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

Percentage Accurate: 93.7% → 99.6%

Time: 16.1s

Alternatives: 18

Speedup: 1.0×

• 31.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 45000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, -x\right), \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 45000000000000.0)
   (/
    (fma
     x
     (+ 0.91893853320467 (fma (log x) (+ x -0.5) (- x)))
     (fma
      z
      (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
      0.083333333333333))
    x)
   (+ (- (* x (log x)) x) (* z (* z (/ (+ 0.0007936500793651 y) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 45000000000000.0) {
		tmp = fma(x, (0.91893853320467 + fma(log(x), (x + -0.5), -x)), fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333)) / x;
	} else {
		tmp = ((x * log(x)) - x) + (z * (z * ((0.0007936500793651 + y) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 45000000000000.0)
		tmp = Float64(fma(x, Float64(0.91893853320467 + fma(log(x), Float64(x + -0.5), Float64(-x))), fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333)) / x);
	else
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(z * Float64(z * Float64(Float64(0.0007936500793651 + y) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 45000000000000.0], N[(N[(x * N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(z * N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5e13

   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

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

   5. Simplified 99.7%

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

   if 4.5e13 < x

   1. Initial program 84.9%

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

   3. Taylor expanded in 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. Simplified 99.6%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. neg-mul-1 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. log-rec N/A

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

      11. remove-double-neg N/A

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

      12. lower-log.f64 99.6

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

   8. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 45000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, -x\right), \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) - \frac{z \cdot \left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) - 0.083333333333333}{x} \leq -5 \cdot 10^{+141}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (-
       (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
       (/
        (-
         (* z (- 0.0027777777777778 (* z (+ 0.0007936500793651 y))))
         0.083333333333333)
        x))
      -5e+141)
   (* z (* y (/ z x)))
   (/
    (fma z (fma z 0.0007936500793651 -0.0027777777777778) 0.083333333333333)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((0.91893853320467 + ((log(x) * (x - 0.5)) - x)) - (((z * (0.0027777777777778 - (z * (0.0007936500793651 + y)))) - 0.083333333333333) / x)) <= -5e+141) {
		tmp = z * (y * (z / x));
	} else {
		tmp = fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) - Float64(Float64(Float64(z * Float64(0.0027777777777778 - Float64(z * Float64(0.0007936500793651 + y)))) - 0.083333333333333) / x)) <= -5e+141)
		tmp = Float64(z * Float64(y * Float64(z / x)));
	else
		tmp = Float64(fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(z * N[(0.0027777777777778 - N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -5e+141], N[(z * N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 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)) < -5.00000000000000025e141

   1. Initial program 90.5%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   8. Simplified 92.6%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 92.6

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

   11. Simplified 92.6%

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

   if -5.00000000000000025e141 < (+.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 94.1%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 88.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Simplified 63.6%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 57.5

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

   11. Simplified 57.5%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) - \frac{z \cdot \left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) - 0.083333333333333}{x} \leq -5 \cdot 10^{+141}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{x} \cdot \left(z + \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{y}\right), \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{1}{x \cdot z} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right), \left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778))))
   (if (<= t_0 -5e-14)
     (fma
      y
      (* (/ z x) (+ z (/ (fma z 0.0007936500793651 -0.0027777777777778) y)))
      (/ 0.083333333333333 x))
     (if (<= t_0 5e-15)
       (+
        (/ 0.083333333333333 x)
        (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))
       (*
        z
        (fma
         z
         (* (/ 1.0 (* x z)) (- (/ 0.083333333333333 z) 0.0027777777777778))
         (* (+ 0.0007936500793651 y) (/ z x))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -5e-14) {
		tmp = fma(y, ((z / x) * (z + (fma(z, 0.0007936500793651, -0.0027777777777778) / y))), (0.083333333333333 / x));
	} else if (t_0 <= 5e-15) {
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	} else {
		tmp = z * fma(z, ((1.0 / (x * z)) * ((0.083333333333333 / z) - 0.0027777777777778)), ((0.0007936500793651 + y) * (z / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -5e-14)
		tmp = fma(y, Float64(Float64(z / x) * Float64(z + Float64(fma(z, 0.0007936500793651, -0.0027777777777778) / y))), Float64(0.083333333333333 / x));
	elseif (t_0 <= 5e-15)
		tmp = Float64(Float64(0.083333333333333 / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)));
	else
		tmp = Float64(z * fma(z, Float64(Float64(1.0 / Float64(x * z)) * Float64(Float64(0.083333333333333 / z) - 0.0027777777777778)), Float64(Float64(0.0007936500793651 + y) * Float64(z / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-14], N[(y * N[(N[(z / x), $MachinePrecision] * N[(z + N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-15], N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision] * N[(N[(0.083333333333333 / z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -5.0000000000000002e-14

