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

Percentage Accurate: 93.7% → 99.6%

Time: 16.6s

Alternatives: 21

Speedup: 1.0×

• 31.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, -x\right), \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x, z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.6e+41)
   (/
    (fma
     x
     (+ 0.91893853320467 (fma (log x) (+ x -0.5) (- x)))
     (fma
      z
      (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
      0.083333333333333))
    x)
   (+
    0.91893853320467
    (fma (log x) x (- (* z (* (/ z x) (+ 0.0007936500793651 y))) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.6e+41) {
		tmp = fma(x, (0.91893853320467 + fma(log(x), (x + -0.5), -x)), fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333)) / x;
	} else {
		tmp = 0.91893853320467 + fma(log(x), x, ((z * ((z / x) * (0.0007936500793651 + y))) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.6e+41)
		tmp = Float64(fma(x, Float64(0.91893853320467 + fma(log(x), Float64(x + -0.5), Float64(-x))), fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333)) / x);
	else
		tmp = Float64(0.91893853320467 + fma(log(x), x, Float64(Float64(z * Float64(Float64(z / x) * Float64(0.0007936500793651 + y))) - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.6e+41], N[(N[(x * N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * x + N[(N[(z * N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e41

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.7%

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

   if 2.6000000000000001e41 < x

   1. Initial program 84.0%

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

   3. 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. Simplified 99.6%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. +-lowering-+.f64 99.6

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, -x\right), \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x, z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+47}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))
           x))))
   (if (<= t_0 -4e+47)
     (* y (* z (/ z x)))
     (if (<= t_0 2e+306)
       (+
        0.91893853320467
        (fma (log x) (+ x -0.5) (- (/ 0.083333333333333 x) x)))
       (+ 0.91893853320467 (* z (* (/ z x) (+ 0.0007936500793651 y))))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -4e+47) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 2e+306) {
		tmp = 0.91893853320467 + fma(log(x), (x + -0.5), ((0.083333333333333 / x) - x));
	} else {
		tmp = 0.91893853320467 + (z * ((z / x) * (0.0007936500793651 + y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_0 <= -4e+47)
		tmp = Float64(y * Float64(z * Float64(z / x)));
	elseif (t_0 <= 2e+306)
		tmp = Float64(0.91893853320467 + fma(log(x), Float64(x + -0.5), Float64(Float64(0.083333333333333 / x) - x)));
	else
		tmp = Float64(0.91893853320467 + Float64(z * Float64(Float64(z / x) * Float64(0.0007936500793651 + y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+47], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+306], N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.91893853320467 + N[(z * N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.0000000000000002e47

   1. Initial program 79.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 89.5

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

   5. Simplified 89.5%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 92.6

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

   7. Applied egg-rr 92.6%

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

   if -4.0000000000000002e47 < (+.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)) < 2.00000000000000003e306

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 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. Simplified 94.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 88.1

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

   8. Simplified 88.1%

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

   if 2.00000000000000003e306 < (+.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 82.5%

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

   3. Taylor expanded in y around 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. Simplified 99.9%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. +-lowering-+.f64 87.2

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

   8. Simplified 87.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \leq -4 \cdot 10^{+47}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

3. Taylor expanded in y around 0

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

5. Simplified 93.5%

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

6. Taylor expanded in z around 0

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

8. Simplified 99.2%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.21:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.21)
   (/
    (fma
     z
     (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
     (fma x (fma -0.5 (log x) 0.91893853320467) 0.083333333333333))
    x)
   (+
    0.91893853320467
    (fma
     (log x)
     (+ x -0.5)
     (- (* z (* (/ z x) (+ 0.0007936500793651 y))) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.21) {
		tmp = fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), fma(x, fma(-0.5, log(x), 0.91893853320467), 0.083333333333333)) / x;
	} else {
		tmp = 0.91893853320467 + fma(log(x), (x + -0.5), ((z * ((z / x) * (0.0007936500793651 + y))) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.21)
		tmp = Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), fma(x, fma(-0.5, log(x), 0.91893853320467), 0.083333333333333)) / x);
	else
		tmp = Float64(0.91893853320467 + fma(log(x), Float64(x + -0.5), Float64(Float64(z * Float64(Float64(z / x) * Float64(0.0007936500793651 + y))) - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.21], N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + N[(x * N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(N[(z * N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.209999999999999992

