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

Percentage Accurate: 94.2% → 99.5%

Time: 15.6s

Alternatives: 18

Speedup: 1.0×

• 31.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(x, \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467\right) - x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 - \left(x + \log x \cdot \left(0.5 - x\right)\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5e+24)
   (/
    1.0
    (/
     x
     (fma
      x
      (- (fma (+ x -0.5) (log x) 0.91893853320467) x)
      (fma
       z
       (fma z (+ y 0.0007936500793651) -0.0027777777777778)
       0.083333333333333))))
   (+
    (- 0.91893853320467 (+ x (* (log x) (- 0.5 x))))
    (* z (* z (/ (+ y 0.0007936500793651) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5e+24) {
		tmp = 1.0 / (x / fma(x, (fma((x + -0.5), log(x), 0.91893853320467) - x), fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333)));
	} else {
		tmp = (0.91893853320467 - (x + (log(x) * (0.5 - x)))) + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5e+24)
		tmp = Float64(1.0 / Float64(x / fma(x, Float64(fma(Float64(x + -0.5), log(x), 0.91893853320467) - x), fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333))));
	else
		tmp = Float64(Float64(0.91893853320467 - Float64(x + Float64(log(x) * Float64(0.5 - x)))) + Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5e+24], N[(1.0 / N[(x / N[(x * N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision] + N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 - N[(x + N[(N[Log[x], $MachinePrecision] * N[(0.5 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000045e24

   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

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

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

   if 5.00000000000000045e24 < x

   1. Initial program 87.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(x, \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467\right) - x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 - \left(x + \log x \cdot \left(0.5 - x\right)\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 - \left(x + \log x \cdot \left(0.5 - x\right)\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+163}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (- 0.91893853320467 (+ x (* (log x) (- 0.5 x))))
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
           x))))
   (if (<= t_0 -1e+163)
     (* (* y z) (/ z x))
     (if (<= t_0 INFINITY)
       (+
        0.91893853320467
        (fma (log x) (+ x -0.5) (- (/ 0.083333333333333 x) x)))
       (* z (* z (+ (/ 0.0007936500793651 x) (/ y x))))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 - (x + (log(x) * (0.5 - x)))) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -1e+163) {
		tmp = (y * z) * (z / x);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.91893853320467 + fma(log(x), (x + -0.5), ((0.083333333333333 / x) - x));
	} else {
		tmp = z * (z * ((0.0007936500793651 / x) + (y / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 - Float64(x + Float64(log(x) * Float64(0.5 - x)))) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_0 <= -1e+163)
		tmp = Float64(Float64(y * z) * Float64(z / x));
	elseif (t_0 <= Inf)
		tmp = Float64(0.91893853320467 + fma(log(x), Float64(x + -0.5), Float64(Float64(0.083333333333333 / x) - x)));
	else
		tmp = Float64(z * Float64(z * Float64(Float64(0.0007936500793651 / x) + Float64(y / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 - N[(x + N[(N[Log[x], $MachinePrecision] * N[(0.5 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+163], N[(N[(y * z), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -9.9999999999999994e162

   1. Initial program 83.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   7. Applied rewrites 83.8%

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

   if -9.9999999999999994e162 < (+.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)) < +inf.0

   1. Initial program 95.7%

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

   3. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if +inf.0 < (+.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 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 1.2%

       \[\leadsto \frac{z \cdot -0.0027777777777778}{x} \]

   12. Taylor expanded in z around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 0.2

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

   14. Applied rewrites 0.2%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 - \left(x + \log x \cdot \left(0.5 - x\right)\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -1 \cdot 10^{+163}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;\left(0.91893853320467 - \left(x + \log x \cdot \left(0.5 - x\right)\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq \infty:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+110}:\\ \;\;\;\;\left(x + -0.5\right) \cdot \log x + \left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} - \left(x + -0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 - \left(x + \log x \cdot \left(0.5 - x\right)\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e+110)
   (+
    (* (+ x -0.5) (log x))
    (-
     (/
      (fma
       z
       (fma (+ y 0.0007936500793651) z -0.0027777777777778)
       0.083333333333333)
      x)
     (+ x -0.91893853320467)))
   (+
    (- 0.91893853320467 (+ x (* (log x) (- 0.5 x))))
    (* z (* z (/ (+ y 0.0007936500793651) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+110) {
		tmp = ((x + -0.5) * log(x)) + ((fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x) - (x + -0.91893853320467));
	} else {
		tmp = (0.91893853320467 - (x + (log(x) * (0.5 - x)))) + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e+110)
		tmp = Float64(Float64(Float64(x + -0.5) * log(x)) + Float64(Float64(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x) - Float64(x + -0.91893853320467)));
	else
		tmp = Float64(Float64(0.91893853320467 - Float64(x + Float64(log(x) * Float64(0.5 - x)))) + Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e+110], N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 - N[(x + N[(N[Log[x], $MachinePrecision] * N[(0.5 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e110