   1. Initial program 91.3%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-sub N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 88.6

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

   8. Simplified 88.6%

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

   if -5.0000000000000002e-14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4.99999999999999999e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 99.5%

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

   if 4.99999999999999999e-15 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 87.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. Simplified 82.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Simplified 74.4%

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

   9. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(-1 \cdot \frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x} - \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot z}}{z} + \frac{y \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 N/A

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

   11. Simplified 75.6%

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

   12. Taylor expanded in z around -inf

       \[\leadsto \color{blue}{{z}^{2} \cdot \left(-1 \cdot \frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x} - \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot z}}{z} + \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. lower-fma.f64 N/A

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

   14. Simplified 80.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{x} \cdot \left(z + \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{y}\right), \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{1}{x \cdot z} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right), \left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{x} \cdot \left(z + \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{y}\right), \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{1}{x \cdot z} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right), \left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778))))
   (if (<= t_0 -5e-14)
     (fma
      y
      (* (/ z x) (+ z (/ (fma z 0.0007936500793651 -0.0027777777777778) y)))
      (/ 0.083333333333333 x))
     (if (<= t_0 5e-15)
       (+
        0.91893853320467
        (fma (log x) (+ x -0.5) (- (/ 0.083333333333333 x) x)))
       (*
        z
        (fma
         z
         (* (/ 1.0 (* x z)) (- (/ 0.083333333333333 z) 0.0027777777777778))
         (* (+ 0.0007936500793651 y) (/ z x))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -5e-14) {
		tmp = fma(y, ((z / x) * (z + (fma(z, 0.0007936500793651, -0.0027777777777778) / y))), (0.083333333333333 / x));
	} else if (t_0 <= 5e-15) {
		tmp = 0.91893853320467 + fma(log(x), (x + -0.5), ((0.083333333333333 / x) - x));
	} else {
		tmp = z * fma(z, ((1.0 / (x * z)) * ((0.083333333333333 / z) - 0.0027777777777778)), ((0.0007936500793651 + y) * (z / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -5e-14)
		tmp = fma(y, Float64(Float64(z / x) * Float64(z + Float64(fma(z, 0.0007936500793651, -0.0027777777777778) / y))), Float64(0.083333333333333 / x));
	elseif (t_0 <= 5e-15)
		tmp = Float64(0.91893853320467 + fma(log(x), Float64(x + -0.5), Float64(Float64(0.083333333333333 / x) - x)));
	else
		tmp = Float64(z * fma(z, Float64(Float64(1.0 / Float64(x * z)) * Float64(Float64(0.083333333333333 / z) - 0.0027777777777778)), Float64(Float64(0.0007936500793651 + y) * Float64(z / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-14], N[(y * N[(N[(z / x), $MachinePrecision] * N[(z + N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-15], 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[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision] * N[(N[(0.083333333333333 / z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -5.0000000000000002e-14

   1. Initial program 91.3%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-sub N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 88.6

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

   8. Simplified 88.6%

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

   if -5.0000000000000002e-14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4.99999999999999999e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   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 99.5

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

   5. Simplified 99.5%

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

   if 4.99999999999999999e-15 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 87.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. Simplified 82.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Simplified 74.4%

