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

Percentage Accurate: 93.8% → 99.0%

Time: 17.3s

Alternatives: 20

Speedup: 1.0×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% 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: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{y}{x}, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\right), \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4e+97)
   (+
    (+ (- (/ (* (fma x x -0.25) (log x)) (+ x 0.5)) x) 0.91893853320467)
    (/
     (+
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
      0.083333333333333)
     x))
   (+
    0.91893853320467
    (fma
     (log x)
     (+ x -0.5)
     (-
      (fma
       z
       (fma z (/ y x) (/ (fma z 0.0007936500793651 -0.0027777777777778) x))
       (/ 0.083333333333333 x))
      x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e+97) {
		tmp = ((((fma(x, x, -0.25) * log(x)) / (x + 0.5)) - x) + 0.91893853320467) + (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x);
	} else {
		tmp = 0.91893853320467 + fma(log(x), (x + -0.5), (fma(z, fma(z, (y / x), (fma(z, 0.0007936500793651, -0.0027777777777778) / x)), (0.083333333333333 / x)) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4e+97)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(x, x, -0.25) * log(x)) / Float64(x + 0.5)) - x) + 0.91893853320467) + Float64(Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x));
	else
		tmp = Float64(0.91893853320467 + fma(log(x), Float64(x + -0.5), Float64(fma(z, fma(z, Float64(y / x), Float64(fma(z, 0.0007936500793651, -0.0027777777777778) / x)), Float64(0.083333333333333 / x)) - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4e+97], N[(N[(N[(N[(N[(N[(x * x + -0.25), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $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[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(N[(z * N[(z * N[(y / x), $MachinePrecision] + N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0000000000000003e97

   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. flip-- N/A

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

      2. lift-log.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{\frac{-1}{4}}\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      10. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 4.0000000000000003e97 < x

   1. Initial program 82.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{x + 0.5} - x\right) + 0.91893853320467\right) + \frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{y}{x}, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\right), \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := \frac{z \cdot \left(t\_0 - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+186}:\\ \;\;\;\;\frac{z \cdot t\_0}{x}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1
         (+
          (/ (+ (* z (- t_0 0.0027777777777778)) 0.083333333333333) x)
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))
   (if (<= t_1 -5e+186)
     (/ (* z t_0) x)
     (if (<= t_1 4e+294)
       (fma
        (+ x -0.5)
        (log x)
        (-
         (+
          0.91893853320467
          (/
           (fma
            z
            (fma z 0.0007936500793651 -0.0027777777777778)
            0.083333333333333)
           x))
         x))
       (* (* z z) (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	double tmp;
	if (t_1 <= -5e+186) {
		tmp = (z * t_0) / x;
	} else if (t_1 <= 4e+294) {
		tmp = fma((x + -0.5), log(x), ((0.91893853320467 + (fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x)) - x));
	} else {
		tmp = (z * z) * ((y / x) + (0.0007936500793651 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(Float64(Float64(Float64(z * Float64(t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)))
	tmp = 0.0
	if (t_1 <= -5e+186)
		tmp = Float64(Float64(z * t_0) / x);
	elseif (t_1 <= 4e+294)
		tmp = fma(Float64(x + -0.5), log(x), Float64(Float64(0.91893853320467 + Float64(fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x)) - x));
	else
		tmp = Float64(Float64(z * z) * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(z * N[(t$95$0 - 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$1, -5e+186], N[(N[(z * t$95$0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$1, 4e+294], N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 + N[(N[(z * N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999954e186

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if -4.99999999999999954e186 < (+.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)) < 4.00000000000000027e294

   1. Initial program 99.5%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 96.6

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

   7. Applied rewrites 96.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 96.6

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

   9. Applied rewrites 96.6%

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

   if 4.00000000000000027e294 < (+.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 82.2%

