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

Percentage Accurate: 93.7% → 99.6%

Time: 19.2s

Alternatives: 22

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x\right) \cdot \left(x + 0.91893853320467\right) + \left(x + 0.5\right) \cdot \left(0.8444480278083504 - x \cdot x\right)}{\left(x + 0.91893853320467\right) \cdot \left(x + 0.5\right)} + \frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8e+15)
   (+
    (/
     (+
      (* (* (fma x x -0.25) (log x)) (+ x 0.91893853320467))
      (* (+ x 0.5) (- 0.8444480278083504 (* x x))))
     (* (+ x 0.91893853320467) (+ x 0.5)))
    (/
     (+
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
      0.083333333333333)
     x))
   (+ (fma x (log x) (- x)) (* z (* z (/ (+ y 0.0007936500793651) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8e+15) {
		tmp = ((((fma(x, x, -0.25) * log(x)) * (x + 0.91893853320467)) + ((x + 0.5) * (0.8444480278083504 - (x * x)))) / ((x + 0.91893853320467) * (x + 0.5))) + (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x);
	} else {
		tmp = fma(x, log(x), -x) + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8e+15)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(x, x, -0.25) * log(x)) * Float64(x + 0.91893853320467)) + Float64(Float64(x + 0.5) * Float64(0.8444480278083504 - Float64(x * x)))) / Float64(Float64(x + 0.91893853320467) * Float64(x + 0.5))) + Float64(Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x));
	else
		tmp = Float64(fma(x, log(x), Float64(-x)) + Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8e+15], N[(N[(N[(N[(N[(N[(x * x + -0.25), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] * N[(x + 0.91893853320467), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 0.5), $MachinePrecision] * N[(0.8444480278083504 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 0.91893853320467), $MachinePrecision] * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e15

   1. Initial program 99.7%

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

      1. associate-+l- N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. flip-- N/A

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

      5. frac-sub N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x\right) \cdot \left(x + 0.91893853320467\right) - \left(x + 0.5\right) \cdot \left(x \cdot x - 0.8444480278083504\right)}{\left(x + 0.5\right) \cdot \left(x + 0.91893853320467\right)}} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 8e15 < x

   1. Initial program 86.0%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 86.1

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 86.1

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

   7. Simplified 86.1%

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

   8. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x, \log x, \mathsf{neg}\left(x\right)\right) + \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-lowering-+.f64 99.7

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

   10. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x\right) \cdot \left(x + 0.91893853320467\right) + \left(x + 0.5\right) \cdot \left(0.8444480278083504 - x \cdot x\right)}{\left(x + 0.91893853320467\right) \cdot \left(x + 0.5\right)} + \frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+124}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{-0.0027777777777778}{x \cdot z}, \left(y + 0.0007936500793651\right) \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
            0.083333333333333)
           x)
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))
   (if (<= t_0 -5e+124)
     (*
      z
      (fma
       z
       (/ -0.0027777777777778 (* x z))
       (* (+ y 0.0007936500793651) (/ z x))))
     (if (<= t_0 4e+306)
       (+
        (fma (+ x -0.5) (log x) (- 0.91893853320467 x))
        (/ 0.083333333333333 x))
       (fma
        (fma z (+ y 0.0007936500793651) -0.0027777777777778)
        (/ z x)
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	double tmp;
	if (t_0 <= -5e+124) {
		tmp = z * fma(z, (-0.0027777777777778 / (x * z)), ((y + 0.0007936500793651) * (z / x)));
	} else if (t_0 <= 4e+306) {
		tmp = fma((x + -0.5), log(x), (0.91893853320467 - x)) + (0.083333333333333 / x);
	} else {
		tmp = fma(fma(z, (y + 0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)))
	tmp = 0.0
	if (t_0 <= -5e+124)
		tmp = Float64(z * fma(z, Float64(-0.0027777777777778 / Float64(x * z)), Float64(Float64(y + 0.0007936500793651) * Float64(z / x))));
	elseif (t_0 <= 4e+306)
		tmp = Float64(fma(Float64(x + -0.5), log(x), Float64(0.91893853320467 - x)) + Float64(0.083333333333333 / x));
	else
		tmp = fma(fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+124], N[(z * N[(z * N[(-0.0027777777777778 / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+306], N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.9999999999999996e124

