Average Error: 29.4 → 1.9
Time: 11.0s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004} \le 6.72312118170133423 \cdot 10^{275}:\\ \;\;\;\;\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), b\right)\right) \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004} \le 6.72312118170133423 \cdot 10^{275}:\\
\;\;\;\;\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), b\right)\right) \cdot y + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r313410 = x;
        double r313411 = y;
        double r313412 = z;
        double r313413 = 3.13060547623;
        double r313414 = r313412 * r313413;
        double r313415 = 11.1667541262;
        double r313416 = r313414 + r313415;
        double r313417 = r313416 * r313412;
        double r313418 = t;
        double r313419 = r313417 + r313418;
        double r313420 = r313419 * r313412;
        double r313421 = a;
        double r313422 = r313420 + r313421;
        double r313423 = r313422 * r313412;
        double r313424 = b;
        double r313425 = r313423 + r313424;
        double r313426 = r313411 * r313425;
        double r313427 = 15.234687407;
        double r313428 = r313412 + r313427;
        double r313429 = r313428 * r313412;
        double r313430 = 31.4690115749;
        double r313431 = r313429 + r313430;
        double r313432 = r313431 * r313412;
        double r313433 = 11.9400905721;
        double r313434 = r313432 + r313433;
        double r313435 = r313434 * r313412;
        double r313436 = 0.607771387771;
        double r313437 = r313435 + r313436;
        double r313438 = r313426 / r313437;
        double r313439 = r313410 + r313438;
        return r313439;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r313440 = y;
        double r313441 = z;
        double r313442 = 3.13060547623;
        double r313443 = r313441 * r313442;
        double r313444 = 11.1667541262;
        double r313445 = r313443 + r313444;
        double r313446 = r313445 * r313441;
        double r313447 = t;
        double r313448 = r313446 + r313447;
        double r313449 = r313448 * r313441;
        double r313450 = a;
        double r313451 = r313449 + r313450;
        double r313452 = r313451 * r313441;
        double r313453 = b;
        double r313454 = r313452 + r313453;
        double r313455 = r313440 * r313454;
        double r313456 = 15.234687407;
        double r313457 = r313441 + r313456;
        double r313458 = r313457 * r313441;
        double r313459 = 31.4690115749;
        double r313460 = r313458 + r313459;
        double r313461 = r313460 * r313441;
        double r313462 = 11.9400905721;
        double r313463 = r313461 + r313462;
        double r313464 = r313463 * r313441;
        double r313465 = 0.607771387771;
        double r313466 = r313464 + r313465;
        double r313467 = r313455 / r313466;
        double r313468 = 6.723121181701334e+275;
        bool r313469 = r313467 <= r313468;
        double r313470 = 1.0;
        double r313471 = fma(r313457, r313441, r313459);
        double r313472 = fma(r313471, r313441, r313462);
        double r313473 = fma(r313472, r313441, r313465);
        double r313474 = r313470 / r313473;
        double r313475 = fma(r313441, r313442, r313444);
        double r313476 = fma(r313475, r313441, r313447);
        double r313477 = fma(r313476, r313441, r313450);
        double r313478 = fma(r313441, r313477, r313453);
        double r313479 = r313474 * r313478;
        double r313480 = r313479 * r313440;
        double r313481 = x;
        double r313482 = r313480 + r313481;
        double r313483 = 2.0;
        double r313484 = pow(r313441, r313483);
        double r313485 = r313447 / r313484;
        double r313486 = r313442 + r313485;
        double r313487 = fma(r313440, r313486, r313481);
        double r313488 = r313469 ? r313482 : r313487;
        return r313488;
}

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

Original29.4
Target1.2
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771)) < 6.723121181701334e+275

    1. Initial program 3.2

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Simplified2.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt2.1

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}}, z, a\right), z, b\right), x\right)\]
    5. Using strategy rm

    6. Applied fma-udef2.1

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}, z, a\right), z, b\right) + x}\]

    7. Simplified2.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left({\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}\right)}^{3}, z, a\right), b\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}{y}}} + x\]
    8. Using strategy rm

    9. Applied associate-/r/1.5

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left({\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}\right)}^{3}, z, a\right), b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot y} + x\]
    10. Using strategy rm

    11. Applied *-un-lft-identity1.5

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left({\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}\right)}^{3}, z, a\right), b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot \color{blue}{\left(1 \cdot y\right)} + x\]

    12. Applied associate-*r*1.5

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left({\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}\right)}^{3}, z, a\right), b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot 1\right) \cdot y} + x\]

    13. Simplified1.5

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), b\right)\right)} \cdot y + x\]

    if 6.723121181701334e+275 < (/ (* 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 62.5

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Simplified60.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)}\]

    3. Taylor expanded around inf 10.4

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.13060547622999996 \cdot y\right)}\]

    4. Simplified2.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004} \le 6.72312118170133423 \cdot 10^{275}:\\ \;\;\;\;\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), b\right)\right) \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))))