   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. sub-neg N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-+.f64 N/A

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

      14. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   if 1e110 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+110}:\\ \;\;\;\;\left(x + -0.5\right) \cdot \log x + \left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} - \left(x + -0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 - \left(x + \log x \cdot \left(0.5 - x\right)\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 - \left(x + \log x \cdot \left(0.5 - x\right)\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.2)
   (fma
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (fma -0.5 (log x) 0.91893853320467))
   (+
    (- 0.91893853320467 (+ x (* (log x) (- 0.5 x))))
    (* z (* z (/ (+ y 0.0007936500793651) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.2) {
		tmp = fma(fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333), (1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	} else {
		tmp = (0.91893853320467 - (x + (log(x) * (0.5 - x)))) + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.2)
		tmp = fma(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = Float64(Float64(0.91893853320467 - Float64(x + Float64(log(x) * Float64(0.5 - x)))) + Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.2], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 - N[(x + N[(N[Log[x], $MachinePrecision] * N[(0.5 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.20000000000000001

   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

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 68.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + \frac{7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 99.0

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

   9. Applied rewrites 99.0%

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

   if 0.20000000000000001 < x

   1. Initial program 87.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 - \left(x + \log x \cdot \left(0.5 - x\right)\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \log x, -x\right)\\ \mathbf{if}\;x \leq 1.06 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), \mathsf{fma}\left(x, t\_0, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x (log x) (- x))))
   (if (<= x 1.06e+111)
     (/
      (fma
       z
       (fma z (+ y 0.0007936500793651) -0.0027777777777778)
       (fma x t_0 0.083333333333333))
      x)
     (+ t_0 (* y (/ (* z z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, log(x), -x);
	double tmp;
	if (x <= 1.06e+111) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), fma(x, t_0, 0.083333333333333)) / x;
	} else {
		tmp = t_0 + (y * ((z * z) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, log(x), Float64(-x))
	tmp = 0.0
	if (x <= 1.06e+111)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), fma(x, t_0, 0.083333333333333)) / x);
	else
		tmp = Float64(t_0 + Float64(y * Float64(Float64(z * z) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]}, If[LessEqual[x, 1.06e+111], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + N[(x * t$95$0 + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(t$95$0 + N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.06e111

   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

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \mathsf{fma}\left(x, x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

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

   if 1.06e111 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 81.5%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 91.3

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. distribute-rgt-in N/A

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

       4. mul-1-neg N/A

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

       5. distribute-lft-neg-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. mul-1-neg N/A

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

       9. neg-mul-1 N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. log-rec N/A

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

       13. remove-double-neg N/A

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

       14. lower-log.f64 N/A

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

       15. lower-neg.f64 91.4

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

   11. Applied rewrites 91.4%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.06 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \log x, -x\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.6e+49)
   (fma
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (fma -0.5 (log x) 0.91893853320467))
   (+ (fma x (log x) (- x)) (* y (/ (* z z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.6e+49) {
		tmp = fma(fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333), (1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	} else {
		tmp = fma(x, log(x), -x) + (y * ((z * z) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.6e+49)
		tmp = fma(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = Float64(fma(x, log(x), Float64(-x)) + Float64(y * Float64(Float64(z * z) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.6e+49], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.60000000000000004e49

   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

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 67.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + \frac{7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 94.2

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

   9. Applied rewrites 94.2%

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

   if 4.60000000000000004e49 < x

   1. Initial program 86.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. distribute-rgt-in N/A

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

       4. mul-1-neg N/A

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

       5. distribute-lft-neg-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. mul-1-neg N/A

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

       9. neg-mul-1 N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. log-rec N/A

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

       13. remove-double-neg N/A

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

       14. lower-log.f64 N/A

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

       15. lower-neg.f64 92.6

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

   11. Applied rewrites 92.6%

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

Alternative 7: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.6e+49)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (+ (fma x (log x) (- x)) (* y (/ (* z z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.6e+49) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma(x, log(x), -x) + (y * ((z * z) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.6e+49)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(fma(x, log(x), Float64(-x)) + Float64(y * Float64(Float64(z * z) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.6e+49], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.60000000000000004e49

   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. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if 4.60000000000000004e49 < x

   1. Initial program 86.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. distribute-rgt-in N/A

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

       4. mul-1-neg N/A

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

       5. distribute-lft-neg-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. mul-1-neg N/A

