Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D

Average Accuracy: 54.1% → 98.3%

Time: 38.3s

Precision: binary64

Cost: 46536

Math

\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) + \left(\frac{y}{z} \cdot \frac{t + 457.9610022158428}{z} + \mathsf{fma}\left(-36.52704169880642, \frac{y}{z}, \frac{y}{{z}^{3}} \cdot \left(t \cdot -15.234687407 + \left(a + -5864.8025282699045\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+25}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b))
   (+
    (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z)
    0.607771387771))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.05e+34)
   (+
    (fma 3.13060547623 y x)
    (+
     (* (/ y z) (/ (+ t 457.9610022158428) z))
     (fma
      -36.52704169880642
      (/ y z)
      (*
       (/ y (pow z 3.0))
       (+ (* t -15.234687407) (+ a -5864.8025282699045))))))
   (if (<= z 1.75e+25)
     (+
      x
      (/
       (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b)
       (/
        (fma
         z
         (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
         0.607771387771)
        y)))
     (fma
      y
      (+
       3.13060547623
       (+
        (+ (/ 457.9610022158428 (* z z)) (/ t (* z z)))
        (/ -36.52704169880642 z)))
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.05e+34) {
		tmp = fma(3.13060547623, y, x) + (((y / z) * ((t + 457.9610022158428) / z)) + fma(-36.52704169880642, (y / z), ((y / pow(z, 3.0)) * ((t * -15.234687407) + (a + -5864.8025282699045)))));
	} else if (z <= 1.75e+25) {
		tmp = x + (fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b) / (fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771) / y));
	} else {
		tmp = fma(y, (3.13060547623 + (((457.9610022158428 / (z * z)) + (t / (z * z))) + (-36.52704169880642 / z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)))
end

↓

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.05e+34)
		tmp = Float64(fma(3.13060547623, y, x) + Float64(Float64(Float64(y / z) * Float64(Float64(t + 457.9610022158428) / z)) + fma(-36.52704169880642, Float64(y / z), Float64(Float64(y / (z ^ 3.0)) * Float64(Float64(t * -15.234687407) + Float64(a + -5864.8025282699045))))));
	elseif (z <= 1.75e+25)
		tmp = Float64(x + Float64(fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b) / Float64(fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771) / y)));
	else
		tmp = fma(y, Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 / Float64(z * z)) + Float64(t / Float64(z * z))) + Float64(-36.52704169880642 / z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.05e+34], N[(N[(3.13060547623 * y + x), $MachinePrecision] + N[(N[(N[(y / z), $MachinePrecision] * N[(N[(t + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(-36.52704169880642 * N[(y / z), $MachinePrecision] + N[(N[(y / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * -15.234687407), $MachinePrecision] + N[(a + -5864.8025282699045), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e+25], N[(x + N[(N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision] / N[(N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(3.13060547623 + N[(N[(N[(457.9610022158428 / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(t / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Target

Original	54.1%
Target	98.5%
Herbie	98.3%

\[\begin{array}{l} \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if z < -1.05000000000000009e34

   1. Initial program 7.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Simplified 11.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)} \]
      Proof

      [Start] 7.2	\[ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
      +-commutative [=>] 7.2	\[ \color{blue}{\frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} + x} \]
      associate-*r/ [<=] 11.6	\[ \color{blue}{y \cdot \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}} + x \]
      fma-def [=>] 11.6	\[ \color{blue}{\mathsf{fma}\left(y, \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}, x\right)} \]

   3. Taylor expanded in z around inf 78.1%

      \[\leadsto \color{blue}{-36.52704169880642 \cdot \frac{y}{z} + \left(\frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}} + \left(3.13060547623 \cdot y + \left(\frac{y \cdot \left(457.9610022158428 + t\right)}{{z}^{2}} + x\right)\right)\right)} \]