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. log-lowering-log.f64 98.6

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

   5. Simplified 98.6%

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

   if 0.209999999999999992 < x

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 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. Simplified 99.6%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. +-lowering-+.f64 99.5

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

   8. Simplified 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.21:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x, z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.0)
   (/
    (fma
     z
     (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
     (fma x (fma -0.5 (log x) 0.91893853320467) 0.083333333333333))
    x)
   (+
    0.91893853320467
    (fma (log x) x (- (* z (* (/ z x) (+ 0.0007936500793651 y))) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.0) {
		tmp = fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), fma(x, fma(-0.5, log(x), 0.91893853320467), 0.083333333333333)) / x;
	} else {
		tmp = 0.91893853320467 + fma(log(x), x, ((z * ((z / x) * (0.0007936500793651 + y))) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.0)
		tmp = Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), fma(x, fma(-0.5, log(x), 0.91893853320467), 0.083333333333333)) / x);
	else
		tmp = Float64(0.91893853320467 + fma(log(x), x, Float64(Float64(z * Float64(Float64(z / x) * Float64(0.0007936500793651 + y))) - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.0], N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + N[(x * N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * x + N[(N[(z * N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. log-lowering-log.f64 98.6

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

   5. Simplified 98.6%

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

   if 3 < x

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 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. Simplified 99.6%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. +-lowering-+.f64 99.5

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

   8. Simplified 99.5%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 98.7%

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

Alternative 6: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in 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. Simplified 87.2%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-lowering-+.f64 97.6

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

   8. Simplified 97.6%

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

   if 1 < x

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 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. Simplified 99.6%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. +-lowering-+.f64 99.5

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

   8. Simplified 99.5%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 98.7%

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

Alternative 7: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 650:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \frac{0.0007936500793651}{x}, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467\right)\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 650.0)
   (+
    0.91893853320467
    (/
     (fma
      z
      (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
      0.083333333333333)
     x))
   (-
    (fma
     z
     (* z (/ 0.0007936500793651 x))
     (fma (log x) (+ x -0.5) 0.91893853320467))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 650.0) {
		tmp = 0.91893853320467 + (fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = fma(z, (z * (0.0007936500793651 / x)), fma(log(x), (x + -0.5), 0.91893853320467)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 650.0)
		tmp = Float64(0.91893853320467 + Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = Float64(fma(z, Float64(z * Float64(0.0007936500793651 / x)), fma(log(x), Float64(x + -0.5), 0.91893853320467)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 650.0], N[(0.91893853320467 + N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 650

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 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. Simplified 87.2%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-lowering-+.f64 97.6

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

   8. Simplified 97.6%

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

   if 650 < x

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 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. Simplified 99.6%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. +-lowering-+.f64 99.5

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

   8. Simplified 99.5%

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

   9. Taylor expanded in y around 0

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

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

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

   11. Simplified 86.0%

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

Alternative 8: 84.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550000:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 550000.0)
   (+
    0.91893853320467
    (/
     (fma
      z
      (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
      0.083333333333333)
     x))
   (- (fma (log x) (+ x -0.5) 0.91893853320467) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 550000.0) {
		tmp = 0.91893853320467 + (fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = fma(log(x), (x + -0.5), 0.91893853320467) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 550000.0)
		tmp = Float64(0.91893853320467 + Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = Float64(fma(log(x), Float64(x + -0.5), 0.91893853320467) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 550000.0], N[(0.91893853320467 + N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e5

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 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. Simplified 87.2%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-lowering-+.f64 97.6

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

   8. Simplified 97.6%

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

   if 5.5e5 < x

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 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. Simplified 99.6%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. +-lowering-+.f64 99.5

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

   8. Simplified 99.5%

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

   9. Taylor expanded in z around 0

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

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

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

       2. +-commutative N/A

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

       3. remove-double-neg N/A

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

       4. log-rec N/A

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

       5. mul-1-neg N/A

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

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

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

       7. mul-1-neg N/A

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

       8. log-rec N/A

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

       9. remove-double-neg N/A

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

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

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

       11. sub-neg N/A

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

       12. metadata-eval N/A

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

       13. +-lowering-+.f64 74.0

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

   11. Simplified 74.0%

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

Alternative 9: 84.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{+21}:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.6e+21)
   (+
    0.91893853320467
    (/
     (fma
      z
      (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
      0.083333333333333)
     x))
   (fma x (log x) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.6e+21) {
		tmp = 0.91893853320467 + (fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = fma(x, log(x), -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.6e+21)
		tmp = Float64(0.91893853320467 + Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = fma(x, log(x), Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.6e+21], N[(0.91893853320467 + N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.6e21