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

   9. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(-1 \cdot \frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x} - \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot z}}{z} + \frac{y \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 N/A

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

   11. Simplified 75.6%

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

   12. Taylor expanded in z around -inf

       \[\leadsto \color{blue}{{z}^{2} \cdot \left(-1 \cdot \frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x} - \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot z}}{z} + \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. lower-fma.f64 N/A

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

   14. Simplified 80.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{x} \cdot \left(z + \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{y}\right), \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{-15}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{1}{x \cdot z} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right), \left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{x} \cdot \left(z + \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{y}\right), \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right) - x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{1}{x \cdot z} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right), \left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778))))
   (if (<= t_0 -5e-14)
     (fma
      y
      (* (/ z x) (+ z (/ (fma z 0.0007936500793651 -0.0027777777777778) y)))
      (/ 0.083333333333333 x))
     (if (<= t_0 5e-15)
       (- (fma (log x) (+ x -0.5) (/ 0.083333333333333 x)) x)
       (*
        z
        (fma
         z
         (* (/ 1.0 (* x z)) (- (/ 0.083333333333333 z) 0.0027777777777778))
         (* (+ 0.0007936500793651 y) (/ z x))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -5e-14) {
		tmp = fma(y, ((z / x) * (z + (fma(z, 0.0007936500793651, -0.0027777777777778) / y))), (0.083333333333333 / x));
	} else if (t_0 <= 5e-15) {
		tmp = fma(log(x), (x + -0.5), (0.083333333333333 / x)) - x;
	} else {
		tmp = z * fma(z, ((1.0 / (x * z)) * ((0.083333333333333 / z) - 0.0027777777777778)), ((0.0007936500793651 + y) * (z / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -5e-14)
		tmp = fma(y, Float64(Float64(z / x) * Float64(z + Float64(fma(z, 0.0007936500793651, -0.0027777777777778) / y))), Float64(0.083333333333333 / x));
	elseif (t_0 <= 5e-15)
		tmp = Float64(fma(log(x), Float64(x + -0.5), Float64(0.083333333333333 / x)) - x);
	else
		tmp = Float64(z * fma(z, Float64(Float64(1.0 / Float64(x * z)) * Float64(Float64(0.083333333333333 / z) - 0.0027777777777778)), Float64(Float64(0.0007936500793651 + y) * Float64(z / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-14], N[(y * N[(N[(z / x), $MachinePrecision] * N[(z + N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-15], N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(z * N[(z * N[(N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision] * N[(N[(0.083333333333333 / z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -5.0000000000000002e-14

   1. Initial program 91.3%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-sub N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 88.6

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

   8. Simplified 88.6%

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

   if -5.0000000000000002e-14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4.99999999999999999e-15

   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. Simplified 93.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 99.5

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

   8. Simplified 99.5%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 96.3

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

   11. Simplified 96.3%

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

   if 4.99999999999999999e-15 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 87.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. Simplified 82.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Simplified 74.4%

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

   9. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(-1 \cdot \frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x} - \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot z}}{z} + \frac{y \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 N/A

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

   11. Simplified 75.6%

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

   12. Taylor expanded in z around -inf

       \[\leadsto \color{blue}{{z}^{2} \cdot \left(-1 \cdot \frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x} - \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot z}}{z} + \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. lower-fma.f64 N/A

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

   14. Simplified 80.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{x} \cdot \left(z + \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{y}\right), \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right) - x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{1}{x \cdot z} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right), \left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.19:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(-0.5, \log x, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right) + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.19)
   (+
    0.91893853320467
    (fma
     -0.5
     (log x)
     (fma
      (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
      (/ z x)
      (/ 0.083333333333333 x))))
   (+
    (* z (* z (/ (+ 0.0007936500793651 y) x)))
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.19) {
		tmp = 0.91893853320467 + fma(-0.5, log(x), fma(fma(z, (0.0007936500793651 + y), -0.0027777777777778), (z / x), (0.083333333333333 / x)));
	} else {
		tmp = (z * (z * ((0.0007936500793651 + y) / x))) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.19)
		tmp = Float64(0.91893853320467 + fma(-0.5, log(x), fma(fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x))));
	else
		tmp = Float64(Float64(z * Float64(z * Float64(Float64(0.0007936500793651 + y) / x))) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.19], N[(0.91893853320467 + N[(-0.5 * N[Log[x], $MachinePrecision] + N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.19