   4. Simplified 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right) + \left(\frac{y}{z} \cdot \frac{t + 457.9610022158428}{z} + \mathsf{fma}\left(-36.52704169880642, \frac{y}{z}, \frac{y}{{z}^{3}} \cdot \left(t \cdot -15.234687407 + \left(a + -5864.8025282699045\right)\right)\right)\right)} \]
      Proof

      [Start] 78.1	\[ -36.52704169880642 \cdot \frac{y}{z} + \left(\frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}} + \left(3.13060547623 \cdot y + \left(\frac{y \cdot \left(457.9610022158428 + t\right)}{{z}^{2}} + x\right)\right)\right) \]
      associate-+r+ [=>] 78.1	\[ \color{blue}{\left(-36.52704169880642 \cdot \frac{y}{z} + \frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right) + \left(3.13060547623 \cdot y + \left(\frac{y \cdot \left(457.9610022158428 + t\right)}{{z}^{2}} + x\right)\right)} \]
      +-commutative [=>] 78.1	\[ \color{blue}{\left(3.13060547623 \cdot y + \left(\frac{y \cdot \left(457.9610022158428 + t\right)}{{z}^{2}} + x\right)\right) + \left(-36.52704169880642 \cdot \frac{y}{z} + \frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)} \]
      +-commutative [=>] 78.1	\[ \left(3.13060547623 \cdot y + \color{blue}{\left(x + \frac{y \cdot \left(457.9610022158428 + t\right)}{{z}^{2}}\right)}\right) + \left(-36.52704169880642 \cdot \frac{y}{z} + \frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right) \]
      associate-+r+ [=>] 78.1	\[ \color{blue}{\left(\left(3.13060547623 \cdot y + x\right) + \frac{y \cdot \left(457.9610022158428 + t\right)}{{z}^{2}}\right)} + \left(-36.52704169880642 \cdot \frac{y}{z} + \frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right) \]
      associate-+l+ [=>] 78.1	\[ \color{blue}{\left(3.13060547623 \cdot y + x\right) + \left(\frac{y \cdot \left(457.9610022158428 + t\right)}{{z}^{2}} + \left(-36.52704169880642 \cdot \frac{y}{z} + \frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)\right)} \]
      fma-def [=>] 78.1	\[ \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} + \left(\frac{y \cdot \left(457.9610022158428 + t\right)}{{z}^{2}} + \left(-36.52704169880642 \cdot \frac{y}{z} + \frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)\right) \]
      unpow2 [=>] 78.1	\[ \mathsf{fma}\left(3.13060547623, y, x\right) + \left(\frac{y \cdot \left(457.9610022158428 + t\right)}{\color{blue}{z \cdot z}} + \left(-36.52704169880642 \cdot \frac{y}{z} + \frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)\right) \]
      times-frac [=>] 77.9	\[ \mathsf{fma}\left(3.13060547623, y, x\right) + \left(\color{blue}{\frac{y}{z} \cdot \frac{457.9610022158428 + t}{z}} + \left(-36.52704169880642 \cdot \frac{y}{z} + \frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)\right) \]
      +-commutative [=>] 77.9	\[ \mathsf{fma}\left(3.13060547623, y, x\right) + \left(\frac{y}{z} \cdot \frac{\color{blue}{t + 457.9610022158428}}{z} + \left(-36.52704169880642 \cdot \frac{y}{z} + \frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)\right) \]
      fma-def [=>] 77.9	\[ \mathsf{fma}\left(3.13060547623, y, x\right) + \left(\frac{y}{z} \cdot \frac{t + 457.9610022158428}{z} + \color{blue}{\mathsf{fma}\left(-36.52704169880642, \frac{y}{z}, \frac{y \cdot \left(\left(1112.0901850848957 + a\right) - 15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)}\right) \]

   if -1.05000000000000009e34 < z < 1.75e25

   1. Initial program 98.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Simplified 98.9%

      \[\leadsto \color{blue}{x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}{y}}} \]
      Proof