   1. Initial program 88.8%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 88.8

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

   4. Applied egg-rr 88.8%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 88.8

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

   7. Simplified 88.8%

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

   8. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. distribute-rgt-in N/A

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

   10. Simplified 96.8%

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

   if -4.9999999999999996e124 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4.00000000000000007e306

   1. Initial program 99.4%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 92.2

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

   7. Simplified 92.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 92.2

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

   9. Applied egg-rr 92.2%

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

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

   1. Initial program 81.7%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 81.7%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 78.0

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

   8. Simplified 78.0%

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

   9. Taylor expanded in z around 0

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

   11. Simplified 85.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+124}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{-0.0027777777777778}{x \cdot z}, \left(y + 0.0007936500793651\right) \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+124}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{-0.0027777777777778}{x \cdot z}, \left(y + 0.0007936500793651\right) \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\left(0.91893853320467 - x\right) + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
            0.083333333333333)
           x)
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))
   (if (<= t_0 -5e+124)
     (*
      z
      (fma
       z
       (/ -0.0027777777777778 (* x z))
       (* (+ y 0.0007936500793651) (/ z x))))
     (if (<= t_0 4e+306)
       (+
        (- 0.91893853320467 x)
        (fma (log x) (+ x -0.5) (/ 0.083333333333333 x)))
       (fma
        (fma z (+ y 0.0007936500793651) -0.0027777777777778)
        (/ z x)
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	double tmp;
	if (t_0 <= -5e+124) {
		tmp = z * fma(z, (-0.0027777777777778 / (x * z)), ((y + 0.0007936500793651) * (z / x)));
	} else if (t_0 <= 4e+306) {
		tmp = (0.91893853320467 - x) + fma(log(x), (x + -0.5), (0.083333333333333 / x));
	} else {
		tmp = fma(fma(z, (y + 0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)))
	tmp = 0.0
	if (t_0 <= -5e+124)
		tmp = Float64(z * fma(z, Float64(-0.0027777777777778 / Float64(x * z)), Float64(Float64(y + 0.0007936500793651) * Float64(z / x))));
	elseif (t_0 <= 4e+306)
		tmp = Float64(Float64(0.91893853320467 - x) + fma(log(x), Float64(x + -0.5), Float64(0.083333333333333 / x)));
	else
		tmp = fma(fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+124], N[(z * N[(z * N[(-0.0027777777777778 / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+306], N[(N[(0.91893853320467 - x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.9999999999999996e124

   1. Initial program 88.8%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 88.8

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

   4. Applied egg-rr 88.8%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 88.8

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

   7. Simplified 88.8%

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

   8. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. distribute-rgt-in N/A

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

   10. Simplified 96.8%

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

   if -4.9999999999999996e124 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4.00000000000000007e306

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. +-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. +-lowering-+.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. accelerator-lowering-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. log-lowering-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. +-lowering-+.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. /-lowering-/.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. --lowering--.f64 92.0

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

   5. Simplified 92.0%

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

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

   1. Initial program 81.7%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 81.7%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 78.0

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

   8. Simplified 78.0%

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

   9. Taylor expanded in z around 0

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

   11. Simplified 85.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+124}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(z, \frac{-0.0027777777777778}{x \cdot z}, \left(y + 0.0007936500793651\right) \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\left(0.91893853320467 - x\right) + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (/
        (+
         (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
         0.083333333333333)
        x)
       (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))
      -5e+124)
   (* y (* z (/ 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 (((((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x))) <= -5e+124) {
		tmp = y * (z * (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(Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x))) <= -5e+124)
		tmp = Float64(y * Float64(z * 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[(N[(N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+124], N[(y * N[(z * 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)) < -4.9999999999999996e124