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

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

      8. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+186}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{elif}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := \frac{z \cdot \left(t\_0 - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+186}:\\ \;\;\;\;\frac{z \cdot t\_0}{x}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - \left(x - \frac{0.083333333333333}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1
         (+
          (/ (+ (* z (- t_0 0.0027777777777778)) 0.083333333333333) x)
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))
   (if (<= t_1 -5e+186)
     (/ (* z t_0) x)
     (if (<= t_1 4e+294)
       (fma
        (+ x -0.5)
        (log x)
        (- 0.91893853320467 (- x (/ 0.083333333333333 x))))
       (* (* z z) (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	double tmp;
	if (t_1 <= -5e+186) {
		tmp = (z * t_0) / x;
	} else if (t_1 <= 4e+294) {
		tmp = fma((x + -0.5), log(x), (0.91893853320467 - (x - (0.083333333333333 / x))));
	} else {
		tmp = (z * z) * ((y / x) + (0.0007936500793651 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(Float64(Float64(Float64(z * Float64(t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)))
	tmp = 0.0
	if (t_1 <= -5e+186)
		tmp = Float64(Float64(z * t_0) / x);
	elseif (t_1 <= 4e+294)
		tmp = fma(Float64(x + -0.5), log(x), Float64(0.91893853320467 - Float64(x - Float64(0.083333333333333 / x))));
	else
		tmp = Float64(Float64(z * z) * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(z * N[(t$95$0 - 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$1, -5e+186], N[(N[(z * t$95$0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$1, 4e+294], N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - N[(x - N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999954e186

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if -4.99999999999999954e186 < (+.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)) < 4.00000000000000027e294

   1. Initial program 99.5%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 86.8

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

   7. Applied rewrites 86.8%

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

   if 4.00000000000000027e294 < (+.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 82.2%

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

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

      8. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+186}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{elif}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - \left(x - \frac{0.083333333333333}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := \frac{z \cdot \left(t\_0 - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+186}:\\ \;\;\;\;\frac{z \cdot t\_0}{x}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1
         (+
          (/ (+ (* z (- t_0 0.0027777777777778)) 0.083333333333333) x)
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))
   (if (<= t_1 -5e+186)
     (/ (* z t_0) x)
     (if (<= t_1 4e+294)
       (+
        (fma (log x) (+ x -0.5) (/ 0.083333333333333 x))
        (- 0.91893853320467 x))
       (* (* z z) (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	double tmp;
	if (t_1 <= -5e+186) {
		tmp = (z * t_0) / x;
	} else if (t_1 <= 4e+294) {
		tmp = fma(log(x), (x + -0.5), (0.083333333333333 / x)) + (0.91893853320467 - x);
	} else {
		tmp = (z * z) * ((y / x) + (0.0007936500793651 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(Float64(Float64(Float64(z * Float64(t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)))
	tmp = 0.0
	if (t_1 <= -5e+186)
		tmp = Float64(Float64(z * t_0) / x);
	elseif (t_1 <= 4e+294)
		tmp = Float64(fma(log(x), Float64(x + -0.5), Float64(0.083333333333333 / x)) + Float64(0.91893853320467 - x));
	else
		tmp = Float64(Float64(z * z) * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(z * N[(t$95$0 - 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$1, -5e+186], N[(N[(z * t$95$0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$1, 4e+294], N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999954e186

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if -4.99999999999999954e186 < (+.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)) < 4.00000000000000027e294

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if 4.00000000000000027e294 < (+.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 82.2%

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

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

      8. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+186}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{elif}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := \frac{z \cdot \left(t\_0 - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+186}:\\ \;\;\;\;\frac{z \cdot t\_0}{x}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(\mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1
         (+
          (/ (+ (* z (- t_0 0.0027777777777778)) 0.083333333333333) x)
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))
   (if (<= t_1 -5e+186)
     (/ (* z t_0) x)
     (if (<= t_1 4e+294)
       (+
        (/ 0.083333333333333 x)
        (- (fma (log x) (+ x -0.5) 0.91893853320467) x))
       (* (* z z) (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	double tmp;
	if (t_1 <= -5e+186) {
		tmp = (z * t_0) / x;
	} else if (t_1 <= 4e+294) {
		tmp = (0.083333333333333 / x) + (fma(log(x), (x + -0.5), 0.91893853320467) - x);
	} else {
		tmp = (z * z) * ((y / x) + (0.0007936500793651 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(Float64(Float64(Float64(z * Float64(t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)))
	tmp = 0.0
	if (t_1 <= -5e+186)
		tmp = Float64(Float64(z * t_0) / x);
	elseif (t_1 <= 4e+294)
		tmp = Float64(Float64(0.083333333333333 / x) + Float64(fma(log(x), Float64(x + -0.5), 0.91893853320467) - x));
	else
		tmp = Float64(Float64(z * z) * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(z * N[(t$95$0 - 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$1, -5e+186], N[(N[(z * t$95$0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$1, 4e+294], N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999954e186