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 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. Simplified 87.8%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-lowering-+.f64 95.0

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

   8. Simplified 95.0%

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

   if 6.6e21 < x

   1. Initial program 85.4%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. neg-mul-1 N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 75.1

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

   5. Simplified 75.1%

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

Alternative 10: 84.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+21}:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.8e+21)
   (+
    0.91893853320467
    (/
     (fma
      z
      (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
      0.083333333333333)
     x))
   (- (* x (log x)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.8e+21) {
		tmp = 0.91893853320467 + (fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = (x * log(x)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.8e+21)
		tmp = Float64(0.91893853320467 + Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = Float64(Float64(x * log(x)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.8e+21], N[(0.91893853320467 + N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.8e21

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 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. Simplified 87.8%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-lowering-+.f64 95.0

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

   8. Simplified 95.0%

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

   if 6.8e21 < x

   1. Initial program 85.4%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. neg-mul-1 N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 75.1

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

   5. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log x, -x\right)} \]
   6. Step-by-step derivation

      1. unsub-neg N/A

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

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

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

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

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

      4. log-lowering-log.f64 74.9

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

   7. Applied egg-rr 74.9%

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

Alternative 11: 64.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-9}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778))))
   (if (<= t_0 -1e+17)
     (* y (* z (/ z x)))
     (if (<= t_0 1e-9)
       (/ 1.0 (* x 12.000000000000048))
       (* z (* (/ z x) (+ 0.0007936500793651 y)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 1e-9) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = z * ((z / x) * (0.0007936500793651 + y));
	}
	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 * (0.0007936500793651d0 + y)) - 0.0027777777777778d0)
    if (t_0 <= (-1d+17)) then
        tmp = y * (z * (z / x))
    else if (t_0 <= 1d-9) then
        tmp = 1.0d0 / (x * 12.000000000000048d0)
    else
        tmp = z * ((z / x) * (0.0007936500793651d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 1e-9) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = z * ((z / x) * (0.0007936500793651 + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)
	tmp = 0
	if t_0 <= -1e+17:
		tmp = y * (z * (z / x))
	elif t_0 <= 1e-9:
		tmp = 1.0 / (x * 12.000000000000048)
	else:
		tmp = z * ((z / x) * (0.0007936500793651 + y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -1e+17)
		tmp = Float64(y * Float64(z * Float64(z / x)));
	elseif (t_0 <= 1e-9)
		tmp = Float64(1.0 / Float64(x * 12.000000000000048));
	else
		tmp = Float64(z * Float64(Float64(z / x) * Float64(0.0007936500793651 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	tmp = 0.0;
	if (t_0 <= -1e+17)
		tmp = y * (z * (z / x));
	elseif (t_0 <= 1e-9)
		tmp = 1.0 / (x * 12.000000000000048);
	else
		tmp = z * ((z / x) * (0.0007936500793651 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+17], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-9], N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $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) < -1e17

   1. Initial program 80.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 83.6

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

   5. Simplified 83.6%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 86.5

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

   7. Applied egg-rr 86.5%

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

   if -1e17 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.00000000000000006e-9

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 77.2%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. associate--l+ N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

   8. Simplified 79.8%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 45.0

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

   11. Simplified 45.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

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

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       3. div-inv N/A

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

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

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

       5. metadata-eval 45.1

          \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

   13. Applied egg-rr 45.1%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]

   if 1.00000000000000006e-9 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.6%

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

   3. Taylor expanded in y around 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. Simplified 98.0%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. +-lowering-+.f64 74.3