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

       9. neg-mul-1 N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. log-rec N/A

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

       13. remove-double-neg N/A

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

       14. lower-log.f64 N/A

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

       15. lower-neg.f64 92.6

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

   11. Applied rewrites 92.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7.8e+49)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (fma x (log x) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7.8e+49) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma(x, log(x), -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7.8e+49)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = fma(x, log(x), Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7.8e+49], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.8000000000000002e49

   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. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if 7.8000000000000002e49 < x

   1. Initial program 86.0%

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

   3. Taylor expanded in 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 97.1%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \log \left(\frac{1}{x}\right), \mathsf{neg}\left(x\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)}, \mathsf{neg}\left(x\right)\right) \]

      12. log-rec N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right), \mathsf{neg}\left(x\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log x}, \mathsf{neg}\left(x\right)\right) \]

      14. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log x}, \mathsf{neg}\left(x\right)\right) \]

      15. lower-neg.f64 79.0

          \[\leadsto \mathsf{fma}\left(x, \log x, \color{blue}{-x}\right) \]

   8. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log x, -x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7.8e+49)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (- (* x (log x)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7.8e+49) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = (x * log(x)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7.8e+49)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(Float64(x * log(x)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7.8e+49], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.8000000000000002e49

   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. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if 7.8000000000000002e49 < x

   1. Initial program 86.0%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \log x + x \cdot -1} \]

      7. *-commutative N/A

         \[\leadsto x \cdot \log x + \color{blue}{-1 \cdot x} \]

      8. neg-mul-1 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 79.0

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

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{x \cdot \log x - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          0.083333333333333
          (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))))
   (if (<= t_0 -1e+33)
     (* (* y z) (/ z x))
     (if (<= t_0 0.1)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       (* (/ (+ y 0.0007936500793651) x) (* z z))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
	double tmp;
	if (t_0 <= -1e+33) {
		tmp = (y * z) * (z / x);
	} else if (t_0 <= 0.1) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = ((y + 0.0007936500793651) / x) * (z * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
	tmp = 0.0
	if (t_0 <= -1e+33)
		tmp = Float64(Float64(y * z) * Float64(z / x));
	elseif (t_0 <= 0.1)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+33], N[(N[(y * z), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -9.9999999999999995e32

   1. Initial program 84.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.1%

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

   if -9.9999999999999995e32 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in 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 95.1%

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

   10. Applied rewrites 98.4%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{\color{blue}{x}} \]
   12. Step-by-step derivation

   13. Applied rewrites 52.2%

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

   if 0.10000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 89.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 71.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          0.083333333333333
          (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))))
   (if (<= t_0 -1e+33)
     (* (* y z) (/ z x))
     (if (<= t_0 5e+33)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       (/ (* z (fma z 0.0007936500793651 -0.0027777777777778)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = 0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
	double tmp;
	if (t_0 <= -1e+33) {
		tmp = (y * z) * (z / x);
	} else if (t_0 <= 5e+33) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = (z * fma(z, 0.0007936500793651, -0.0027777777777778)) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
	tmp = 0.0
	if (t_0 <= -1e+33)
		tmp = Float64(Float64(y * z) * Float64(z / x));
	elseif (t_0 <= 5e+33)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(z * fma(z, 0.0007936500793651, -0.0027777777777778)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+33], N[(N[(y * z), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+33], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -9.9999999999999995e32

   1. Initial program 84.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.1%

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

   if -9.9999999999999995e32 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.99999999999999973e33

   1. Initial program 99.5%

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

   3. Taylor expanded in 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 94.4%

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

   10. Applied rewrites 97.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{\color{blue}{x}} \]
   12. Step-by-step derivation

   13. Applied rewrites 50.8%

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

   if 4.99999999999999973e33 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 89.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 59.7%

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

   12. Taylor expanded in z around 0

       \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x} \]
   13. Step-by-step derivation

   14. Applied rewrites 59.7%

       \[\leadsto \frac{z \cdot \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(0.0007936500793651 \cdot z\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          0.083333333333333
          (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))))
   (if (<= t_0 -1e+33)
     (* (* y z) (/ z x))
     (if (<= t_0 0.1)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       (/ (* z (* 0.0007936500793651 z)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = 0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
	double tmp;
	if (t_0 <= -1e+33) {
		tmp = (y * z) * (z / x);
	} else if (t_0 <= 0.1) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = (z * (0.0007936500793651 * z)) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
	tmp = 0.0
	if (t_0 <= -1e+33)
		tmp = Float64(Float64(y * z) * Float64(z / x));
	elseif (t_0 <= 0.1)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(z * Float64(0.0007936500793651 * z)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+33], N[(N[(y * z), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * N[(0.0007936500793651 * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -9.9999999999999995e32

   1. Initial program 84.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.1%

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

   if -9.9999999999999995e32 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in 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 95.1%