   1. Initial program 88.8%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 95.6

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

   5. Simplified 95.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 95.6

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

   7. Applied egg-rr 95.6%

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

   if -4.9999999999999996e124 < (+.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 93.0%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 91.3%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 53.6

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

   8. Simplified 53.6%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. accelerator-lowering-fma.f64 51.0

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

   11. Simplified 51.0%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+14}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{1}{\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.9e+14)
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (/
     1.0
     (/
      x
      (fma
       z
       (fma (+ y 0.0007936500793651) z -0.0027777777777778)
       0.083333333333333))))
   (+ (fma x (log x) (- x)) (* z (* z (/ (+ y 0.0007936500793651) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.9e+14) {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (1.0 / (x / fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333)));
	} else {
		tmp = fma(x, log(x), -x) + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.9e+14)
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(1.0 / Float64(x / fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333))));
	else
		tmp = Float64(fma(x, log(x), Float64(-x)) + Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.9e+14], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x / N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9e14

   1. Initial program 99.7%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval 99.7

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

   4. Applied egg-rr 99.7%

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

   if 2.9e14 < x

   1. Initial program 86.0%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 86.1

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 86.1

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

   7. Simplified 86.1%

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

   8. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x, \log x, \mathsf{neg}\left(x\right)\right) + \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-lowering-+.f64 99.7

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

   10. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+14}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{1}{\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} + \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.5e+15)
   (+
    (/
     (fma
      z
      (fma (+ y 0.0007936500793651) z -0.0027777777777778)
      0.083333333333333)
     x)
    (fma (+ x -0.5) (log x) (- 0.91893853320467 x)))
   (+ (fma x (log x) (- x)) (* z (* z (/ (+ y 0.0007936500793651) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.5e+15) {
		tmp = (fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x) + fma((x + -0.5), log(x), (0.91893853320467 - x));
	} else {
		tmp = fma(x, log(x), -x) + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.5e+15)
		tmp = Float64(Float64(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x) + fma(Float64(x + -0.5), log(x), Float64(0.91893853320467 - x)));
	else
		tmp = Float64(fma(x, log(x), Float64(-x)) + Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.5e+15], N[(N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5e15

   1. Initial program 99.7%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 99.7

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

   4. Applied egg-rr 99.7%

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

   if 4.5e15 < x

   1. Initial program 86.0%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 86.1

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 86.1

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

   7. Simplified 86.1%

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

   8. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x, \log x, \mathsf{neg}\left(x\right)\right) + \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-lowering-+.f64 99.7

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

   10. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} + \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.7)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (+ (fma x (log x) (- x)) (* z (* z (/ (+ y 0.0007936500793651) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.7) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma(x, log(x), -x) + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(fma(x, log(x), Float64(-x)) + Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.7], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. +-lowering-+.f64 97.9

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

   5. Simplified 97.9%

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

   if 2.7000000000000002 < x

   1. Initial program 86.2%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 86.4

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

   4. Applied egg-rr 86.4%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 86.2

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

   7. Simplified 86.2%

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

   8. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x, \log x, \mathsf{neg}\left(x\right)\right) + \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-lowering-+.f64 99.5

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

   10. Simplified 99.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 360000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \frac{z}{x}, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467 - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 360000000.0)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (fma (* y z) (/ z x) (fma (log x) (+ x -0.5) (- 0.91893853320467 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 360000000.0) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma((y * z), (z / x), fma(log(x), (x + -0.5), (0.91893853320467 - x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 360000000.0)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = fma(Float64(y * z), Float64(z / x), fma(log(x), Float64(x + -0.5), Float64(0.91893853320467 - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 360000000.0], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6e8

   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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. +-lowering-+.f64 97.9

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

   5. Simplified 97.9%

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

   if 3.6e8 < x

   1. Initial program 86.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 87.4

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

   5. Simplified 87.4%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. log-lowering-log.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. +-lowering-+.f64 N/A