      [Start] 98.2	\[ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
      *-commutative [=>] 98.2	\[ x + \frac{\color{blue}{\left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right) \cdot y}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
      associate-/l* [=>] 98.8	\[ x + \color{blue}{\frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{y}}} \]

   if 1.75e25 < z

   1. Initial program 9.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Simplified 14.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)} \]
      Proof

      [Start] 9.5	\[ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
      +-commutative [=>] 9.5	\[ \color{blue}{\frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} + x} \]
      associate-*r/ [<=] 14.6	\[ \color{blue}{y \cdot \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}} + x \]
      fma-def [=>] 14.6	\[ \color{blue}{\mathsf{fma}\left(y, \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}, x\right)} \]

   3. Taylor expanded in z around inf 97.1%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{t}{{z}^{2}}\right)\right) - 36.52704169880642 \cdot \frac{1}{z}}, x\right) \]

   4. Simplified 97.1%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) - \frac{36.52704169880642}{z}\right)}, x\right) \]
      Proof

      [Start] 97.1	\[ \mathsf{fma}\left(y, \left(3.13060547623 + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{t}{{z}^{2}}\right)\right) - 36.52704169880642 \cdot \frac{1}{z}, x\right) \]
      associate--l+ [=>] 97.1	\[ \mathsf{fma}\left(y, \color{blue}{3.13060547623 + \left(\left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right)}, x\right) \]
      associate-*r/ [=>] 97.1	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\color{blue}{\frac{457.9610022158428 \cdot 1}{{z}^{2}}} + \frac{t}{{z}^{2}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right), x\right) \]
      metadata-eval [=>] 97.1	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{\color{blue}{457.9610022158428}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right), x\right) \]
      unpow2 [=>] 97.1	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{\color{blue}{z \cdot z}} + \frac{t}{{z}^{2}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right), x\right) \]
      unpow2 [=>] 97.1	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{\color{blue}{z \cdot z}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right), x\right) \]
      associate-*r/ [=>] 97.1	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) - \color{blue}{\frac{36.52704169880642 \cdot 1}{z}}\right), x\right) \]
      metadata-eval [=>] 97.1	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) - \frac{\color{blue}{36.52704169880642}}{z}\right), x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) + \left(\frac{y}{z} \cdot \frac{t + 457.9610022158428}{z} + \mathsf{fma}\left(-36.52704169880642, \frac{y}{z}, \frac{y}{{z}^{3}} \cdot \left(t \cdot -15.234687407 + \left(a + -5864.8025282699045\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+25}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.1%
Cost	14852

\[\begin{array}{l} t_1 := 0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)\\ t_2 := \frac{t}{z \cdot z}\\ t_3 := \frac{457.9610022158428}{z \cdot z}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(t_3 + \left(\left(t_2 - \frac{\left(t \cdot 15.234687407 + 5864.8025282699045\right) - a}{{z}^{3}}\right) + \frac{-36.52704169880642}{z}\right)\right), x\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+25}:\\ \;\;\;\;\frac{y \cdot \left(b + {z}^{2} \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)}{t_1} + \left(x + \frac{y \cdot \left(z \cdot a\right)}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(t_3 + t_2\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	97.6%
Cost	10249

\[\begin{array}{l} t_1 := 0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{+23} \lor \neg \left(z \leq 1.6 \cdot 10^{+25}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(b + {z}^{2} \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)}{t_1} + \left(x + \frac{y \cdot \left(z \cdot a\right)}{t_1}\right)\\ \end{array} \]

Alternative 3
Accuracy	97.8%
Cost	7881

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+21} \lor \neg \left(z \leq 1.9 \cdot 10^{+25}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	95.8%
Cost	2632

\[\begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+25}:\\ \;\;\;\;x + 3.13060547623 \cdot y\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+25}:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(0.31942702700572795 + \frac{3.7269864963038164}{z}\right) - t \cdot \frac{0.10203362558171805}{z \cdot z}}\\ \end{array} \]