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

   8. Simplified 74.3%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq -1 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 10^{-9}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-9}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778))))
   (if (<= t_0 -1e+17)
     (* y (* z (/ z x)))
     (if (<= t_0 1e-9)
       (/ 1.0 (* x 12.000000000000048))
       (* z (* z (/ (+ 0.0007936500793651 y) x)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 1e-9) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = z * (z * ((0.0007936500793651 + y) / 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 * (0.0007936500793651d0 + y)) - 0.0027777777777778d0)
    if (t_0 <= (-1d+17)) then
        tmp = y * (z * (z / x))
    else if (t_0 <= 1d-9) then
        tmp = 1.0d0 / (x * 12.000000000000048d0)
    else
        tmp = z * (z * ((0.0007936500793651d0 + y) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 1e-9) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = z * (z * ((0.0007936500793651 + y) / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)
	tmp = 0
	if t_0 <= -1e+17:
		tmp = y * (z * (z / x))
	elif t_0 <= 1e-9:
		tmp = 1.0 / (x * 12.000000000000048)
	else:
		tmp = z * (z * ((0.0007936500793651 + y) / x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -1e+17)
		tmp = Float64(y * Float64(z * Float64(z / x)));
	elseif (t_0 <= 1e-9)
		tmp = Float64(1.0 / Float64(x * 12.000000000000048));
	else
		tmp = Float64(z * Float64(z * Float64(Float64(0.0007936500793651 + y) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	tmp = 0.0;
	if (t_0 <= -1e+17)
		tmp = y * (z * (z / x));
	elseif (t_0 <= 1e-9)
		tmp = 1.0 / (x * 12.000000000000048);
	else
		tmp = z * (z * ((0.0007936500793651 + y) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+17], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-9], N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] / 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) < -1e17

   1. Initial program 80.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 83.6

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

   5. Simplified 83.6%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 86.5

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

   7. Applied egg-rr 86.5%

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

   if -1e17 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.00000000000000006e-9

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 77.2%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. associate--l+ N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

   8. Simplified 79.8%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 45.0

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

   11. Simplified 45.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

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

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       3. div-inv N/A

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

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

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

       5. metadata-eval 45.1

          \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

   13. Applied egg-rr 45.1%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]

   if 1.00000000000000006e-9 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. log-lowering-log.f64 69.2

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

   5. Simplified 69.2%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-in N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. +-lowering-+.f64 72.5

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

   8. Simplified 72.5%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq -1 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 10^{-9}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778))))
   (if (<= t_0 -1e+17)
     (* y (* z (/ z x)))
     (if (<= t_0 4e+18)
       (/ 1.0 (* x 12.000000000000048))
       (* z (* z (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 4e+18) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = z * (z * (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) :: tmp
    t_0 = z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0)
    if (t_0 <= (-1d+17)) then
        tmp = y * (z * (z / x))
    else if (t_0 <= 4d+18) then
        tmp = 1.0d0 / (x * 12.000000000000048d0)
    else
        tmp = z * (z * (0.0007936500793651d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 4e+18) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = z * (z * (0.0007936500793651 / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)
	tmp = 0
	if t_0 <= -1e+17:
		tmp = y * (z * (z / x))
	elif t_0 <= 4e+18:
		tmp = 1.0 / (x * 12.000000000000048)
	else:
		tmp = z * (z * (0.0007936500793651 / x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -1e+17)
		tmp = Float64(y * Float64(z * Float64(z / x)));
	elseif (t_0 <= 4e+18)
		tmp = Float64(1.0 / Float64(x * 12.000000000000048));
	else
		tmp = Float64(z * Float64(z * Float64(0.0007936500793651 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	tmp = 0.0;
	if (t_0 <= -1e+17)
		tmp = y * (z * (z / x));
	elseif (t_0 <= 4e+18)
		tmp = 1.0 / (x * 12.000000000000048);
	else
		tmp = z * (z * (0.0007936500793651 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+17], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+18], N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(0.0007936500793651 / 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) < -1e17

   1. Initial program 80.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 83.6

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

   5. Simplified 83.6%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 86.5

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

   7. Applied egg-rr 86.5%

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

   if -1e17 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4e18

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 77.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. associate--l+ N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

   8. Simplified 78.2%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 43.8

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

   11. Simplified 43.8%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

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

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       3. div-inv N/A

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

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

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

       5. metadata-eval 43.8

          \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

   13. Applied egg-rr 43.8%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]

   if 4e18 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.2%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. log-lowering-log.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in z around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-lowering-+.f64 71.2

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

   8. Simplified 71.2%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

          \[\leadsto {z}^{2} \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} \cdot 1}}{x} \]

       5. associate-*r/ N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. associate-*r/ N/A

