y / x) + (0.0007936500793651 / x)) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = 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))
	tmp = 0.0
	if (t_0 <= -1e+122)
		tmp = fma(Float64(z * Float64(z / x)), y, Float64(0.083333333333333 / x));
	elseif (t_0 <= 4e+306)
		tmp = Float64(Float64(log(x) * x) - Float64(Float64(x - 0.91893853320467) - Float64(0.083333333333333 / x)));
	else
		tmp = Float64(Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -1e+122], N[(N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision] * y + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+306], N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -1.00000000000000001e122

   1. Initial program 84.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} + \frac{{z}^{2}}{x}\right)\right)\right)\right) - \frac{x}{y}\right)} \]

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}, y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 8.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{{z}^{2}}{x}, y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.7%

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

   if -1.00000000000000001e122 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4.00000000000000007e306

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      11. lower--.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around 0

      \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 90.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \log \left(\frac{1}{x}\right)\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\log \left(\frac{1}{x}\right) \cdot x\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right) \cdot x} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right) \cdot x} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} \cdot x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      5. log-rec N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) \cdot x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \color{blue}{\log x} \cdot x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      7. lower-log.f64 86.3

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

   10. Applied rewrites 86.3%

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

   if 4.00000000000000007e306 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 87.9

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

   5. Applied rewrites 87.9%

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

Alternative 6: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\log x \cdot \left(x - 0.5\right) - \left(\left(x - 0.91893853320467\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right)\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e+77)
   (-
    (* (log x) (- x 0.5))
    (-
     (- x 0.91893853320467)
     (/
      (fma
       (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
       z
       0.083333333333333)
      x)))
   (-
    (+
     (fma
      (fma (/ y x) z (/ (fma 0.0007936500793651 z -0.0027777777777778) x))
      z
      (/ 0.083333333333333 x))
     (fma (- x 0.5) (log x) 0.91893853320467))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+77) {
		tmp = (log(x) * (x - 0.5)) - ((x - 0.91893853320467) - (fma(fma(z, (0.0007936500793651 + y), -0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = (fma(fma((y / x), z, (fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, (0.083333333333333 / x)) + fma((x - 0.5), log(x), 0.91893853320467)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e+77)
		tmp = Float64(Float64(log(x) * Float64(x - 0.5)) - Float64(Float64(x - 0.91893853320467) - Float64(fma(fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = Float64(Float64(fma(fma(Float64(y / x), z, Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, Float64(0.083333333333333 / x)) + fma(Float64(x - 0.5), log(x), 0.91893853320467)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e+77], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y / x), $MachinePrecision] * z + N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999997e77

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 1.99999999999999997e77 < x

   1. Initial program 79.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) - x} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right)\right) - x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log x \cdot \left(x - 0.5\right)\\ \mathbf{if}\;x \leq 2300:\\ \;\;\;\;t\_0 - \left(\left(x - 0.91893853320467\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(\left(x - 0.91893853320467\right) - z \cdot \frac{\left(0.0007936500793651 + y\right) \cdot z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log x) (- x 0.5))))
   (if (<= x 2300.0)
     (-
      t_0
      (-
       (- x 0.91893853320467)
       (/
        (fma
         (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
         z
         0.083333333333333)
        x)))
     (-
      t_0
      (- (- x 0.91893853320467) (* z (/ (* (+ 0.0007936500793651 y) z) x)))))))

C

double code(double x, double y, double z) {
	double t_0 = log(x) * (x - 0.5);
	double tmp;
	if (x <= 2300.0) {
		tmp = t_0 - ((x - 0.91893853320467) - (fma(fma(z, (0.0007936500793651 + y), -0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = t_0 - ((x - 0.91893853320467) - (z * (((0.0007936500793651 + y) * z) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(x) * Float64(x - 0.5))
	tmp = 0.0
	if (x <= 2300.0)
		tmp = Float64(t_0 - Float64(Float64(x - 0.91893853320467) - Float64(fma(fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = Float64(t_0 - Float64(Float64(x - 0.91893853320467) - Float64(z * Float64(Float64(Float64(0.0007936500793651 + y) * z) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2300.0], N[(t$95$0 - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(z * N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2300