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if -4.99999999999999954e186 < (+.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)) < 4.00000000000000027e294

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.7%

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

   if 4.00000000000000027e294 < (+.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 82.2%

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

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

      8. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+186}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{elif}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(\mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := \frac{z \cdot \left(t\_0 - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+186}:\\ \;\;\;\;\frac{z \cdot t\_0}{x}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + \left(x \cdot \log x - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1
         (+
          (/ (+ (* z (- t_0 0.0027777777777778)) 0.083333333333333) x)
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))
   (if (<= t_1 -5e+186)
     (/ (* z t_0) x)
     (if (<= t_1 4e+294)
       (+ (/ 0.083333333333333 x) (+ 0.91893853320467 (- (* x (log x)) x)))
       (* (* z z) (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	double tmp;
	if (t_1 <= -5e+186) {
		tmp = (z * t_0) / x;
	} else if (t_1 <= 4e+294) {
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((x * log(x)) - x));
	} else {
		tmp = (z * z) * ((y / x) + (0.0007936500793651 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * (y + 0.0007936500793651d0)
    t_1 = (((z * (t_0 - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x))
    if (t_1 <= (-5d+186)) then
        tmp = (z * t_0) / x
    else if (t_1 <= 4d+294) then
        tmp = (0.083333333333333d0 / x) + (0.91893853320467d0 + ((x * log(x)) - x))
    else
        tmp = (z * z) * ((y / x) + (0.0007936500793651d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x));
	double tmp;
	if (t_1 <= -5e+186) {
		tmp = (z * t_0) / x;
	} else if (t_1 <= 4e+294) {
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((x * Math.log(x)) - x));
	} else {
		tmp = (z * z) * ((y / x) + (0.0007936500793651 / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y + 0.0007936500793651)
	t_1 = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x))
	tmp = 0
	if t_1 <= -5e+186:
		tmp = (z * t_0) / x
	elif t_1 <= 4e+294:
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((x * math.log(x)) - x))
	else:
		tmp = (z * z) * ((y / x) + (0.0007936500793651 / x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(Float64(Float64(Float64(z * Float64(t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)))
	tmp = 0.0
	if (t_1 <= -5e+186)
		tmp = Float64(Float64(z * t_0) / x);
	elseif (t_1 <= 4e+294)
		tmp = Float64(Float64(0.083333333333333 / x) + Float64(0.91893853320467 + Float64(Float64(x * log(x)) - x)));
	else
		tmp = Float64(Float64(z * z) * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y + 0.0007936500793651);
	t_1 = (((z * (t_0 - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	tmp = 0.0;
	if (t_1 <= -5e+186)
		tmp = (z * t_0) / x;
	elseif (t_1 <= 4e+294)
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((x * log(x)) - x));
	else
		tmp = (z * z) * ((y / x) + (0.0007936500793651 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(z * N[(t$95$0 - 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$1, -5e+186], N[(N[(z * t$95$0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$1, 4e+294], N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999954e186

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if -4.99999999999999954e186 < (+.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)) < 4.00000000000000027e294

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. lower-log.f64 85.0

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

   8. Applied rewrites 85.0%

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

   if 4.00000000000000027e294 < (+.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 82.2%

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

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

      8. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+186}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{elif}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + \left(x \cdot \log x - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+186}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

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+186)
   (* 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+186) {
		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+186)
		tmp = Float64(y * Float64(Float64(z * 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+186], N[(y * N[(N[(z * z), $MachinePrecision] / x), $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)) < -4.99999999999999954e186

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if -4.99999999999999954e186 < (+.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 93.8%

