Average Error: 28.8 → 1.1
Time: 43.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.0597997332978914 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \mathbf{elif}\;z \le 8.317246802743899 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \end{array}\]
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}
\mathbf{if}\;z \le -1.0597997332978914 \cdot 10^{+45}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\

\mathbf{elif}\;z \le 8.317246802743899 \cdot 10^{+42}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r18035391 = x;
        double r18035392 = y;
        double r18035393 = z;
        double r18035394 = 3.13060547623;
        double r18035395 = r18035393 * r18035394;
        double r18035396 = 11.1667541262;
        double r18035397 = r18035395 + r18035396;
        double r18035398 = r18035397 * r18035393;
        double r18035399 = t;
        double r18035400 = r18035398 + r18035399;
        double r18035401 = r18035400 * r18035393;
        double r18035402 = a;
        double r18035403 = r18035401 + r18035402;
        double r18035404 = r18035403 * r18035393;
        double r18035405 = b;
        double r18035406 = r18035404 + r18035405;
        double r18035407 = r18035392 * r18035406;
        double r18035408 = 15.234687407;
        double r18035409 = r18035393 + r18035408;
        double r18035410 = r18035409 * r18035393;
        double r18035411 = 31.4690115749;
        double r18035412 = r18035410 + r18035411;
        double r18035413 = r18035412 * r18035393;
        double r18035414 = 11.9400905721;
        double r18035415 = r18035413 + r18035414;
        double r18035416 = r18035415 * r18035393;
        double r18035417 = 0.607771387771;
        double r18035418 = r18035416 + r18035417;
        double r18035419 = r18035407 / r18035418;
        double r18035420 = r18035391 + r18035419;
        return r18035420;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r18035421 = z;
        double r18035422 = -1.0597997332978914e+45;
        bool r18035423 = r18035421 <= r18035422;
        double r18035424 = y;
        double r18035425 = t;
        double r18035426 = r18035425 / r18035421;
        double r18035427 = r18035426 / r18035421;
        double r18035428 = 3.13060547623;
        double r18035429 = r18035427 + r18035428;
        double r18035430 = x;
        double r18035431 = fma(r18035424, r18035429, r18035430);
        double r18035432 = 8.317246802743899e+42;
        bool r18035433 = r18035421 <= r18035432;
        double r18035434 = 11.1667541262;
        double r18035435 = fma(r18035428, r18035421, r18035434);
        double r18035436 = fma(r18035435, r18035421, r18035425);
        double r18035437 = a;
        double r18035438 = fma(r18035421, r18035436, r18035437);
        double r18035439 = b;
        double r18035440 = fma(r18035438, r18035421, r18035439);
        double r18035441 = 15.234687407;
        double r18035442 = r18035441 + r18035421;
        double r18035443 = 31.4690115749;
        double r18035444 = fma(r18035421, r18035442, r18035443);
        double r18035445 = 11.9400905721;
        double r18035446 = fma(r18035421, r18035444, r18035445);
        double r18035447 = 0.607771387771;
        double r18035448 = fma(r18035421, r18035446, r18035447);
        double r18035449 = sqrt(r18035448);
        double r18035450 = sqrt(r18035449);
        double r18035451 = r18035440 / r18035450;
        double r18035452 = r18035451 / r18035450;
        double r18035453 = r18035452 / r18035449;
        double r18035454 = fma(r18035424, r18035453, r18035430);
        double r18035455 = r18035433 ? r18035454 : r18035431;
        double r18035456 = r18035423 ? r18035431 : r18035455;
        return r18035456;
}

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

Original28.8
Target1.0
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.0597997332978914e+45 or 8.317246802743899e+42 < z

    1. Initial program 58.8

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

    2. Simplified56.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}\]

    3. Taylor expanded around inf 7.9

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.13060547623 \cdot y\right)}\]

    4. Simplified1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623 + \frac{\frac{t}{z}}{z}, x\right)}\]

    if -1.0597997332978914e+45 < z < 8.317246802743899e+42

    1. Initial program 1.6

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

    2. Simplified0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt1.2

      \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \cdot \sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}, x\right)\]

    5. Applied associate-/r*1.1

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}, x\right)\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt1.1

      \[\leadsto \mathsf{fma}\left(y, \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \cdot \sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\]

    8. Applied sqrt-prod1.1

      \[\leadsto \mathsf{fma}\left(y, \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\]

    9. Applied associate-/r*1.0

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.0597997332978914 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \mathbf{elif}\;z \le 8.317246802743899 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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))) (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))))