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

       11. associate-*l/ N/A

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

       12. *-commutative N/A

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

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

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

       14. *-commutative N/A

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

       15. associate-*l/ N/A

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

       16. associate-*r/ N/A

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

       17. metadata-eval N/A

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

       18. associate-*r/ N/A

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

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

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

       20. associate-*r/ N/A

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

       21. metadata-eval N/A

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

       22. /-lowering-/.f64 60.2

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

   11. Simplified 60.2%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq -1 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;0.0007936500793651 \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778))))
   (if (<= t_0 -1e+17)
     (* y (* z (/ z x)))
     (if (<= t_0 4e+18)
       (/ 1.0 (* x 12.000000000000048))
       (* 0.0007936500793651 (/ (* z z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 4e+18) {
		tmp = 1.0 / (x * 12.000000000000048);
	} 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 * (0.0007936500793651d0 + y)) - 0.0027777777777778d0)
    if (t_0 <= (-1d+17)) then
        tmp = y * (z * (z / x))
    else if (t_0 <= 4d+18) then
        tmp = 1.0d0 / (x * 12.000000000000048d0)
    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 * (0.0007936500793651 + y)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 4e+18) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = 0.0007936500793651 * ((z * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)
	tmp = 0
	if t_0 <= -1e+17:
		tmp = y * (z * (z / x))
	elif t_0 <= 4e+18:
		tmp = 1.0 / (x * 12.000000000000048)
	else:
		tmp = 0.0007936500793651 * ((z * z) / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -1e+17)
		tmp = Float64(y * Float64(z * Float64(z / x)));
	elseif (t_0 <= 4e+18)
		tmp = Float64(1.0 / Float64(x * 12.000000000000048));
	else
		tmp = Float64(0.0007936500793651 * Float64(Float64(z * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778);
	tmp = 0.0;
	if (t_0 <= -1e+17)
		tmp = y * (z * (z / x));
	elseif (t_0 <= 4e+18)
		tmp = 1.0 / (x * 12.000000000000048);
	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[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+17], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+18], N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision], N[(0.0007936500793651 * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -1e17

   1. Initial program 80.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 83.6

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

   5. Simplified 83.6%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 86.5

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

   7. Applied egg-rr 86.5%

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

   if -1e17 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4e18

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 77.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. associate--l+ N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

   8. Simplified 78.2%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 43.8

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

   11. Simplified 43.8%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

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

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       3. div-inv N/A

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

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

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

       5. metadata-eval 43.8

          \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

   13. Applied egg-rr 43.8%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]

   if 4e18 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.2%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. log-lowering-log.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in z around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-lowering-+.f64 71.2

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

   8. Simplified 71.2%

      \[\leadsto \frac{\color{blue}{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}}{x} \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. flip-+ N/A

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

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

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

   10. Applied egg-rr 22.3%

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

   11. Taylor expanded in y around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 59.6

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

   13. Simplified 59.6%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq -1 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;0.0007936500793651 \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;0.0007936500793651 \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       0.083333333333333
       (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))
      4e+18)
   (/ 1.0 (* x 12.000000000000048))
   (* 0.0007936500793651 (/ (* z z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) <= 4e+18) {
		tmp = 1.0 / (x * 12.000000000000048);
	} 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) :: tmp
    if ((0.083333333333333d0 + (z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0))) <= 4d+18) then
        tmp = 1.0d0 / (x * 12.000000000000048d0)
    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 tmp;
	if ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) <= 4e+18) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = 0.0007936500793651 * ((z * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) <= 4e+18:
		tmp = 1.0 / (x * 12.000000000000048)
	else:
		tmp = 0.0007936500793651 * ((z * z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))) <= 4e+18)
		tmp = Float64(1.0 / Float64(x * 12.000000000000048));
	else
		tmp = Float64(0.0007936500793651 * Float64(Float64(z * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) <= 4e+18)
		tmp = 1.0 / (x * 12.000000000000048);
	else
		tmp = 0.0007936500793651 * ((z * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(0.083333333333333 + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+18], N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision], N[(0.0007936500793651 * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4e18

   1. Initial program 95.8%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 81.5%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. associate--l+ N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

   8. Simplified 65.6%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 35.5

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

   11. Simplified 35.5%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   12. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

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

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       3. div-inv N/A

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

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

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

       5. metadata-eval 35.5

          \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

   13. Applied egg-rr 35.5%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]

   if 4e18 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 88.2%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. log-lowering-log.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in z around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-lowering-+.f64 71.2

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

   8. Simplified 71.2%

      \[\leadsto \frac{\color{blue}{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}}{x} \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. flip-+ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{7936500793651}{10000000000000000} \cdot \left(z \cdot z\right)\right) \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \left(z \cdot z\right)\right) - \left(y \cdot \left(z \cdot z\right)\right) \cdot \left(y \cdot \left(z \cdot z\right)\right)}{\frac{7936500793651}{10000000000000000} \cdot \left(z \cdot z\right) - y \cdot \left(z \cdot z\right)}}}{x} \]

