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

Average Accuracy: 53.6% → 98.5%

Time: 47.6s

Precision: binary64

Cost: 52808

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{z \cdot z}\\ t_2 := \frac{457.9610022158428}{z \cdot z}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(t_2 + \left(\left(t_1 - \frac{\left(5864.8025282699045 + t \cdot 15.234687407\right) - a}{{z}^{3}}\right) + \frac{-36.52704169880642}{z}\right)\right), x\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(t_2 + t_1\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \end{array} \]

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
 (let* ((t_1 (/ t (* z z))) (t_2 (/ 457.9610022158428 (* z z))))
   (if (<= z -1.05e+20)
     (fma
      y
      (+
       3.13060547623
       (+
        t_2
        (+
         (-
          t_1
          (/ (- (+ 5864.8025282699045 (* t 15.234687407)) a) (pow z 3.0)))
         (/ -36.52704169880642 z))))
      x)
     (if (<= z 4.5e+37)
       (fma
        (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b)
        (/
         y
         (fma
          z
          (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
          0.607771387771))
        x)
       (fma y (+ 3.13060547623 (+ (+ t_2 t_1) (/ -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 t_1 = t / (z * z);
	double t_2 = 457.9610022158428 / (z * z);
	double tmp;
	if (z <= -1.05e+20) {
		tmp = fma(y, (3.13060547623 + (t_2 + ((t_1 - (((5864.8025282699045 + (t * 15.234687407)) - a) / pow(z, 3.0))) + (-36.52704169880642 / z)))), x);
	} else if (z <= 4.5e+37) {
		tmp = fma(fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b), (y / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), x);
	} else {
		tmp = fma(y, (3.13060547623 + ((t_2 + t_1) + (-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)
	t_1 = Float64(t / Float64(z * z))
	t_2 = Float64(457.9610022158428 / Float64(z * z))
	tmp = 0.0
	if (z <= -1.05e+20)
		tmp = fma(y, Float64(3.13060547623 + Float64(t_2 + Float64(Float64(t_1 - Float64(Float64(Float64(5864.8025282699045 + Float64(t * 15.234687407)) - a) / (z ^ 3.0))) + Float64(-36.52704169880642 / z)))), x);
	elseif (z <= 4.5e+37)
		tmp = fma(fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b), Float64(y / fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), x);
	else
		tmp = fma(y, Float64(3.13060547623 + Float64(Float64(t_2 + t_1) + 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_] := Block[{t$95$1 = N[(t / N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(457.9610022158428 / N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05e+20], N[(y * N[(3.13060547623 + N[(t$95$2 + N[(N[(t$95$1 - N[(N[(N[(5864.8025282699045 + N[(t * 15.234687407), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.5e+37], N[(N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision] * N[(y / N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(3.13060547623 + N[(N[(t$95$2 + t$95$1), $MachinePrecision] + N[(-36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]

Target

Original	53.6%
Target	98.2%
Herbie	98.5%

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

Derivation

1. Split input into 3 regimes

2. if z < -1.05e20

   1. Initial program 10.4%

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

   2. Simplified 14.5%

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

      [Start] 10.4	\[ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
      +-commutative [=>] 10.4	\[ \color{blue}{\frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} + x} \]
      associate-*r/ [<=] 14.5	\[ \color{blue}{y \cdot \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}} + x \]
      fma-def [=>] 14.5	\[ \color{blue}{\mathsf{fma}\left(y, \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}, x\right)} \]

   3. Taylor expanded in z around -inf 98.7%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \left(\frac{t}{{z}^{2}} + -1 \cdot \frac{-1 \cdot a - \left(1112.0901850848957 + -15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)\right)\right) - 36.52704169880642 \cdot \frac{1}{z}}, x\right) \]

   4. Simplified 98.7%

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

      [Start] 98.7	\[ \mathsf{fma}\left(y, \left(3.13060547623 + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \left(\frac{t}{{z}^{2}} + -1 \cdot \frac{-1 \cdot a - \left(1112.0901850848957 + -15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)\right)\right) - 36.52704169880642 \cdot \frac{1}{z}, x\right) \]
      associate--l+ [=>] 98.7	\[ \mathsf{fma}\left(y, \color{blue}{3.13060547623 + \left(\left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \left(\frac{t}{{z}^{2}} + -1 \cdot \frac{-1 \cdot a - \left(1112.0901850848957 + -15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)\right) - 36.52704169880642 \cdot \frac{1}{z}\right)}, x\right) \]
      associate--l+ [=>] 98.7	\[ \mathsf{fma}\left(y, 3.13060547623 + \color{blue}{\left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \left(\left(\frac{t}{{z}^{2}} + -1 \cdot \frac{-1 \cdot a - \left(1112.0901850848957 + -15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right)\right)}, x\right) \]
      associate-*r/ [=>] 98.7	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\color{blue}{\frac{457.9610022158428 \cdot 1}{{z}^{2}}} + \left(\left(\frac{t}{{z}^{2}} + -1 \cdot \frac{-1 \cdot a - \left(1112.0901850848957 + -15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right)\right), x\right) \]
      metadata-eval [=>] 98.7	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\frac{\color{blue}{457.9610022158428}}{{z}^{2}} + \left(\left(\frac{t}{{z}^{2}} + -1 \cdot \frac{-1 \cdot a - \left(1112.0901850848957 + -15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right)\right), x\right) \]
      unpow2 [=>] 98.7	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428}{\color{blue}{z \cdot z}} + \left(\left(\frac{t}{{z}^{2}} + -1 \cdot \frac{-1 \cdot a - \left(1112.0901850848957 + -15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right)\right), x\right) \]

