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

Average Error: 29.8 → 2.0

Time: 1.8min

Precision: binary64

Cost: 46536

Math

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

↓

\[\begin{array}{l} t_1 := x + 3.13060547623 \cdot y\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+63}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), z, t\right), a\right), b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (+ x (* 3.13060547623 y))))
   (if (<= z -5.4e+46)
     t_1
     (if (<= z 4.4e+63)
       (+
        x
        (*
         y
         (/
          (fma z (fma z (fma (fma z 3.13060547623 11.1667541262) z t) a) b)
          (fma
           (fma (fma (+ 15.234687407 z) z 31.4690115749) z 11.9400905721)
           z
           0.607771387771))))
       t_1))))

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 = x + (3.13060547623 * y);
	double tmp;
	if (z <= -5.4e+46) {
		tmp = t_1;
	} else if (z <= 4.4e+63) {
		tmp = x + (y * (fma(z, fma(z, fma(fma(z, 3.13060547623, 11.1667541262), z, t), a), b) / fma(fma(fma((15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)));
	} else {
		tmp = t_1;
	}
	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(x + Float64(3.13060547623 * y))
	tmp = 0.0
	if (z <= -5.4e+46)
		tmp = t_1;
	elseif (z <= 4.4e+63)
		tmp = Float64(x + Float64(y * Float64(fma(z, fma(z, fma(fma(z, 3.13060547623, 11.1667541262), z, t), a), b) / fma(fma(fma(Float64(15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771))));
	else
		tmp = t_1;
	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[(x + N[(3.13060547623 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.4e+46], t$95$1, If[LessEqual[z, 4.4e+63], N[(x + N[(y * N[(N[(z * N[(z * N[(N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision] / N[(N[(N[(N[(15.234687407 + z), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	29.8
Target	1.1
Herbie	2.0

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

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000003e46 or 4.3999999999999997e63 < z

   1. Initial program 61.5

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

   2. Taylor expanded in z around inf 2.6

      \[\leadsto x + \color{blue}{3.13060547623 \cdot y} \]

   if -5.4000000000000003e46 < z < 4.3999999999999997e63

   1. Initial program 3.5

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

   2. Applied egg-rr 1.5

      \[\leadsto x + \color{blue}{y \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), z, t\right), a\right), b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	2.6
Cost	16772

\[\begin{array}{l} t_1 := \frac{y \cdot 36.52704169880642}{z}\\ t_2 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;x + \left(\mathsf{fma}\left(3.13060547623, y, \frac{{z}^{-1}}{z} \cdot \left(y \cdot \left(t - 98.5170599679272\right) - \left(y \cdot 36.52704169880642\right) \cdot -15.234687407\right)\right) - t_1\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;x + t_2\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y - t_1\right)\\ \end{array} \]

Alternative 2
Error	2.6
Cost	11268

\[\begin{array}{l} t_1 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \left(\left(11.1667541262 \cdot \frac{y}{z} + \left(3.13060547623 \cdot y + \left(y \cdot t - \left(15.234687407 \cdot \left(11.1667541262 \cdot y - 47.69379582500642 \cdot y\right) + 98.5170599679272 \cdot y\right)\right) \cdot {\left(\frac{1}{z}\right)}^{2}\right)\right) - 47.69379582500642 \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y - \frac{y \cdot 36.52704169880642}{z}\right)\\ \end{array} \]

Alternative 3
Error	2.7
Cost	6984

\[\begin{array}{l} t_1 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + 3.13060547623 \cdot y\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y - \frac{y \cdot 36.52704169880642}{z}\right)\\ \end{array} \]

Alternative 4
Error	3.1
Cost	2376

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+16}:\\ \;\;\;\;x + \left(\left(11.1667541262 \cdot \frac{y}{z} + 3.13060547623 \cdot y\right) - 47.69379582500642 \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(11.1667541262 \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y - \frac{y \cdot 36.52704169880642}{z}\right)\\ \end{array} \]

Alternative 5
Error	3.6
Cost	1992

\[\begin{array}{l} \mathbf{if}\;z \leq -0.41:\\ \;\;\;\;x + \left(\left(11.1667541262 \cdot \frac{y}{z} + 3.13060547623 \cdot y\right) - 47.69379582500642 \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 5300:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{11.9400905721 \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y - \frac{y \cdot 36.52704169880642}{z}\right)\\ \end{array} \]

Alternative 6
Error	5.1
Cost	1864

\[\begin{array}{l} \mathbf{if}\;z \leq -64000000000:\\ \;\;\;\;x + \left(\left(11.1667541262 \cdot \frac{y}{z} + 3.13060547623 \cdot y\right) - 47.69379582500642 \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{y \cdot \left(b + a \cdot z\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y - \frac{y \cdot 36.52704169880642}{z}\right)\\ \end{array} \]

Alternative 7
Error	7.5
Cost	1608

\[\begin{array}{l} \mathbf{if}\;z \leq -4100000000:\\ \;\;\;\;x + \left(\left(11.1667541262 \cdot \frac{y}{z} + 3.13060547623 \cdot y\right) - 47.69379582500642 \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 12.5:\\ \;\;\;\;x + \left(\left(1.6453555072203998 \cdot \left(a \cdot y\right) - 32.324150453290734 \cdot \left(y \cdot b\right)\right) \cdot z + 1.6453555072203998 \cdot \left(y \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y - \frac{y \cdot 36.52704169880642}{z}\right)\\ \end{array} \]

Alternative 8
Error	8.9
Cost	1224

\[\begin{array}{l} t_1 := x + \left(3.13060547623 \cdot y - \frac{y \cdot 36.52704169880642}{z}\right)\\ \mathbf{if}\;z \leq -350000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;x + \left(-32.324150453290734 \cdot \left(y \cdot \left(b \cdot z\right)\right) + 1.6453555072203998 \cdot \left(y \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	8.9
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -240000000:\\ \;\;\;\;x + \left(\left(11.1667541262 \cdot \frac{y}{z} + 3.13060547623 \cdot y\right) - 47.69379582500642 \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;x + \left(-32.324150453290734 \cdot \left(y \cdot \left(b \cdot z\right)\right) + 1.6453555072203998 \cdot \left(y \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y - \frac{y \cdot 36.52704169880642}{z}\right)\\ \end{array} \]

Alternative 10
Error	9.0
Cost	968

\[\begin{array}{l} t_1 := x + \left(3.13060547623 \cdot y - \frac{y \cdot 36.52704169880642}{z}\right)\\ \mathbf{if}\;z \leq -240000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;x + \left(1.6453555072203998 \cdot b\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	9.0
Cost	712

\[\begin{array}{l} t_1 := x + 3.13060547623 \cdot y\\ \mathbf{if}\;z \leq -240000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;x + \left(1.6453555072203998 \cdot b\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	27.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-67}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-123}:\\ \;\;\;\;3.13060547623 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	19.4
Cost	320

\[x + 3.13060547623 \cdot y \]

Alternative 14
Error	31.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023033 
(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))))