   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

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. log-rec N/A

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

      3. distribute-rgt-neg-out N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   8. Simplified 97.9%

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

   if 0.19 < x

   1. Initial program 85.4%

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

   3. Taylor expanded in z around 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. Simplified 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.19:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(-0.5, \log x, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right) + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.205:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right) + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.205)
   (/
    (fma
     z
     (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
     (fma x (fma -0.5 (log x) 0.91893853320467) 0.083333333333333))
    x)
   (+
    (* z (* z (/ (+ 0.0007936500793651 y) x)))
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.205) {
		tmp = fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), fma(x, fma(-0.5, log(x), 0.91893853320467), 0.083333333333333)) / x;
	} else {
		tmp = (z * (z * ((0.0007936500793651 + y) / x))) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.205)
		tmp = Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), fma(x, fma(-0.5, log(x), 0.91893853320467), 0.083333333333333)) / x);
	else
		tmp = Float64(Float64(z * Float64(z * Float64(Float64(0.0007936500793651 + y) / x))) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.205], N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + N[(x * N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.204999999999999988

   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

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 97.9

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

   5. Simplified 97.9%

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

   if 0.204999999999999988 < x

   1. Initial program 85.4%

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

   3. Taylor expanded in z around 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. Simplified 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.205:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right) + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   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

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 97.9

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

   5. Simplified 97.9%

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

   if 1 < x

   1. Initial program 85.4%

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

   3. Taylor expanded in z around 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. Simplified 99.6%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. neg-mul-1 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. log-rec N/A

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

      11. remove-double-neg N/A

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

      12. lower-log.f64 99.2

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

   8. Simplified 99.2%

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

Alternative 9: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   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 97.2

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

   5. Simplified 97.2%

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

   if 1 < x

   1. Initial program 85.4%

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

   3. Taylor expanded in z around 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. Simplified 99.6%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. neg-mul-1 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. log-rec N/A

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

      11. remove-double-neg N/A

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

      12. lower-log.f64 99.2

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

   8. Simplified 99.2%

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

Alternative 10: 84.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.8e+65)
   (/
    (fma
     z
     (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
     0.083333333333333)
    x)
   (fma x (log x) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.8e+65) {
		tmp = fma(z, fma(z, (0.0007936500793651 + y), -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 <= 6.8e+65)
		tmp = Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -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, 6.8e+65], N[(N[(z * N[(z * N[(0.0007936500793651 + y), $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 < 6.7999999999999999e65

   1. Initial program 98.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 93.1

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

   5. Simplified 93.1%

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

   if 6.7999999999999999e65 < x

   1. Initial program 84.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. Simplified 88.4%

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

   6. Taylor expanded in 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. 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. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-neg.f64 72.8

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

   8. Simplified 72.8%

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

Alternative 11: 84.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.8e+65)
   (/
    (fma
     z
     (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
     0.083333333333333)
    x)
   (- (* x (log x)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.8e+65) {
		tmp = fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = (x * log(x)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.8e+65)
		tmp = Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -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, 6.8e+65], N[(N[(z * N[(z * N[(0.0007936500793651 + y), $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 < 6.7999999999999999e65