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 88.5

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

   7. Applied rewrites 88.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 56.5

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

   10. Applied rewrites 56.5%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) \leq -5 \cdot 10^{+186}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2850000:\\ \;\;\;\;\log x \cdot \left(x + -0.5\right) + \left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} - \left(x + -0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2850000.0)
   (+
    (* (log x) (+ x -0.5))
    (-
     (/
      (fma
       z
       (fma (+ y 0.0007936500793651) z -0.0027777777777778)
       0.083333333333333)
      x)
     (+ x -0.91893853320467)))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2850000.0) {
		tmp = (log(x) * (x + -0.5)) + ((fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x) - (x + -0.91893853320467));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * ((z * (y + 0.0007936500793651)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2850000.0)
		tmp = Float64(Float64(log(x) * Float64(x + -0.5)) + Float64(Float64(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x) - Float64(x + -0.91893853320467)));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2850000.0], N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.85e6

   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

   4. Applied rewrites 99.7%

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

   if 2.85e6 < x

   1. Initial program 88.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2850000:\\ \;\;\;\;\log x \cdot \left(x + -0.5\right) + \left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} - \left(x + -0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8e+51)
   (fma
    (+ x -0.5)
    (log x)
    (-
     (+
      0.91893853320467
      (/
       (fma
        z
        (fma (+ y 0.0007936500793651) z -0.0027777777777778)
        0.083333333333333)
       x))
     x))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8e+51) {
		tmp = fma((x + -0.5), log(x), ((0.91893853320467 + (fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x)) - x));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * ((z * (y + 0.0007936500793651)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8e+51)
		tmp = fma(Float64(x + -0.5), log(x), Float64(Float64(0.91893853320467 + Float64(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x)) - x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8e+51], N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 + N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e51

   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

   4. Applied rewrites 99.7%

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

   if 8e51 < x

   1. Initial program 85.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.0135)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0135) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * ((z * (y + 0.0007936500793651)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.0135)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.0135], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0134999999999999998

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.0134999999999999998 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.35e+46)
   (/
    (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 <= 1.35e+46) {
		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 <= 1.35e+46)
		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, 1.35e+46], 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 < 1.3500000000000001e46

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if 1.3500000000000001e46 < x

   1. Initial program 85.7%

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

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

      10. lower-log.f64 N/A

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

      11. lower-neg.f64 68.2

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

   5. Applied rewrites 68.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.35e+46)
   (/
    (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 <= 1.35e+46) {
		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 <= 1.35e+46)
		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, 1.35e+46], 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 < 1.3500000000000001e46

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if 1.3500000000000001e46 < x

   1. Initial program 85.7%

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      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. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 68.1

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

   7. Applied rewrites 68.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := \frac{z \cdot t\_0}{x}\\ t_2 := z \cdot \left(t\_0 - 0.0027777777777778\right) + 0.083333333333333\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1 (/ (* z t_0) x))
        (t_2 (+ (* z (- t_0 0.0027777777777778)) 0.083333333333333)))
   (if (<= t_2 -5e+31)
     t_1
     (if (<= t_2 2e+14)
       (/
        (fma
         z
         (fma z 0.0007936500793651 -0.0027777777777778)
         0.083333333333333)
        x)
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = (z * t_0) / x;
	double t_2 = (z * (t_0 - 0.0027777777777778)) + 0.083333333333333;
	double tmp;
	if (t_2 <= -5e+31) {
		tmp = t_1;
	} else if (t_2 <= 2e+14) {
		tmp = fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(Float64(z * t_0) / x)
	t_2 = Float64(Float64(z * Float64(t_0 - 0.0027777777777778)) + 0.083333333333333)
	tmp = 0.0
	if (t_2 <= -5e+31)
		tmp = t_1;
	elseif (t_2 <= 2e+14)
		tmp = Float64(fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * t$95$0), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * N[(t$95$0 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+31], t$95$1, If[LessEqual[t$95$2, 2e+14], N[(N[(z * N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], t$95$1]]]]]

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)) < -5.00000000000000027e31 or 2e14 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 73.6

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

   8. Applied rewrites 73.6%

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

   if -5.00000000000000027e31 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2e14