   10. Applied egg-rr 22.3%

       \[\leadsto \frac{\color{blue}{\frac{\left(z \cdot \left(z \cdot 0.0007936500793651\right)\right) \cdot \left(z \cdot \left(z \cdot 0.0007936500793651\right)\right) - \left(z \cdot \left(z \cdot y\right)\right) \cdot \left(z \cdot \left(z \cdot y\right)\right)}{z \cdot \left(z \cdot 0.0007936500793651\right) - z \cdot \left(z \cdot y\right)}}}{x} \]

   11. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]

       3. unpow2 N/A

          \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{\color{blue}{z \cdot z}}{x} \]

       4. *-lowering-*.f64 59.6

          \[\leadsto 0.0007936500793651 \cdot \frac{\color{blue}{z \cdot z}}{x} \]

   13. Simplified 59.6%

       \[\leadsto \color{blue}{0.0007936500793651 \cdot \frac{z \cdot z}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;0.0007936500793651 \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+41}:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.55e+41)
   (+
    0.91893853320467
    (/
     (fma
      z
      (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
      0.083333333333333)
     x))
   (+ 0.91893853320467 (* z (* (/ z x) (+ 0.0007936500793651 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.55e+41) {
		tmp = 0.91893853320467 + (fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = 0.91893853320467 + (z * ((z / x) * (0.0007936500793651 + y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.55e+41)
		tmp = Float64(0.91893853320467 + Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = Float64(0.91893853320467 + Float64(z * Float64(Float64(z / x) * Float64(0.0007936500793651 + y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.55e+41], N[(0.91893853320467 + N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.91893853320467 + N[(z * N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.54999999999999989e41

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 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. Simplified 88.7%

      \[\leadsto \color{blue}{0.91893853320467 + \mathsf{fma}\left(\log x, -0.5 + x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{y}{x}, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\right), \frac{0.083333333333333}{x}\right) - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot z + y \cdot z\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   7. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)} - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      3. +-commutative N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \frac{z \cdot \left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot z + y \cdot z\right)} - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \frac{\color{blue}{\mathsf{fma}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot z + y \cdot z\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      6. sub-neg N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \frac{\mathsf{fma}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot z + y \cdot z\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. distribute-rgt-in N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)} + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \frac{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      10. +-lowering-+.f64 90.8

          \[\leadsto 0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{0.0007936500793651 + y}, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

   8. Simplified 90.8%

      \[\leadsto 0.91893853320467 + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]

   if 2.54999999999999989e41 < x

   1. Initial program 84.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. 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. Simplified 99.6%

      \[\leadsto \color{blue}{0.91893853320467 + \mathsf{fma}\left(\log x, -0.5 + x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{y}{x}, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\right), \frac{0.083333333333333}{x}\right) - x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \frac{y}{x} \cdot z\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} \cdot z + \frac{y}{x} \cdot z\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} \cdot z + \frac{y}{x} \cdot z\right) \]

      7. associate-*l/ N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x}} + \frac{y}{x} \cdot z\right) \]

      8. associate-*r/ N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}} + \frac{y}{x} \cdot z\right) \]

      9. associate-*l/ N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{\frac{y \cdot z}{x}}\right) \]

      10. associate-/l* N/A

          \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{y \cdot \frac{z}{x}}\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \color{blue}{\left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \color{blue}{\left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\color{blue}{\frac{z}{x}} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \]

      14. +-lowering-+.f64 27.3

          \[\leadsto 0.91893853320467 + z \cdot \left(\frac{z}{x} \cdot \color{blue}{\left(0.0007936500793651 + y\right)}\right) \]

   8. Simplified 27.3%

      \[\leadsto 0.91893853320467 + \color{blue}{z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.5)
   (/
    (fma
     z
     (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
     0.083333333333333)
    x)
   (+ 0.91893853320467 (* z (* (/ z x) (+ 0.0007936500793651 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.5) {
		tmp = fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = 0.91893853320467 + (z * ((z / x) * (0.0007936500793651 + y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(0.91893853320467 + Float64(z * Float64(Float64(z / x) * Float64(0.0007936500793651 + y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.5], N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(0.91893853320467 + N[(z * N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. +-lowering-+.f64 97.6