   1. Initial program 98.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 93.1

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

   5. Simplified 93.1%

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

   if 6.7999999999999999e65 < x

   1. Initial program 84.3%

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

   3. Taylor expanded in 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 72.8

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

   5. Simplified 72.8%

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

Alternative 12: 64.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ t_1 := 0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z (/ (+ 0.0007936500793651 y) x))))
        (t_1
         (+
          0.083333333333333
          (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))))
   (if (<= t_1 -2e+75)
     t_0
     (if (<= t_1 5e+27)
       (/
        (fma
         z
         (fma z 0.0007936500793651 -0.0027777777777778)
         0.083333333333333)
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * ((0.0007936500793651 + y) / x));
	double t_1 = 0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778));
	double tmp;
	if (t_1 <= -2e+75) {
		tmp = t_0;
	} else if (t_1 <= 5e+27) {
		tmp = fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * Float64(Float64(0.0007936500793651 + y) / x)))
	t_1 = Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778)))
	tmp = 0.0
	if (t_1 <= -2e+75)
		tmp = t_0;
	elseif (t_1 <= 5e+27)
		tmp = Float64(fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.083333333333333 + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+75], t$95$0, If[LessEqual[t$95$1, 5e+27], N[(N[(z * N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]]

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)) < -1.99999999999999985e75 or 4.99999999999999979e27 < (+.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 88.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 81.5

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

   8. Simplified 81.5%

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

   if -1.99999999999999985e75 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.99999999999999979e27

   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. Simplified 93.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Simplified 57.6%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 57.6

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

   11. Simplified 57.6%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+75}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \mathbf{elif}\;0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+75}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -0.0027777777777778, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot 0.0007936500793651}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          0.083333333333333
          (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))))
   (if (<= t_0 -2e+75)
     (* z (* y (/ z x)))
     (if (<= t_0 0.1)
       (/ (fma z -0.0027777777777778 0.083333333333333) x)
       (* z (/ (* z 0.0007936500793651) x))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778));
	double tmp;
	if (t_0 <= -2e+75) {
		tmp = z * (y * (z / x));
	} else if (t_0 <= 0.1) {
		tmp = fma(z, -0.0027777777777778, 0.083333333333333) / x;
	} else {
		tmp = z * ((z * 0.0007936500793651) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778)))
	tmp = 0.0
	if (t_0 <= -2e+75)
		tmp = Float64(z * Float64(y * Float64(z / x)));
	elseif (t_0 <= 0.1)
		tmp = Float64(fma(z, -0.0027777777777778, 0.083333333333333) / x);
	else
		tmp = Float64(z * Float64(Float64(z * 0.0007936500793651) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.083333333333333 + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+75], N[(z * N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(N[(z * -0.0027777777777778 + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(N[(z * 0.0007936500793651), $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)) < -1.99999999999999985e75

   1. Initial program 91.1%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   8. Simplified 86.2%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 86.2

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

   11. Simplified 86.2%

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

   if -1.99999999999999985e75 < (+.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. Simplified 93.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Simplified 56.5%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}}{x} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{z \cdot \frac{-13888888888889}{5000000000000000}} + \frac{83333333333333}{1000000000000000}}{x} \]

       3. lower-fma.f64 56.5

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

   11. Simplified 56.5%

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

   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 87.3%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 76.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   8. Simplified 62.1%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r/ N/A

          \[\leadsto z \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x}} \]

       2. lower-/.f64 N/A

          \[\leadsto z \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x}} \]

       3. *-commutative N/A

          \[\leadsto z \cdot \frac{\color{blue}{z \cdot \frac{7936500793651}{10000000000000000}}}{x} \]

       4. lower-*.f64 59.7

          \[\leadsto z \cdot \frac{\color{blue}{z \cdot 0.0007936500793651}}{x} \]

   11. Simplified 59.7%

       \[\leadsto z \cdot \color{blue}{\frac{z \cdot 0.0007936500793651}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+75}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -0.0027777777777778, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot 0.0007936500793651}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot \frac{z}{x}\right)\\ t_1 := 0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -0.0027777777777778, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y (/ z x))))
        (t_1
         (+
          0.083333333333333
          (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))))
   (if (<= t_1 -2e+75)
     t_0
     (if (<= t_1 5e+27)
       (/ (fma z -0.0027777777777778 0.083333333333333) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * (z / x));
	double t_1 = 0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778));
	double tmp;
	if (t_1 <= -2e+75) {
		tmp = t_0;
	} else if (t_1 <= 5e+27) {
		tmp = fma(z, -0.0027777777777778, 0.083333333333333) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y * Float64(z / x)))
	t_1 = Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778)))
	tmp = 0.0
	if (t_1 <= -2e+75)
		tmp = t_0;
	elseif (t_1 <= 5e+27)
		tmp = Float64(fma(z, -0.0027777777777778, 0.083333333333333) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.083333333333333 + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+75], t$95$0, If[LessEqual[t$95$1, 5e+27], N[(N[(z * -0.0027777777777778 + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]]