   1. Initial program 99.5%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 55.7

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

   10. Applied rewrites 55.7%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651 \cdot \left(z \cdot z\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))
   (if (<= t_0 -5e+31)
     (* y (/ (* z z) x))
     (if (<= t_0 5e-6)
       (+ 0.91893853320467 (/ 0.083333333333333 x))
       (/ (* 0.0007936500793651 (* 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 <= -5e+31) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 5e-6) {
		tmp = 0.91893853320467 + (0.083333333333333 / x);
	} else {
		tmp = (0.0007936500793651 * (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 <= (-5d+31)) then
        tmp = y * ((z * z) / x)
    else if (t_0 <= 5d-6) then
        tmp = 0.91893853320467d0 + (0.083333333333333d0 / x)
    else
        tmp = (0.0007936500793651d0 * (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 <= -5e+31) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 5e-6) {
		tmp = 0.91893853320467 + (0.083333333333333 / x);
	} else {
		tmp = (0.0007936500793651 * (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 <= -5e+31:
		tmp = y * ((z * z) / x)
	elif t_0 <= 5e-6:
		tmp = 0.91893853320467 + (0.083333333333333 / x)
	else:
		tmp = (0.0007936500793651 * (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 <= -5e+31)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 5e-6)
		tmp = Float64(0.91893853320467 + Float64(0.083333333333333 / x));
	else
		tmp = Float64(Float64(0.0007936500793651 * 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 <= -5e+31)
		tmp = y * ((z * z) / x);
	elseif (t_0 <= 5e-6)
		tmp = 0.91893853320467 + (0.083333333333333 / x);
	else
		tmp = (0.0007936500793651 * (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, -5e+31], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-6], N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.0007936500793651 * N[(z * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -5.00000000000000027e31

   1. Initial program 95.0%

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

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -5.00000000000000027e31 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.00000000000000041e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. distribute-neg-frac N/A

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

      3. distribute-neg-frac2 N/A

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

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

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 96.7

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

   10. Applied rewrites 96.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 56.2%

       \[\leadsto \color{blue}{0.91893853320467} + \frac{0.083333333333333}{x} \]

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

   1. Initial program 88.3%

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 78.2

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

   7. Applied rewrites 78.2%

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

   8. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.0

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

   10. Applied rewrites 59.0%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -5 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651 \cdot \left(z \cdot z\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+26}:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))
   (if (<= t_0 -5e+31)
     (* y (/ (* z z) x))
     (if (<= t_0 5e+26)
       (+ 0.91893853320467 (/ 0.083333333333333 x))
       (* 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 <= -5e+31) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 5e+26) {
		tmp = 0.91893853320467 + (0.083333333333333 / x);
	} 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 <= (-5d+31)) then
        tmp = y * ((z * z) / x)
    else if (t_0 <= 5d+26) then
        tmp = 0.91893853320467d0 + (0.083333333333333d0 / x)
    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 <= -5e+31) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 5e+26) {
		tmp = 0.91893853320467 + (0.083333333333333 / x);
	} 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 <= -5e+31:
		tmp = y * ((z * z) / x)
	elif t_0 <= 5e+26:
		tmp = 0.91893853320467 + (0.083333333333333 / x)
	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 <= -5e+31)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 5e+26)
		tmp = Float64(0.91893853320467 + Float64(0.083333333333333 / x));
	else
		tmp = Float64(y * Float64(z * Float64(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 <= -5e+31)
		tmp = y * ((z * z) / x);
	elseif (t_0 <= 5e+26)
		tmp = 0.91893853320467 + (0.083333333333333 / x);
	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, -5e+31], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+26], N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -5.00000000000000027e31

   1. Initial program 95.0%

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

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -5.00000000000000027e31 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.0000000000000001e26

   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

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

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 96.6%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. distribute-neg-frac N/A

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

      3. distribute-neg-frac2 N/A

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

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

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 94.3

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

   10. Applied rewrites 94.3%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 53.7%

       \[\leadsto \color{blue}{0.91893853320467} + \frac{0.083333333333333}{x} \]

   if 5.0000000000000001e26 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 87.6%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 50.0

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

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{y \cdot \frac{z \cdot z}{x}} \]
   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. lower-*.f64 N/A