         \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{0.0007936500793651 + y}, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]

   if 2.5 < x

   1. Initial program 86.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 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. Simplified 99.6%

      \[\leadsto \color{blue}{0.91893853320467 + \mathsf{fma}\left(\log x, -0.5 + x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{y}{x}, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\right), \frac{0.083333333333333}{x}\right) - x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \frac{y}{x} \cdot z\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} \cdot z + \frac{y}{x} \cdot z\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} \cdot z + \frac{y}{x} \cdot z\right) \]

      7. associate-*l/ N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x}} + \frac{y}{x} \cdot z\right) \]

      8. associate-*r/ N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}} + \frac{y}{x} \cdot z\right) \]

      9. associate-*l/ N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{\frac{y \cdot z}{x}}\right) \]

      10. associate-/l* N/A

          \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{y \cdot \frac{z}{x}}\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \color{blue}{\left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \color{blue}{\left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{91893853320467}{100000000000000} + z \cdot \left(\color{blue}{\frac{z}{x}} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \]

      14. +-lowering-+.f64 29.5

          \[\leadsto 0.91893853320467 + z \cdot \left(\frac{z}{x} \cdot \color{blue}{\left(0.0007936500793651 + y\right)}\right) \]

   8. Simplified 29.5%

      \[\leadsto 0.91893853320467 + \color{blue}{z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 64.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.8e+108)
   (/
    (fma
     z
     (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
     0.083333333333333)
    x)
   (* z (* z (/ (+ 0.0007936500793651 y) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.8e+108) {
		tmp = fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = z * (z * ((0.0007936500793651 + y) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.8e+108)
		tmp = Float64(fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(z * Float64(z * Float64(Float64(0.0007936500793651 + y) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.8e+108], N[(N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.80000000000000015e108

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. +-lowering-+.f64 80.0

         \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{0.0007936500793651 + y}, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

   5. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]

   if 5.80000000000000015e108 < x

   1. Initial program 77.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{83333333333333}{1000000000000000} + x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}}{x} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \left(\frac{83333333333333}{1000000000000000} + x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right)\right)}}{x} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000} + x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right)\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000} + x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right)\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000} + x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right)\right)}{x} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000} + x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right)\right)}{x} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} + y}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000} + x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right)\right)}{x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \color{blue}{x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right) + \frac{83333333333333}{1000000000000000}}\right)}{x} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \color{blue}{\mathsf{fma}\left(x, \frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x, \frac{83333333333333}{1000000000000000}\right)}\right)}{x} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \mathsf{fma}\left(x, \color{blue}{\frac{-1}{2} \cdot \log x + \frac{91893853320467}{100000000000000}}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      13. log-lowering-log.f64 11.5

          \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, \color{blue}{\log x}, 0.91893853320467\right), 0.083333333333333\right)\right)}{x} \]

   5. Simplified 11.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), 0.083333333333333\right)\right)}{x}} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      4. associate-/l* N/A

         \[\leadsto z \cdot \color{blue}{\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]

      5. distribute-rgt-in N/A

         \[\leadsto z \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} \cdot z + y \cdot z}}{x} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{\frac{7936500793651}{10000000000000000} \cdot z + y \cdot z}{x}} \]

      7. distribute-rgt-in N/A

         \[\leadsto z \cdot \frac{\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}}{x} \]

      8. associate-/l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto z \cdot \left(z \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x}}\right) \]

      11. +-lowering-+.f64 23.3

          \[\leadsto z \cdot \left(z \cdot \frac{\color{blue}{0.0007936500793651 + y}}{x}\right) \]

   8. Simplified 23.3%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 23.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot 12.000000000000048} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 (* x 12.000000000000048)))

C

double code(double x, double y, double z) {
	return 1.0 / (x * 12.000000000000048);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / (x * 12.000000000000048d0)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / (x * 12.000000000000048);
}

Python

def code(x, y, z):
	return 1.0 / (x * 12.000000000000048)

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(x * 12.000000000000048))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / (x * 12.000000000000048);
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.8%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} + \frac{{z}^{2}}{x}\right)\right)\right)\right) - \frac{x}{y}\right)} \]

5. Simplified 81.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(\log x, \frac{-0.5 + x}{y}, \frac{z}{x} \cdot \left(z + \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{y}\right)\right) + \left(\frac{0.91893853320467}{y} - \frac{x}{y}\right), \frac{0.083333333333333}{x}\right)} \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right)} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}} + y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} + y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} + y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right) \]