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

      20. +-commutative N/A

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

      21. unsub-neg N/A

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

      22. --lowering--.f64 89.1

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

   7. Applied egg-rr 89.1%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 360000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \frac{z}{x}, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467 - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 360000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot z}{x}, z, \mathsf{fma}\left(x, \log x, -x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 360000000.0)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (fma (/ (* y z) x) z (fma x (log x) (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 360000000.0) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma(((y * z) / x), z, fma(x, log(x), -x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 360000000.0)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = fma(Float64(Float64(y * z) / x), z, fma(x, log(x), Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 360000000.0], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision] * z + N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6e8

   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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. +-lowering-+.f64 97.9

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

   5. Simplified 97.9%

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

   if 3.6e8 < x

   1. Initial program 86.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 87.4

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

   8. Simplified 87.4%

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 88.9

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

   10. Applied egg-rr 88.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 360000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot z}{x}, z, \mathsf{fma}\left(x, \log x, -x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3e+42)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (fma x (log x) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e+42) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma(x, log(x), -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3e+42)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = fma(x, log(x), Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3e+42], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.00000000000000029e42

   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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. +-lowering-+.f64 96.5

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

   5. Simplified 96.5%

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

   if 3.00000000000000029e42 < x

   1. Initial program 85.5%

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

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

      7. neg-mul-1 N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 73.9

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

   5. Simplified 73.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.1e+42)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (- (* x (log x)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.1e+42) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = (x * log(x)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.1e+42)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(Float64(x * log(x)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.1e+42], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e42

   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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. +-lowering-+.f64 96.5

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

   5. Simplified 96.5%

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

   if 1.1000000000000001e42 < x

   1. Initial program 85.5%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 85.7

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

   4. Applied egg-rr 85.7%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. sub-neg N/A

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

      6. div-inv N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 85.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   8. 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. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. log-lowering-log.f64 73.7

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

   9. Simplified 73.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+126}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))
   (if (<= t_0 -2e+126)
     (* y (* z (/ z x)))
     (if (<= t_0 5e-6) (/ 1.0 (* x 12.000000000000048)) (* y (/ (* z z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -2e+126) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 5e-6) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0)
    if (t_0 <= (-2d+126)) then
        tmp = y * (z * (z / x))
    else if (t_0 <= 5d-6) then
        tmp = 1.0d0 / (x * 12.000000000000048d0)
    else
        tmp = y * ((z * z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -2e+126) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 5e-6) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)
	tmp = 0
	if t_0 <= -2e+126:
		tmp = y * (z * (z / x))
	elif t_0 <= 5e-6:
		tmp = 1.0 / (x * 12.000000000000048)
	else:
		tmp = y * ((z * z) / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -2e+126)
		tmp = Float64(y * Float64(z * Float64(z / x)));
	elseif (t_0 <= 5e-6)
		tmp = Float64(1.0 / Float64(x * 12.000000000000048));
	else
		tmp = Float64(y * Float64(Float64(z * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	tmp = 0.0;
	if (t_0 <= -2e+126)
		tmp = y * (z * (z / x));
	elseif (t_0 <= 5e-6)
		tmp = 1.0 / (x * 12.000000000000048);
	else
		tmp = y * ((z * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+126], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-6], N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] / x), $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) < -1.99999999999999985e126

   1. Initial program 89.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 92.9

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

   5. Simplified 92.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 92.9

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

   7. Applied egg-rr 92.9%

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

   if -1.99999999999999985e126 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.00000000000000041e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 42.1

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

   8. Simplified 42.1%

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

   9. Taylor expanded in z around 0

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

       1. /-lowering-/.f64 42.2

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

   11. Simplified 42.2%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       3. div-inv N/A

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

       5. metadata-eval 42.4

          \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

   13. Applied egg-rr 42.4%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]

   if 5.00000000000000041e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 46.0