Alternative 5
Accuracy	95.3%
Cost	2376

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20}:\\ \;\;\;\;x + 3.13060547623 \cdot y\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+25}:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot 11.1667541262\right)\right)\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(0.31942702700572795 + \frac{3.7269864963038164}{z}\right) - t \cdot \frac{0.10203362558171805}{z \cdot z}}\\ \end{array} \]

Alternative 6
Accuracy	94.2%
Cost	1992

\[\begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;x + 3.13060547623 \cdot y\\ \mathbf{elif}\;z \leq 95:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)\right)}{0.607771387771 + z \cdot 11.9400905721}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(0.31942702700572795 + \frac{3.7269864963038164}{z}\right) - t \cdot \frac{0.10203362558171805}{z \cdot z}}\\ \end{array} \]

Alternative 7
Accuracy	92.0%
Cost	1864

\[\begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+33}:\\ \;\;\;\;x + 3.13060547623 \cdot y\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+25}:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot a\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(0.31942702700572795 + \frac{3.7269864963038164}{z}\right) - t \cdot \frac{0.10203362558171805}{z \cdot z}}\\ \end{array} \]

Alternative 8
Accuracy	91.3%
Cost	1480

\[\begin{array}{l} \mathbf{if}\;z \leq -31:\\ \;\;\;\;x + 3.13060547623 \cdot y\\ \mathbf{elif}\;z \leq 0.16:\\ \;\;\;\;y \cdot \left(z \cdot \left(b \cdot -32.324150453290734 - a \cdot -1.6453555072203998\right)\right) + \left(x + 1.6453555072203998 \cdot \left(y \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(0.31942702700572795 + \frac{3.7269864963038164}{z}\right) - t \cdot \frac{0.10203362558171805}{z \cdot z}}\\ \end{array} \]

Alternative 9
Accuracy	85.6%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;z \leq -0.019:\\ \;\;\;\;x + \left(3.13060547623 \cdot y - \frac{y}{\frac{z}{36.52704169880642}}\right)\\ \mathbf{elif}\;z \leq 0.0155:\\ \;\;\;\;x + 1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(0.31942702700572795 + \frac{3.7269864963038164}{z}\right) - t \cdot \frac{0.10203362558171805}{z \cdot z}}\\ \end{array} \]

Alternative 10
Accuracy	69.5%
Cost	980

\[\begin{array}{l} t_1 := x + 3.13060547623 \cdot y\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-149}:\\ \;\;\;\;y \cdot \left(b \cdot 1.6453555072203998\right)\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-263}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{-269}:\\ \;\;\;\;1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-169}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	85.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -0.019 \lor \neg \left(z \leq 0.16\right):\\ \;\;\;\;x + \left(3.13060547623 \cdot y - \frac{y}{\frac{z}{36.52704169880642}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1.6453555072203998 \cdot \left(y \cdot b\right)\\ \end{array} \]

Alternative 12
Accuracy	85.5%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -0.019:\\ \;\;\;\;x + 3.13060547623 \cdot y\\ \mathbf{elif}\;z \leq 0.81:\\ \;\;\;\;x + 1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{0.31942702700572795 + \frac{3.7269864963038164}{z}}\\ \end{array} \]

Alternative 13
Accuracy	85.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.018 \lor \neg \left(z \leq 105\right):\\ \;\;\;\;x + 3.13060547623 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + b \cdot \left(y \cdot 1.6453555072203998\right)\\ \end{array} \]

Alternative 14
Accuracy	85.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0155 \lor \neg \left(z \leq 90000\right):\\ \;\;\;\;x + 3.13060547623 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + 1.6453555072203998 \cdot \left(y \cdot b\right)\\ \end{array} \]

Alternative 15
Accuracy	56.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-114}:\\ \;\;\;\;3.13060547623 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Accuracy	49.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))