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      11. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 2300 < x

   1. Initial program 83.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      11. lower--.f64 83.5

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

   4. Applied rewrites 83.5%

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

   5. Taylor expanded in z around inf

      \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}}\right) \]

      2. unpow2 N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(z \cdot z\right)} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right) \]

      3. associate-*l* N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \color{blue}{z \cdot \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \color{blue}{\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot 1} + y\right)}{x}\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\left(\frac{1}{y} \cdot y\right)} + y\right)}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right) \cdot y} + y\right)}{x}\right) \]

      8. *-lft-identity N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right) \cdot y + \color{blue}{1 \cdot y}\right)}{x}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \color{blue}{\left(y \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y} + 1\right)\right)}}{x}\right) \]

      10. +-commutative N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \left(y \cdot \color{blue}{\left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}\right)}{x}\right) \]

      11. associate-*l* N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{\color{blue}{\left(z \cdot y\right) \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}}{x}\right) \]

      12. *-commutative N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{\color{blue}{\left(y \cdot z\right)} \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x}\right) \]

      13. associate-*r* N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{\color{blue}{y \cdot \left(z \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)\right)}}{x}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \color{blue}{z \cdot \frac{y \cdot \left(z \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)\right)}{x}}\right) \]

   7. Applied rewrites 96.6%

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

Alternative 8: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\log x \cdot \left(x - 0.5\right) - \left(\left(x - 0.91893853320467\right) - z \cdot \frac{\left(0.0007936500793651 + y\right) \cdot z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.038)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (-
    (* (log x) (- x 0.5))
    (- (- x 0.91893853320467) (* z (/ (* (+ 0.0007936500793651 y) z) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.038) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) * (x - 0.5)) - ((x - 0.91893853320467) - (z * (((0.0007936500793651 + y) * z) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.038)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) * Float64(x - 0.5)) - Float64(Float64(x - 0.91893853320467) - Float64(z * Float64(Float64(Float64(0.0007936500793651 + y) * z) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.038], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(z * N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0379999999999999991

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 0.0379999999999999991 < x

   1. Initial program 84.0%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      11. lower--.f64 84.0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}}\right) \]

      2. unpow2 N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(z \cdot z\right)} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right) \]

      3. associate-*l* N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \color{blue}{z \cdot \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \color{blue}{\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot 1} + y\right)}{x}\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\left(\frac{1}{y} \cdot y\right)} + y\right)}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right) \cdot y} + y\right)}{x}\right) \]

      8. *-lft-identity N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right) \cdot y + \color{blue}{1 \cdot y}\right)}{x}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \color{blue}{\left(y \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y} + 1\right)\right)}}{x}\right) \]

      10. +-commutative N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{z \cdot \left(y \cdot \color{blue}{\left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}\right)}{x}\right) \]

      11. associate-*l* N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{\color{blue}{\left(z \cdot y\right) \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}}{x}\right) \]

      12. *-commutative N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{\color{blue}{\left(y \cdot z\right)} \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x}\right) \]

      13. associate-*r* N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - z \cdot \frac{\color{blue}{y \cdot \left(z \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)\right)}}{x}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) - \left(\left(x - \frac{91893853320467}{100000000000000}\right) - \color{blue}{z \cdot \frac{y \cdot \left(z \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)\right)}{x}}\right) \]

   7. Applied rewrites 95.5%

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

Alternative 9: 84.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\log x - 1}{y} \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7e+40)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (/ (- (log x) 1.0) y) y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7e+40) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (((log(x) - 1.0) / y) * y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7e+40)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(log(x) - 1.0) / y) * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7e+40], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.9999999999999998e40

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if 6.9999999999999998e40 < x

   1. Initial program 81.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} + \frac{{z}^{2}}{x}\right)\right)\right)\right) - \frac{x}{y}\right)} \]