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

   5. Simplified 46.0%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+126}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ t_1 := y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
        (t_1 (* y (* z (/ z x)))))
   (if (<= t_0 -2e+126)
     t_1
     (if (<= t_0 5e-6) (/ 1.0 (* x 12.000000000000048)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double t_1 = y * (z * (z / x));
	double tmp;
	if (t_0 <= -2e+126) {
		tmp = t_1;
	} else if (t_0 <= 5e-6) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0)
    t_1 = y * (z * (z / x))
    if (t_0 <= (-2d+126)) then
        tmp = t_1
    else if (t_0 <= 5d-6) then
        tmp = 1.0d0 / (x * 12.000000000000048d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double t_1 = y * (z * (z / x));
	double tmp;
	if (t_0 <= -2e+126) {
		tmp = t_1;
	} else if (t_0 <= 5e-6) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)
	t_1 = y * (z * (z / x))
	tmp = 0
	if t_0 <= -2e+126:
		tmp = t_1
	elif t_0 <= 5e-6:
		tmp = 1.0 / (x * 12.000000000000048)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))
	t_1 = Float64(y * Float64(z * Float64(z / x)))
	tmp = 0.0
	if (t_0 <= -2e+126)
		tmp = t_1;
	elseif (t_0 <= 5e-6)
		tmp = Float64(1.0 / Float64(x * 12.000000000000048));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	t_1 = y * (z * (z / x));
	tmp = 0.0;
	if (t_0 <= -2e+126)
		tmp = t_1;
	elseif (t_0 <= 5e-6)
		tmp = 1.0 / (x * 12.000000000000048);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+126], t$95$1, If[LessEqual[t$95$0, 5e-6], N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -1.99999999999999985e126 or 5.00000000000000041e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 87.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 57.1

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

   5. Simplified 57.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 57.0

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

   7. Applied egg-rr 57.0%

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

   if -1.99999999999999985e126 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.00000000000000041e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 42.1

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

   8. Simplified 42.1%

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

   9. Taylor expanded in z around 0

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

       1. /-lowering-/.f64 42.2

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

   11. Simplified 42.2%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       3. div-inv N/A

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

       5. metadata-eval 42.4

          \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

   13. Applied egg-rr 42.4%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+126}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.7% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
       0.083333333333333)
      5.0)
   (/ (fma z (* y z) 0.083333333333333) x)
   (* (+ y 0.0007936500793651) (/ (* z z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) <= 5.0) {
		tmp = fma(z, (y * z), 0.083333333333333) / x;
	} else {
		tmp = (y + 0.0007936500793651) * ((z * z) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) <= 5.0)
		tmp = Float64(fma(z, Float64(y * z), 0.083333333333333) / x);
	else
		tmp = Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 5.0], N[(N[(z * N[(y * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 5

   1. Initial program 96.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 96.9%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 53.3

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

   8. Simplified 53.3%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{z \cdot y}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

       2. *-lowering-*.f64 53.0

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

   11. Simplified 53.0%

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

   if 5 < (+.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 86.7%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 86.8

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

   4. Applied egg-rr 86.8%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 86.8

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

   7. Simplified 86.8%

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

   8. Taylor expanded in z around inf

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

      1. distribute-rgt-in N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 66.3

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

   10. Simplified 66.3%

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

4. Final simplification 58.8%

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

Alternative 15: 52.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333 \leq 5 \cdot 10^{+158}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, y \cdot z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
       0.083333333333333)
      5e+158)
   (/ (fma z (* y z) 0.083333333333333) x)
   (* y (/ (* z z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) <= 5e+158) {
		tmp = fma(z, (y * z), 0.083333333333333) / x;
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) <= 5e+158)
		tmp = Float64(fma(z, Float64(y * z), 0.083333333333333) / x);
	else
		tmp = Float64(y * Float64(Float64(z * z) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 5e+158], N[(N[(z * N[(y * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.9999999999999996e158

   1. Initial program 97.3%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 96.2%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 49.9

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

   8. Simplified 49.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{z \cdot y}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

       2. *-lowering-*.f64 46.9

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

   11. Simplified 46.9%

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

   if 4.9999999999999996e158 < (+.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 82.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 56.5

