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

Average Error: 29.2 → 1.1

Time: 25.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.038261497315181869126238031565883904229 \cdot 10^{47} \lor \neg \left(z \le 177470294739602374656\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right) \cdot z + t, z, a\right), z, b\right), x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r225310 = x;
        double r225311 = y;
        double r225312 = z;
        double r225313 = 3.13060547623;
        double r225314 = r225312 * r225313;
        double r225315 = 11.1667541262;
        double r225316 = r225314 + r225315;
        double r225317 = r225316 * r225312;
        double r225318 = t;
        double r225319 = r225317 + r225318;
        double r225320 = r225319 * r225312;
        double r225321 = a;
        double r225322 = r225320 + r225321;
        double r225323 = r225322 * r225312;
        double r225324 = b;
        double r225325 = r225323 + r225324;
        double r225326 = r225311 * r225325;
        double r225327 = 15.234687407;
        double r225328 = r225312 + r225327;
        double r225329 = r225328 * r225312;
        double r225330 = 31.4690115749;
        double r225331 = r225329 + r225330;
        double r225332 = r225331 * r225312;
        double r225333 = 11.9400905721;
        double r225334 = r225332 + r225333;
        double r225335 = r225334 * r225312;
        double r225336 = 0.607771387771;
        double r225337 = r225335 + r225336;
        double r225338 = r225326 / r225337;
        double r225339 = r225310 + r225338;
        return r225339;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r225340 = z;
        double r225341 = -7.038261497315182e+47;
        bool r225342 = r225340 <= r225341;
        double r225343 = 1.7747029473960237e+20;
        bool r225344 = r225340 <= r225343;
        double r225345 = !r225344;
        bool r225346 = r225342 || r225345;
        double r225347 = y;
        double r225348 = t;
        double r225349 = 2.0;
        double r225350 = pow(r225340, r225349);
        double r225351 = r225348 / r225350;
        double r225352 = 3.13060547623;
        double r225353 = r225351 + r225352;
        double r225354 = x;
        double r225355 = fma(r225347, r225353, r225354);
        double r225356 = 15.234687407;
        double r225357 = r225340 + r225356;
        double r225358 = 31.4690115749;
        double r225359 = fma(r225357, r225340, r225358);
        double r225360 = 11.9400905721;
        double r225361 = fma(r225359, r225340, r225360);
        double r225362 = 0.607771387771;
        double r225363 = fma(r225361, r225340, r225362);
        double r225364 = r225347 / r225363;
        double r225365 = 11.1667541262;
        double r225366 = fma(r225340, r225352, r225365);
        double r225367 = r225366 * r225340;
        double r225368 = r225367 + r225348;
        double r225369 = a;
        double r225370 = fma(r225368, r225340, r225369);
        double r225371 = b;
        double r225372 = fma(r225370, r225340, r225371);
        double r225373 = fma(r225364, r225372, r225354);
        double r225374 = r225346 ? r225355 : r225373;
        return r225374;
}

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	29.2
Target	0.9
Herbie	1.1

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

Derivation

1. Split input into 2 regimes

2. if z < -7.038261497315182e+47 or 1.7747029473960237e+20 < z

   1. Initial program 59.5

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

   2. Simplified 57.6

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

   4. Applied fma-udef 57.6

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

   6. Applied add-cube-cbrt 57.6

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

   7. Applied associate-/r* 57.6

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

   9. Applied add-cube-cbrt 57.6

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

   10. Taylor expanded around inf 9.2

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

   11. Simplified 1.6

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

   if -7.038261497315182e+47 < z < 1.7747029473960237e+20

   1. Initial program 1.2

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

   2. Simplified 0.8

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

   4. Applied fma-udef 0.8

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.038261497315181869126238031565883904229 \cdot 10^{47} \lor \neg \left(z \le 177470294739602374656\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right) \cdot z + t, z, a\right), z, b\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))