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

   10. Applied rewrites 98.4%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{\color{blue}{x}} \]
   12. Step-by-step derivation

   13. Applied rewrites 52.2%

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

   if 0.10000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 89.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 57.0%

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

   12. Taylor expanded in z around inf

       \[\leadsto \frac{\frac{7936500793651}{10000000000000000} \cdot {z}^{2}}{x} \]
   13. Step-by-step derivation

   14. Applied rewrites 57.0%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(0.0007936500793651 \cdot z\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          0.083333333333333
          (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))))
   (if (<= t_0 -1e+33)
     (* (* y z) (/ z x))
     (if (<= t_0 5e+29)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       (* y (/ (* z z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
	double tmp;
	if (t_0 <= -1e+33) {
		tmp = (y * z) * (z / x);
	} else if (t_0 <= 5e+29) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
	tmp = 0.0
	if (t_0 <= -1e+33)
		tmp = Float64(Float64(y * z) * Float64(z / x));
	elseif (t_0 <= 5e+29)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(y * Float64(Float64(z * z) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+33], N[(N[(y * z), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+29], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -9.9999999999999995e32

   1. Initial program 84.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.1%

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

   if -9.9999999999999995e32 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 5.0000000000000001e29

   1. Initial program 99.5%

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

   3. Taylor expanded in 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 94.4%

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

   10. Applied rewrites 98.4%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{\color{blue}{x}} \]
   12. Step-by-step derivation

   13. Applied rewrites 51.1%

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

   if 5.0000000000000001e29 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 89.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 91.5%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.3

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

   8. Applied rewrites 51.3%

      \[\leadsto \color{blue}{y \cdot \frac{z \cdot z}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ t_1 := y \cdot \frac{z \cdot z}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
        (t_1 (* y (/ (* z z) x))))
   (if (<= t_0 -1e+33)
     t_1
     (if (<= t_0 5e+29)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double t_1 = y * ((z * z) / x);
	double tmp;
	if (t_0 <= -1e+33) {
		tmp = t_1;
	} else if (t_0 <= 5e+29) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))
	t_1 = Float64(y * Float64(Float64(z * z) / x))
	tmp = 0.0
	if (t_0 <= -1e+33)
		tmp = t_1;
	elseif (t_0 <= 5e+29)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+33], t$95$1, If[LessEqual[t$95$0, 5e+29], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -9.9999999999999995e32 or 5.0000000000000001e29 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 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 94.4%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.7

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

   8. Applied rewrites 57.7%

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

   if -9.9999999999999995e32 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.0000000000000001e29

   1. Initial program 99.5%

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

   3. Taylor expanded in 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 94.4%

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

   10. Applied rewrites 98.4%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{\color{blue}{x}} \]
   12. Step-by-step derivation

   13. Applied rewrites 51.1%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -1 \cdot 10^{+33}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       0.083333333333333
       (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
      2e+251)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (* z (* z (+ (/ 0.0007936500793651 x) (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) <= 2e+251) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = z * (z * ((0.0007936500793651 / x) + (y / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) <= 2e+251)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(z * Float64(z * Float64(Float64(0.0007936500793651 / x) + Float64(y / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+251], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2.0000000000000001e251

   1. Initial program 96.2%

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

   3. Taylor expanded in x around 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. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 56.7

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

   5. Applied rewrites 56.7%

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

   if 2.0000000000000001e251 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 83.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 77.4%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 14.2%

       \[\leadsto \frac{z \cdot -0.0027777777777778}{x} \]

   12. Taylor expanded in z around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 86.7

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

   14. Applied rewrites 86.7%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.3% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.6%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 60.8

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

5. Applied rewrites 60.8%

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

6. Final simplification 60.8%

   \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
7. Add Preprocessing

Alternative 17: 29.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (fma -0.0027777777777778 z 0.083333333333333) x))

C

double code(double x, double y, double z) {
	return fma(-0.0027777777777778, z, 0.083333333333333) / x;
}

Julia

function code(x, y, z)
	return Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x)
end

Wolfram

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

Derivation

1. Initial program 93.6%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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 94.4%

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

7. Applied rewrites 94.5%

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

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

10. Applied rewrites 68.8%

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

11. Taylor expanded in x around 0

    \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{\color{blue}{x}} \]
12. Step-by-step derivation

13. Applied rewrites 32.4%

    \[\leadsto \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{\color{blue}{x}} \]
14. Add Preprocessing

Alternative 18: 8.1% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 73.9%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 35.4%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 7.5%

    \[\leadsto \frac{z \cdot -0.0027777777777778}{x} \]
12. Add Preprocessing

Developer Target 1: 98.7% 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 2024233 
(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)))