   if -1.05e20 < z < 4.49999999999999962e37

   1. Initial program 98.3%

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

   2. Simplified 99.0%

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

      [Start] 98.3	\[ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
      +-commutative [=>] 98.3	\[ \color{blue}{\frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} + x} \]
      associate-*l/ [<=] 99.0	\[ \color{blue}{\frac{y}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)} + x \]
      *-commutative [=>] 99.0	\[ \color{blue}{\left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right) \cdot \frac{y}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}} + x \]
      fma-def [=>] 99.0	\[ \color{blue}{\mathsf{fma}\left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b, \frac{y}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}, x\right)} \]

   if 4.49999999999999962e37 < z

   1. Initial program 6.3%

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

   2. Simplified 10.1%

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

      [Start] 6.3	\[ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
      +-commutative [=>] 6.3	\[ \color{blue}{\frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} + x} \]
      associate-*r/ [<=] 10.1	\[ \color{blue}{y \cdot \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}} + x \]
      fma-def [=>] 10.1	\[ \color{blue}{\mathsf{fma}\left(y, \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}, x\right)} \]

   3. Taylor expanded in z around inf 97.3%

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

   4. Simplified 97.3%

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

      [Start] 97.3	\[ \mathsf{fma}\left(y, \left(3.13060547623 + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{t}{{z}^{2}}\right)\right) - 36.52704169880642 \cdot \frac{1}{z}, x\right) \]
      associate--l+ [=>] 97.3	\[ \mathsf{fma}\left(y, \color{blue}{3.13060547623 + \left(\left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right)}, x\right) \]
      associate-*r/ [=>] 97.3	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\color{blue}{\frac{457.9610022158428 \cdot 1}{{z}^{2}}} + \frac{t}{{z}^{2}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right), x\right) \]
      metadata-eval [=>] 97.3	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{\color{blue}{457.9610022158428}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right), x\right) \]
      unpow2 [=>] 97.3	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{\color{blue}{z \cdot z}} + \frac{t}{{z}^{2}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right), x\right) \]
      unpow2 [=>] 97.3	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{\color{blue}{z \cdot z}}\right) - 36.52704169880642 \cdot \frac{1}{z}\right), x\right) \]
      associate-*r/ [=>] 97.3	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) - \color{blue}{\frac{36.52704169880642 \cdot 1}{z}}\right), x\right) \]
      metadata-eval [=>] 97.3	\[ \mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) - \frac{\color{blue}{36.52704169880642}}{z}\right), x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428}{z \cdot z} + \left(\left(\frac{t}{z \cdot z} - \frac{\left(5864.8025282699045 + t \cdot 15.234687407\right) - a}{{z}^{3}}\right) + \frac{-36.52704169880642}{z}\right)\right), x\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.7%
Cost	46536

\[\begin{array}{l} t_1 := \frac{t}{z \cdot z}\\ t_2 := \frac{457.9610022158428}{z \cdot z}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(t_2 + \left(\left(t_1 - \frac{\left(5864.8025282699045 + t \cdot 15.234687407\right) - a}{{z}^{3}}\right) + \frac{-36.52704169880642}{z}\right)\right), x\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), z, t\right), z, a\right), z, b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(t_2 + t_1\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	98.9%
Cost	14985

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+20} \lor \neg \left(z \leq 52000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428}{z \cdot z} + \left(\left(\frac{t}{z \cdot z} - \frac{\left(5864.8025282699045 + t \cdot 15.234687407\right) - a}{{z}^{3}}\right) + \frac{-36.52704169880642}{z}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	97.2%
Cost	12233

\[\begin{array}{l} t_1 := \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 4 \cdot 10^{+268}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_1\\ \end{array} \]

Alternative 4
Accuracy	97.0%
Cost	12232

\[\begin{array}{l} t_1 := \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-36.52704169880642 \cdot \frac{y}{z} + \left(y \cdot 3.13060547623 + \left(x + \frac{y \cdot \left(457.9610022158428 + t\right)}{{z}^{2}}\right)\right)\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+268}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \end{array} \]