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

      4. lower-/.f64 50.6

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

   7. Applied rewrites 50.6%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -5 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{+26}:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ t_1 := y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+26}:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
        (t_1 (* y (* z (/ z x)))))
   (if (<= t_0 -5e+31)
     t_1
     (if (<= t_0 5e+26) (+ 0.91893853320467 (/ 0.083333333333333 x)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double t_1 = y * (z * (z / x));
	double tmp;
	if (t_0 <= -5e+31) {
		tmp = t_1;
	} else if (t_0 <= 5e+26) {
		tmp = 0.91893853320467 + (0.083333333333333 / x);
	} 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 <= (-5d+31)) then
        tmp = t_1
    else if (t_0 <= 5d+26) then
        tmp = 0.91893853320467d0 + (0.083333333333333d0 / x)
    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 <= -5e+31) {
		tmp = t_1;
	} else if (t_0 <= 5e+26) {
		tmp = 0.91893853320467 + (0.083333333333333 / x);
	} 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 <= -5e+31:
		tmp = t_1
	elif t_0 <= 5e+26:
		tmp = 0.91893853320467 + (0.083333333333333 / x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))
	t_1 = Float64(y * Float64(z * Float64(z / x)))
	tmp = 0.0
	if (t_0 <= -5e+31)
		tmp = t_1;
	elseif (t_0 <= 5e+26)
		tmp = Float64(0.91893853320467 + Float64(0.083333333333333 / x));
	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 <= -5e+31)
		tmp = t_1;
	elseif (t_0 <= 5e+26)
		tmp = 0.91893853320467 + (0.083333333333333 / x);
	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, -5e+31], t$95$1, If[LessEqual[t$95$0, 5e+26], N[(0.91893853320467 + N[(0.083333333333333 / x), $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) < -5.00000000000000027e31 or 5.0000000000000001e26 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.7%

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

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.7

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

   5. Applied rewrites 57.7%

      \[\leadsto \color{blue}{y \cdot \frac{z \cdot z}{x}} \]
   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. lower-*.f64 N/A

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

      4. lower-/.f64 58.2

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

   7. Applied rewrites 58.2%

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

   if -5.00000000000000027e31 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.0000000000000001e26

   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

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

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 96.6%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. distribute-neg-frac N/A

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

      3. distribute-neg-frac2 N/A

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

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

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 94.3

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

   10. Applied rewrites 94.3%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 53.7%

       \[\leadsto \color{blue}{0.91893853320467} + \frac{0.083333333333333}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -5 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{+26}:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.7% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 66.3

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

5. Applied rewrites 66.3%

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

6. Final simplification 66.3%

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

Alternative 18: 23.0% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ 0.91893853320467 + \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 0.91893853320467 (/ 0.083333333333333 x)))

C

double code(double x, double y, double z) {
	return 0.91893853320467 + (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 = 0.91893853320467d0 + (0.083333333333333d0 / x)
end function

Java

public static double code(double x, double y, double z) {
	return 0.91893853320467 + (0.083333333333333 / x);
}

Python

def code(x, y, z):
	return 0.91893853320467 + (0.083333333333333 / x)

Julia

function code(x, y, z)
	return Float64(0.91893853320467 + Float64(0.083333333333333 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.91893853320467 + (0.083333333333333 / x);
end

Wolfram

code[x_, y_, z_] := N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 55.7%

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

8. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. distribute-neg-frac N/A

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

   3. distribute-neg-frac2 N/A

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

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

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

   5. log-rec N/A

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

   6. remove-double-neg N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-log.f64 54.7

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

10. Applied rewrites 54.7%

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

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]
12. Step-by-step derivation

13. Applied rewrites 25.1%

    \[\leadsto \color{blue}{0.91893853320467} + \frac{0.083333333333333}{x} \]
14. Add Preprocessing

Alternative 19: 22.4% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 55.7%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 24.5

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

8. Applied rewrites 24.5%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
9. Add Preprocessing

Alternative 20: 1.3% accurate, 49.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 55.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 23.7%

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

11. Taylor expanded in x around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 1.2

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

13. Applied rewrites 1.2%

    \[\leadsto \color{blue}{-x} \]
14. Add Preprocessing

Developer Target 1: 98.6% 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 2024216 
(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)))