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{y} - \frac{1}{y}\right), y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 49.1%

      \[\leadsto \mathsf{fma}\left(\left(\frac{\log x}{y} - \frac{1}{y}\right) \cdot x, y, \frac{0.083333333333333}{x}\right) \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 71.5%

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

Alternative 10: 66.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.083333333333333, \frac{-1}{x}, \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e-40)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (fma
    -0.083333333333333
    (/ -1.0 x)
    (* (* (/ z x) (+ 0.0007936500793651 y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e-40) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = fma(-0.083333333333333, (-1.0 / x), (((z / x) * (0.0007936500793651 + y)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e-40)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = fma(-0.083333333333333, Float64(-1.0 / x), Float64(Float64(Float64(z / x) * Float64(0.0007936500793651 + y)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e-40], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(-0.083333333333333 * N[(-1.0 / x), $MachinePrecision] + N[(N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999993e-41

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 9.9999999999999993e-41 < x

   1. Initial program 85.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) - x} \]

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \frac{1}{x \cdot 12.000000000000048}\right) + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right)\right) - x \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(\frac{-83333333333333}{1000000000000000}, \frac{1}{-x}, {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 40.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.083333333333333, \frac{-1}{x}, \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4e+78)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (+ (/ y x) (/ 0.0007936500793651 x)) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e+78) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (((y / x) + (0.0007936500793651 / x)) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4e+78)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4e+78], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000003e78

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if 4.00000000000000003e78 < x

   1. Initial program 79.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 29.9

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

   5. Applied rewrites 29.9%

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

Alternative 12: 65.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7.2e+72)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (/ z x) (+ 0.0007936500793651 y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7.2e+72) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((z / x) * (0.0007936500793651 + y)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7.2e+72)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * Float64(0.0007936500793651 + y)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7.2e+72], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.20000000000000069e72

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if 7.20000000000000069e72 < x

   1. Initial program 80.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{y \cdot \left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \frac{{z}^{2}}{x}\right)\right)} \]

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \frac{y}{x} \cdot z\right)} \cdot z \]

      7. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \left(\frac{1}{x} \cdot z\right)} + \frac{y}{x} \cdot z\right) \cdot z \]

      8. associate-*l/ N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1 \cdot z}{x}} + \frac{y}{x} \cdot z\right) \cdot z \]

      9. *-lft-identity N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{\color{blue}{z}}{x} + \frac{y}{x} \cdot z\right) \cdot z \]

      10. associate-*l/ N/A

          \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{\frac{y \cdot z}{x}}\right) \cdot z \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{y \cdot \frac{z}{x}}\right) \cdot z \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)} \cdot z \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)} \cdot z \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{z}{x}} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot z \]

      15. lower-+.f64 30.3

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

   8. Applied rewrites 30.3%

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

Alternative 13: 44.0% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* (* (/ z x) (+ 0.0007936500793651 y)) z))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return ((z / x) * (0.0007936500793651 + y)) * z;
}

Python

def code(x, y, z):
	return ((z / x) * (0.0007936500793651 + y)) * z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(z / x) * Float64(0.0007936500793651 + y)) * z)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in y around inf

   \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{y \cdot \left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \frac{{z}^{2}}{x}\right)\right)} \]

5. Applied rewrites 93.0%

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

6. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

   6. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \frac{y}{x} \cdot z\right)} \cdot z \]

   7. associate-*l* N/A

      \[\leadsto \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \left(\frac{1}{x} \cdot z\right)} + \frac{y}{x} \cdot z\right) \cdot z \]

   8. associate-*l/ N/A

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1 \cdot z}{x}} + \frac{y}{x} \cdot z\right) \cdot z \]

   9. *-lft-identity N/A

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{\color{blue}{z}}{x} + \frac{y}{x} \cdot z\right) \cdot z \]

   10. associate-*l/ N/A

       \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{\frac{y \cdot z}{x}}\right) \cdot z \]

   11. associate-/l* N/A

       \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{y \cdot \frac{z}{x}}\right) \cdot z \]

   12. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)} \cdot z \]

   13. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)} \cdot z \]