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

   5. Simplified 56.5%

      \[\leadsto \color{blue}{y \cdot \frac{z \cdot z}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333 \leq 5 \cdot 10^{+158}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, y \cdot z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.55e+75)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (* (* z z) (+ (/ 0.0007936500793651 x) (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.55e+75) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = (z * z) * ((0.0007936500793651 / x) + (y / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.55e+75)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(Float64(z * z) * Float64(Float64(0.0007936500793651 / x) + Float64(y / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.55e+75], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5500000000000001e75

   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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. +-lowering-+.f64 89.9

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

   5. Simplified 89.9%

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

   if 1.5500000000000001e75 < x

   1. Initial program 84.2%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-lowering-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 26.6

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

   5. Simplified 26.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in y around inf

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

5. Simplified 91.0%

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

6. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. /-lowering-/.f64 58.2

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

8. Simplified 58.2%

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

9. Taylor expanded in z around 0

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

11. Simplified 61.9%

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

12. Final simplification 61.9%

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

Alternative 18: 63.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.85e+75)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (* (+ y 0.0007936500793651) (/ (* z z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.85e+75) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = (y + 0.0007936500793651) * ((z * z) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.85e+75)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.85e+75], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85000000000000005e75

   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. /-lowering-/.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. accelerator-lowering-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. accelerator-lowering-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. +-lowering-+.f64 89.9

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

   5. Simplified 89.9%

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

   if 1.85000000000000005e75 < x

   1. Initial program 84.2%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 84.4

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

   4. Applied egg-rr 84.4%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. log-lowering-log.f64 N/A

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

      11. neg-lowering-neg.f64 84.4

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

   7. Simplified 84.4%

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

   8. Taylor expanded in z around inf

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

      1. distribute-rgt-in N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 26.6

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

   10. Simplified 26.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.62 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z \cdot \left(y + 0.0007936500793651\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.62e+75)
   (/ (fma z (* z (+ y 0.0007936500793651)) 0.083333333333333) x)
   (* (+ y 0.0007936500793651) (/ (* z z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.62e+75) {
		tmp = fma(z, (z * (y + 0.0007936500793651)), 0.083333333333333) / x;
	} else {
		tmp = (y + 0.0007936500793651) * ((z * z) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.62e+75)
		tmp = Float64(fma(z, Float64(z * Float64(y + 0.0007936500793651)), 0.083333333333333) / x);
	else
		tmp = Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.62e+75], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6200000000000001e75

   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 y around inf

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 89.2

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

   8. Simplified 89.2%

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

   9. Taylor expanded in z around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. associate-*l* N/A

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

       7. lft-mult-inverse N/A

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

       8. metadata-eval N/A

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

       9. *-lft-identity N/A

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

       10. *-lowering-*.f64 N/A

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

       11. +-lowering-+.f64 89.3

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

   11. Simplified 89.3%

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

   if 1.6200000000000001e75 < x

   1. Initial program 84.2%

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

      1. +-lowering-+.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-lowering-+.f64 N/A

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

      17. metadata-eval 84.4

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

   4. Applied egg-rr 84.4%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{-1}\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right)\right) + x \cdot -1\right)} + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right)\right) + \color{blue}{-1 \cdot x}\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

      5. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \log \left(\frac{1}{x}\right), \mathsf{neg}\left(x\right)\right)} + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)}, \mathsf{neg}\left(x\right)\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. log-rec N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right), \mathsf{neg}\left(x\right)\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log x}, \mathsf{neg}\left(x\right)\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log x}, \mathsf{neg}\left(x\right)\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

      11. neg-lowering-neg.f64 84.4

          \[\leadsto \mathsf{fma}\left(x, \log x, \color{blue}{-x}\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

   7. Simplified 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log x, -x\right)} + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot {z}^{2} + \frac{y}{x} \cdot {z}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot {z}^{2} + \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} \cdot {z}^{2} + \frac{y \cdot {z}^{2}}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} \cdot {z}^{2} + \frac{y \cdot {z}^{2}}{x} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot {z}^{2}}{x}} + \frac{y \cdot {z}^{2}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x}} + \frac{y \cdot {z}^{2}}{x} \]