   6. associate--l+ N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + y \cdot \color{blue}{\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right)} \]

   7. distribute-lft-in N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \color{blue}{\left(y \cdot \left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y}\right) + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(y \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{91893853320467}{100000000000000}\right)} + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\color{blue}{\left(y \cdot \frac{1}{y}\right) \cdot \frac{91893853320467}{100000000000000}} + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right) \]

   10. rgt-mult-inverse N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\color{blue}{1} \cdot \frac{91893853320467}{100000000000000} + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\color{blue}{\frac{91893853320467}{100000000000000}} + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \color{blue}{\left(\frac{91893853320467}{100000000000000} + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right)} \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} + \color{blue}{y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)}\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} + y \cdot \color{blue}{\left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)}\right) \]

8. Simplified 47.6%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x} + \left(0.91893853320467 + y \cdot \left(\log x \cdot \frac{x + -0.5}{y} - \frac{x}{y}\right)\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 23.0

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

11. Simplified 23.0%

    \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
12. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

    3. div-inv N/A

       \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

    5. metadata-eval 23.0

       \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

13. Applied egg-rr 23.0%

    \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]
14. Add Preprocessing

Alternative 20: 23.7% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

1. Initial program 92.8%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x \cdot y} + \left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x \cdot y} + \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} + \frac{{z}^{2}}{x}\right)\right)\right)\right) - \frac{x}{y}\right)} \]

5. Simplified 81.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(\log x, \frac{-0.5 + x}{y}, \frac{z}{x} \cdot \left(z + \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{y}\right)\right) + \left(\frac{0.91893853320467}{y} - \frac{x}{y}\right), \frac{0.083333333333333}{x}\right)} \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right)} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}} + y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} + y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} + y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \frac{x}{y}\right) \]

   6. associate--l+ N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + y \cdot \color{blue}{\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right)} \]

   7. distribute-lft-in N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \color{blue}{\left(y \cdot \left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y}\right) + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(y \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{91893853320467}{100000000000000}\right)} + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\color{blue}{\left(y \cdot \frac{1}{y}\right) \cdot \frac{91893853320467}{100000000000000}} + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right) \]

   10. rgt-mult-inverse N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\color{blue}{1} \cdot \frac{91893853320467}{100000000000000} + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\color{blue}{\frac{91893853320467}{100000000000000}} + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \color{blue}{\left(\frac{91893853320467}{100000000000000} + y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)\right)} \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} + \color{blue}{y \cdot \left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)}\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} + y \cdot \color{blue}{\left(\frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y} - \frac{x}{y}\right)}\right) \]

8. Simplified 47.6%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x} + \left(0.91893853320467 + y \cdot \left(\log x \cdot \frac{x + -0.5}{y} - \frac{x}{y}\right)\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 23.0

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

11. Simplified 23.0%

    \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
12. Add Preprocessing

Alternative 21: 4.1% accurate, 148.0× speedup

Math

\[\begin{array}{l} \\ 0.91893853320467 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.91893853320467)

C

double code(double x, double y, double z) {
	return 0.91893853320467;
}

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
end function

Java

public static double code(double x, double y, double z) {
	return 0.91893853320467;
}

Python

def code(x, y, z):
	return 0.91893853320467

Julia

function code(x, y, z)
	return 0.91893853320467
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.91893853320467;
end

Wolfram

code[x_, y_, z_] := 0.91893853320467

Derivation

1. Initial program 92.8%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) - x} \]

5. Simplified 93.5%

   \[\leadsto \color{blue}{0.91893853320467 + \mathsf{fma}\left(\log x, -0.5 + x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{y}{x}, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\right), \frac{0.083333333333333}{x}\right) - x\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{91893853320467}{100000000000000} + \frac{\color{blue}{y \cdot {z}^{2}}}{x} \]

   3. unpow2 N/A

      \[\leadsto \frac{91893853320467}{100000000000000} + \frac{y \cdot \color{blue}{\left(z \cdot z\right)}}{x} \]

   4. *-lowering-*.f64 29.5

      \[\leadsto 0.91893853320467 + \frac{y \cdot \color{blue}{\left(z \cdot z\right)}}{x} \]

8. Simplified 29.5%

   \[\leadsto 0.91893853320467 + \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000}} \]
10. Step-by-step derivation

11. Simplified 4.3%

    \[\leadsto \color{blue}{0.91893853320467} \]
12. 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 2024204 
(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)))