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

Average Error: 29.5 → 0.9

Time: 22.8s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.292408313613421589028349220039609902128 \cdot 10^{44} \lor \neg \left(z \le 5.016237580989981461752426694136856045492 \cdot 10^{48}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right)}} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r203078 = x;
        double r203079 = y;
        double r203080 = z;
        double r203081 = 3.13060547623;
        double r203082 = r203080 * r203081;
        double r203083 = 11.1667541262;
        double r203084 = r203082 + r203083;
        double r203085 = r203084 * r203080;
        double r203086 = t;
        double r203087 = r203085 + r203086;
        double r203088 = r203087 * r203080;
        double r203089 = a;
        double r203090 = r203088 + r203089;
        double r203091 = r203090 * r203080;
        double r203092 = b;
        double r203093 = r203091 + r203092;
        double r203094 = r203079 * r203093;
        double r203095 = 15.234687407;
        double r203096 = r203080 + r203095;
        double r203097 = r203096 * r203080;
        double r203098 = 31.4690115749;
        double r203099 = r203097 + r203098;
        double r203100 = r203099 * r203080;
        double r203101 = 11.9400905721;
        double r203102 = r203100 + r203101;
        double r203103 = r203102 * r203080;
        double r203104 = 0.607771387771;
        double r203105 = r203103 + r203104;
        double r203106 = r203094 / r203105;
        double r203107 = r203078 + r203106;
        return r203107;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r203108 = z;
        double r203109 = -2.2924083136134216e+44;
        bool r203110 = r203108 <= r203109;
        double r203111 = 5.0162375809899815e+48;
        bool r203112 = r203108 <= r203111;
        double r203113 = !r203112;
        bool r203114 = r203110 || r203113;
        double r203115 = y;
        double r203116 = t;
        double r203117 = 2.0;
        double r203118 = pow(r203108, r203117);
        double r203119 = r203116 / r203118;
        double r203120 = 3.13060547623;
        double r203121 = r203119 + r203120;
        double r203122 = x;
        double r203123 = fma(r203115, r203121, r203122);
        double r203124 = 15.234687407;
        double r203125 = r203108 + r203124;
        double r203126 = 31.4690115749;
        double r203127 = fma(r203125, r203108, r203126);
        double r203128 = 11.9400905721;
        double r203129 = fma(r203127, r203108, r203128);
        double r203130 = 0.607771387771;
        double r203131 = fma(r203129, r203108, r203130);
        double r203132 = 11.1667541262;
        double r203133 = fma(r203108, r203120, r203132);
        double r203134 = fma(r203133, r203108, r203116);
        double r203135 = a;
        double r203136 = fma(r203134, r203108, r203135);
        double r203137 = b;
        double r203138 = fma(r203136, r203108, r203137);
        double r203139 = r203131 / r203138;
        double r203140 = r203115 / r203139;
        double r203141 = r203140 + r203122;
        double r203142 = r203114 ? r203123 : r203141;
        return r203142;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	29.5
Target	1.0
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -2.2924083136134216e+44 or 5.0162375809899815e+48 < z

   1. Initial program 60.8

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

   2. Simplified 59.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)}\]
   3. Using strategy rm

   4. Applied div-inv 59.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\]

   5. Taylor expanded around inf 8.2

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.130605476229999961645944495103321969509 \cdot y\right)}\]

   6. Simplified 1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509, x\right)}\]

   if -2.2924083136134216e+44 < z < 5.0162375809899815e+48

   1. Initial program 2.2

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

   2. Simplified 1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 1.3

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right)}}, z, b\right), x\right)\]
   5. Using strategy rm

   6. Applied fma-udef 1.3

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)} \cdot \mathsf{fma}\left(\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right)}, z, b\right) + x}\]

   7. Simplified 0.8

      \[\leadsto \color{blue}{\frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right)}}} + x\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.292408313613421589028349220039609902128 \cdot 10^{44} \lor \neg \left(z \le 5.016237580989981461752426694136856045492 \cdot 10^{48}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right)}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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))) (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)))))

  (+ 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))))