Alternative 5
Accuracy	95.2%
Cost	6985

\[\begin{array}{l} t_1 := \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 4 \cdot 10^{+268}\right):\\ \;\;\;\;x + y \cdot 3.13060547623\\ \mathbf{else}:\\ \;\;\;\;x + t_1\\ \end{array} \]

Alternative 6
Accuracy	94.9%
Cost	2376

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+26}:\\ \;\;\;\;x + y \cdot 3.13060547623\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot 11.1667541262\right)\right)\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot \left(31.4690115749 + z \cdot \left(z + 15.234687407\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{3.7269864963038164}{z} + \left(0.31942702700572795 + \frac{t \cdot -0.10203362558171805}{z \cdot z}\right)}\\ \end{array} \]

Alternative 7
Accuracy	94.1%
Cost	2248

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;x + y \cdot 3.13060547623\\ \mathbf{elif}\;z \leq 420000:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot 31.4690115749\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{3.7269864963038164}{z} + \left(0.31942702700572795 + \frac{t \cdot -0.10203362558171805}{z \cdot z}\right)}\\ \end{array} \]

Alternative 8
Accuracy	94.0%
Cost	1992

\[\begin{array}{l} \mathbf{if}\;z \leq -18500000000:\\ \;\;\;\;x + y \cdot 3.13060547623\\ \mathbf{elif}\;z \leq 6500000:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(11.1667541262 + z \cdot 3.13060547623\right)\right)\right)\right)}{0.607771387771 + z \cdot 11.9400905721}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{3.7269864963038164}{z} + \left(0.31942702700572795 + \frac{t \cdot -0.10203362558171805}{z \cdot z}\right)}\\ \end{array} \]

Alternative 9
Accuracy	92.0%
Cost	1608

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0029:\\ \;\;\;\;-36.52704169880642 \cdot \frac{y}{z} + \left(x + y \cdot 3.13060547623\right)\\ \mathbf{elif}\;z \leq 26000:\\ \;\;\;\;x + \frac{y}{\frac{0.607771387771 + z \cdot 11.9400905721}{b + z \cdot \left(a + \left(z \cdot z\right) \cdot 11.1667541262\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{3.7269864963038164}{z} + \left(0.31942702700572795 + \frac{t \cdot -0.10203362558171805}{z \cdot z}\right)}\\ \end{array} \]

Alternative 10
Accuracy	91.6%
Cost	1480

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;x + y \cdot 3.13060547623\\ \mathbf{elif}\;z \leq 6.2:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot a\right)}{0.607771387771 + z \cdot \left(11.9400905721 + z \cdot 31.4690115749\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{3.7269864963038164}{z} + \left(0.31942702700572795 + \frac{t \cdot -0.10203362558171805}{z \cdot z}\right)}\\ \end{array} \]

Alternative 11
Accuracy	86.1%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{y}{0.31942702700572795 + \frac{3.7269864963038164 + \frac{-3.241970391368047}{z}}{z}}\\ \mathbf{elif}\;z \leq 17:\\ \;\;\;\;x + b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{3.7269864963038164}{z} + \left(0.31942702700572795 + \frac{t \cdot -0.10203362558171805}{z \cdot z}\right)}\\ \end{array} \]

Alternative 12
Accuracy	70.1%
Cost	980

\[\begin{array}{l} t_1 := b \cdot \left(y \cdot 1.6453555072203998\right)\\ t_2 := x + y \cdot 3.13060547623\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-207}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Accuracy	86.1%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{y}{0.31942702700572795 + \frac{3.7269864963038164 + \frac{-3.241970391368047}{z}}{z}}\\ \mathbf{elif}\;z \leq 0.3:\\ \;\;\;\;x + b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 3.13060547623\\ \end{array} \]

Alternative 14
Accuracy	51.7%
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1400000000000 \lor \neg \left(x \leq -4 \cdot 10^{-51}\right) \land x \leq 1.92 \cdot 10^{-187}:\\ \;\;\;\;1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Accuracy	51.0%
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+55}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1050000000000 \lor \neg \left(x \leq -4.1 \cdot 10^{-51}\right) \land x \leq 1.2 \cdot 10^{-188}:\\ \;\;\;\;b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Accuracy	85.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+18} \lor \neg \left(z \leq 55\right):\\ \;\;\;\;x + y \cdot 3.13060547623\\ \mathbf{else}:\\ \;\;\;\;x + b \cdot \left(y \cdot 1.6453555072203998\right)\\ \end{array} \]

Alternative 17
Accuracy	86.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{y}{\frac{3.7269864963038164}{z} + 0.31942702700572795}\\ \mathbf{elif}\;z \leq 8.5:\\ \;\;\;\;x + b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 3.13060547623\\ \end{array} \]

Alternative 18
Accuracy	49.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(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))))