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)) < -1.99999999999999985e75 or 4.99999999999999979e27 < (+.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 88.2%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 84.0%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   8. Simplified 70.5%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 56.8

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

   11. Simplified 56.8%

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

   if -1.99999999999999985e75 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.99999999999999979e27

   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. Simplified 93.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Simplified 57.6%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}}{x} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{z \cdot \frac{-13888888888889}{5000000000000000}} + \frac{83333333333333}{1000000000000000}}{x} \]

       3. lower-fma.f64 55.4

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

   11. Simplified 55.4%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+75}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -0.0027777777777778, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{1}{x \cdot z} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right), \left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.35e+44)
   (/
    (fma
     z
     (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
     0.083333333333333)
    x)
   (*
    z
    (fma
     z
     (* (/ 1.0 (* x z)) (- (/ 0.083333333333333 z) 0.0027777777777778))
     (* (+ 0.0007936500793651 y) (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.35e+44) {
		tmp = fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = z * fma(z, ((1.0 / (x * z)) * ((0.083333333333333 / z) - 0.0027777777777778)), ((0.0007936500793651 + y) * (z / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.35e+44)
		tmp = Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(z * fma(z, Float64(Float64(1.0 / Float64(x * z)) * Float64(Float64(0.083333333333333 / z) - 0.0027777777777778)), Float64(Float64(0.0007936500793651 + y) * Float64(z / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.35e+44], N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(z * N[(N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision] * N[(N[(0.083333333333333 / z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e44

   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 94.6

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

   5. Simplified 94.6%

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

   if 1.35e44 < x

   1. Initial program 83.4%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 88.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Simplified 25.2%

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

   9. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(-1 \cdot \frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x} - \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot z}}{z} + \frac{y \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 N/A

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

   11. Simplified 26.5%

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

   12. Taylor expanded in z around -inf

       \[\leadsto \color{blue}{{z}^{2} \cdot \left(-1 \cdot \frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x} - \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot z}}{z} + \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. lower-fma.f64 N/A

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

   14. Simplified 32.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{1}{x \cdot z} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right), \left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5e+44)
   (/
    (fma
     z
     (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
     0.083333333333333)
    x)
   (* z (* z (/ (+ 0.0007936500793651 y) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5e+44) {
		tmp = fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = z * (z * ((0.0007936500793651 + y) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5e+44)
		tmp = Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(z * Float64(z * Float64(Float64(0.0007936500793651 + y) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5e+44], N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999996e44

   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 94.6

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

   5. Simplified 94.6%

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

   if 4.9999999999999996e44 < x

   1. Initial program 83.4%

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

   3. Taylor expanded in z around 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 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 32.1

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

   8. Simplified 32.1%

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

Alternative 17: 29.1% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(z * -0.0027777777777778 + 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. Simplified 90.6%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

8. Simplified 67.8%

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

9. Taylor expanded in z around 0

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

    1. +-commutative N/A

       \[\leadsto \frac{\color{blue}{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}}{x} \]

    2. *-commutative N/A

       \[\leadsto \frac{\color{blue}{z \cdot \frac{-13888888888889}{5000000000000000}} + \frac{83333333333333}{1000000000000000}}{x} \]

    3. lower-fma.f64 31.9

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

11. Simplified 31.9%

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

Alternative 18: 23.5% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Simplified 90.6%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

8. Simplified 67.8%

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

9. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
10. Step-by-step derivation

    1. lower-/.f64 27.8

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

11. Simplified 27.8%

    \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
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 2024208 
(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)))