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

Average Error: 29.6 → 1.7

Time: 22.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(a + \left(t + \left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z\right) \cdot z\right) \cdot z + b\right)}{0.6077713877710000378584709324059076607227 + z \cdot \left(11.94009057210000079862766142468899488449 + z \cdot \left(31.46901157490000144889563671313226222992 + \left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right)\right)} \le 4.483865611994518803897357548829768572791 \cdot 10^{294}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\sqrt[3]{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}}}{\sqrt[3]{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{t}{z}}{z}} \cdot \sqrt[3]{\frac{\frac{t}{z}}{z}}, \sqrt[3]{\frac{\frac{t}{z}}{z}}, 3.130605476229999961645944495103321969509\right), x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r15367297 = x;
        double r15367298 = y;
        double r15367299 = z;
        double r15367300 = 3.13060547623;
        double r15367301 = r15367299 * r15367300;
        double r15367302 = 11.1667541262;
        double r15367303 = r15367301 + r15367302;
        double r15367304 = r15367303 * r15367299;
        double r15367305 = t;
        double r15367306 = r15367304 + r15367305;
        double r15367307 = r15367306 * r15367299;
        double r15367308 = a;
        double r15367309 = r15367307 + r15367308;
        double r15367310 = r15367309 * r15367299;
        double r15367311 = b;
        double r15367312 = r15367310 + r15367311;
        double r15367313 = r15367298 * r15367312;
        double r15367314 = 15.234687407;
        double r15367315 = r15367299 + r15367314;
        double r15367316 = r15367315 * r15367299;
        double r15367317 = 31.4690115749;
        double r15367318 = r15367316 + r15367317;
        double r15367319 = r15367318 * r15367299;
        double r15367320 = 11.9400905721;
        double r15367321 = r15367319 + r15367320;
        double r15367322 = r15367321 * r15367299;
        double r15367323 = 0.607771387771;
        double r15367324 = r15367322 + r15367323;
        double r15367325 = r15367313 / r15367324;
        double r15367326 = r15367297 + r15367325;
        return r15367326;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r15367327 = y;
        double r15367328 = a;
        double r15367329 = t;
        double r15367330 = 3.13060547623;
        double r15367331 = z;
        double r15367332 = r15367330 * r15367331;
        double r15367333 = 11.1667541262;
        double r15367334 = r15367332 + r15367333;
        double r15367335 = r15367334 * r15367331;
        double r15367336 = r15367329 + r15367335;
        double r15367337 = r15367336 * r15367331;
        double r15367338 = r15367328 + r15367337;
        double r15367339 = r15367338 * r15367331;
        double r15367340 = b;
        double r15367341 = r15367339 + r15367340;
        double r15367342 = r15367327 * r15367341;
        double r15367343 = 0.607771387771;
        double r15367344 = 11.9400905721;
        double r15367345 = 31.4690115749;
        double r15367346 = 15.234687407;
        double r15367347 = r15367331 + r15367346;
        double r15367348 = r15367347 * r15367331;
        double r15367349 = r15367345 + r15367348;
        double r15367350 = r15367331 * r15367349;
        double r15367351 = r15367344 + r15367350;
        double r15367352 = r15367331 * r15367351;
        double r15367353 = r15367343 + r15367352;
        double r15367354 = r15367342 / r15367353;
        double r15367355 = 4.483865611994519e+294;
        bool r15367356 = r15367354 <= r15367355;
        double r15367357 = fma(r15367330, r15367331, r15367333);
        double r15367358 = fma(r15367357, r15367331, r15367329);
        double r15367359 = fma(r15367331, r15367358, r15367328);
        double r15367360 = fma(r15367359, r15367331, r15367340);
        double r15367361 = fma(r15367331, r15367347, r15367345);
        double r15367362 = fma(r15367331, r15367361, r15367344);
        double r15367363 = fma(r15367331, r15367362, r15367343);
        double r15367364 = cbrt(r15367363);
        double r15367365 = r15367364 * r15367364;
        double r15367366 = r15367360 / r15367365;
        double r15367367 = r15367366 / r15367364;
        double r15367368 = x;
        double r15367369 = fma(r15367327, r15367367, r15367368);
        double r15367370 = r15367329 / r15367331;
        double r15367371 = r15367370 / r15367331;
        double r15367372 = cbrt(r15367371);
        double r15367373 = r15367372 * r15367372;
        double r15367374 = fma(r15367373, r15367372, r15367330);
        double r15367375 = fma(r15367327, r15367374, r15367368);
        double r15367376 = r15367356 ? r15367369 : r15367375;
        return r15367376;
}

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	29.6
Target	1.2
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771)) < 4.483865611994519e+294

   1. Initial program 2.9

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

   2. Simplified 1.4

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

   4. Applied add-cube-cbrt 1.5

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

   5. Applied associate-/r* 1.5

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

   if 4.483865611994519e+294 < (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))

   1. Initial program 63.4

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

   2. Simplified 61.3

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

   3. Taylor expanded around inf 10.0

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

   4. Simplified 1.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.130605476229999961645944495103321969509, x\right)}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 1.9

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\sqrt[3]{\frac{\frac{t}{z}}{z}} \cdot \sqrt[3]{\frac{\frac{t}{z}}{z}}\right) \cdot \sqrt[3]{\frac{\frac{t}{z}}{z}}} + 3.130605476229999961645944495103321969509, x\right)\]

   7. Applied fma-def 1.9

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{\frac{t}{z}}{z}} \cdot \sqrt[3]{\frac{\frac{t}{z}}{z}}, \sqrt[3]{\frac{\frac{t}{z}}{z}}, 3.130605476229999961645944495103321969509\right)}, x\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(a + \left(t + \left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z\right) \cdot z\right) \cdot z + b\right)}{0.6077713877710000378584709324059076607227 + z \cdot \left(11.94009057210000079862766142468899488449 + z \cdot \left(31.46901157490000144889563671313226222992 + \left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right)\right)} \le 4.483865611994518803897357548829768572791 \cdot 10^{294}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\sqrt[3]{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}}}{\sqrt[3]{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{t}{z}}{z}} \cdot \sqrt[3]{\frac{\frac{t}{z}}{z}}, \sqrt[3]{\frac{\frac{t}{z}}{z}}, 3.130605476229999961645944495103321969509\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"

  :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))))