   14. lower-/.f64 N/A

       \[\leadsto \left(\color{blue}{\frac{z}{x}} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot z \]

   15. lower-+.f64 43.5

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

8. Applied rewrites 43.5%

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

Alternative 14: 43.7% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* (/ (* (+ 0.0007936500793651 y) z) x) z))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return (((0.0007936500793651 + y) * z) / x) * z;
}

Python

def code(x, y, z):
	return (((0.0007936500793651 + y) * z) / x) * z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) / x) * z)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} + \frac{{z}^{2}}{x}\right)\right)\right)\right) - \frac{x}{y}\right)} \]

5. Applied rewrites 80.3%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x \cdot \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{y} - \frac{1}{y}\right), y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
7. Step-by-step derivation

8. Applied rewrites 44.6%

   \[\leadsto \mathsf{fma}\left(\left(\frac{\log x}{y} - \frac{1}{y}\right) \cdot x, y, \frac{0.083333333333333}{x}\right) \]

9. Taylor expanded in z around inf

   \[\leadsto y \cdot \color{blue}{\left({z}^{2} \cdot \left(\frac{1}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right)\right)} \]

11. Applied rewrites 42.8%

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

Alternative 15: 31.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(\frac{z}{x} \cdot z\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

   4. unpow2 N/A

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

   5. lower-*.f64 32.4

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

5. Applied rewrites 32.4%

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

7. Applied rewrites 35.1%

   \[\leadsto y \cdot \color{blue}{\left(\frac{z}{x} \cdot z\right)} \]
8. Add Preprocessing

Alternative 16: 8.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ -0.0027777777777778 \cdot \frac{z}{x} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return -0.0027777777777778 * (z / x);
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return -0.0027777777777778 * (z / x);
}

Python

def code(x, y, z):
	return -0.0027777777777778 * (z / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right) \cdot {z}^{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right) \cdot {z}^{2}} \]

   3. associate-*r/ N/A

      \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x \cdot z}}\right) \cdot {z}^{2} \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x \cdot z}\right) \cdot {z}^{2} \]

   5. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right)} \cdot {z}^{2} \]

   6. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   8. lower-/.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   9. associate-*r/ N/A

      \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   11. lower-/.f64 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   12. *-commutative N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{\color{blue}{z \cdot x}}\right) \cdot {z}^{2} \]

   13. associate-/r* N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \color{blue}{\frac{\frac{\frac{13888888888889}{5000000000000000}}{z}}{x}}\right) \cdot {z}^{2} \]

   14. metadata-eval N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{\color{blue}{\frac{13888888888889}{5000000000000000} \cdot 1}}{z}}{x}\right) \cdot {z}^{2} \]

   15. associate-*r/ N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\color{blue}{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}}}{x}\right) \cdot {z}^{2} \]

   16. lower-/.f64 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}}{x}}\right) \cdot {z}^{2} \]

   17. associate-*r/ N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{z}}}{x}\right) \cdot {z}^{2} \]

   18. metadata-eval N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{z}}{x}\right) \cdot {z}^{2} \]

   19. lower-/.f64 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\color{blue}{\frac{\frac{13888888888889}{5000000000000000}}{z}}}{x}\right) \cdot {z}^{2} \]

   20. unpow2 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{\frac{13888888888889}{5000000000000000}}{z}}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

   21. lower-*.f64 39.8

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

5. Applied rewrites 39.8%

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

6. Taylor expanded in z around 0

   \[\leadsto \frac{-13888888888889}{5000000000000000} \cdot \color{blue}{\frac{z}{x}} \]
7. Step-by-step derivation

8. Applied rewrites 7.4%

   \[\leadsto -0.0027777777777778 \cdot \color{blue}{\frac{z}{x}} \]
9. Add Preprocessing

Developer Target 1: 98.8% accurate, 0.9× 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 2024313 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (* (- x 1/2) (log x)) (- 91893853320467/100000000000000 x)) (/ 83333333333333/1000000000000000 x)) (* (/ z x) (- (* z (+ y 7936500793651/10000000000000000)) 13888888888889/5000000000000000))))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))