      7. associate-/l* N/A

         \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x} + \color{blue}{y \cdot \frac{{z}^{2}}{x}} \]

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\frac{{z}^{2}}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{z}^{2}}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{z}^{2}}{x}} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\color{blue}{z \cdot z}}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{z \cdot z}}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) \]

      13. +-lowering-+.f64 26.6

          \[\leadsto \frac{z \cdot z}{x} \cdot \color{blue}{\left(0.0007936500793651 + y\right)} \]

   10. Simplified 26.6%

       \[\leadsto \color{blue}{\frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.62 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z \cdot \left(y + 0.0007936500793651\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.4% 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 92.4%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{y \cdot \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{y} + {z}^{2}\right)\right)}}{x} \]

5. Simplified 91.0%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \mathsf{fma}\left(z, \frac{0.0007936500793651}{y}, z\right), -0.0027777777777778\right), 0.083333333333333\right)}}{x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{z \cdot \left(y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

   4. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(y, z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{z}{y} + z}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{z}{y}, z\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

   9. /-lowering-/.f64 58.2

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \mathsf{fma}\left(0.0007936500793651, \color{blue}{\frac{z}{y}}, z\right), -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

8. Simplified 58.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \mathsf{fma}\left(0.0007936500793651, \frac{z}{y}, z\right), -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]

9. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]
10. Step-by-step derivation

11. Simplified 26.6%

    \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{-0.0027777777777778}, 0.083333333333333\right)}{x} \]
12. Add Preprocessing

Alternative 21: 23.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot 12.000000000000048} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 (* x 12.000000000000048)))

C

double code(double x, double y, double z) {
	return 1.0 / (x * 12.000000000000048);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / (x * 12.000000000000048d0)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / (x * 12.000000000000048);
}

Python

def code(x, y, z):
	return 1.0 / (x * 12.000000000000048)

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(x * 12.000000000000048))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / (x * 12.000000000000048);
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.4%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{y \cdot \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{y} + {z}^{2}\right)\right)}}{x} \]

5. Simplified 91.0%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \mathsf{fma}\left(z, \frac{0.0007936500793651}{y}, z\right), -0.0027777777777778\right), 0.083333333333333\right)}}{x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{z \cdot \left(y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

   4. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(y, z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{z}{y} + z}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{z}{y}, z\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

   9. /-lowering-/.f64 58.2

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \mathsf{fma}\left(0.0007936500793651, \color{blue}{\frac{z}{y}}, z\right), -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

8. Simplified 58.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \mathsf{fma}\left(0.0007936500793651, \frac{z}{y}, z\right), -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]

9. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 19.6

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

11. Simplified 19.6%

    \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
12. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

    3. div-inv N/A

       \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

    5. metadata-eval 19.7

       \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

13. Applied egg-rr 19.7%

    \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]
14. Add Preprocessing

Alternative 22: 23.8% 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 92.4%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{y \cdot \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{y} + {z}^{2}\right)\right)}}{x} \]

5. Simplified 91.0%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \mathsf{fma}\left(z, \frac{0.0007936500793651}{y}, z\right), -0.0027777777777778\right), 0.083333333333333\right)}}{x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{z \cdot \left(y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

   4. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, y \cdot \left(z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(y, z + \frac{7936500793651}{10000000000000000} \cdot \frac{z}{y}, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{z}{y} + z}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{z}{y}, z\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

   9. /-lowering-/.f64 58.2

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \mathsf{fma}\left(0.0007936500793651, \color{blue}{\frac{z}{y}}, z\right), -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

8. Simplified 58.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \mathsf{fma}\left(0.0007936500793651, \frac{z}{y}, z\right), -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]

9. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 19.6

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

11. Simplified 19.6%

    \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
12. Add Preprocessing

Developer Target 1: 98.5% 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 2024199 
(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)))