Average Error: 19.8 → 0.2
Time: 16.2s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -220354992.6475144922733306884765625 \lor \neg \left(z \le 7.619103618905890033200650757194694051577 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}
↓
\begin{array}{l}
\mathbf{if}\;z \le -220354992.6475144922733306884765625 \lor \neg \left(z \le 7.619103618905890033200650757194694051577 \cdot 10^{-5}\right):\\
\;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\

\end{array}
double f(double x, double y, double z) {
        double r227448 = x;
        double r227449 = y;
        double r227450 = z;
        double r227451 = 0.0692910599291889;
        double r227452 = r227450 * r227451;
        double r227453 = 0.4917317610505968;
        double r227454 = r227452 + r227453;
        double r227455 = r227454 * r227450;
        double r227456 = 0.279195317918525;
        double r227457 = r227455 + r227456;
        double r227458 = r227449 * r227457;
        double r227459 = 6.012459259764103;
        double r227460 = r227450 + r227459;
        double r227461 = r227460 * r227450;
        double r227462 = 3.350343815022304;
        double r227463 = r227461 + r227462;
        double r227464 = r227458 / r227463;
        double r227465 = r227448 + r227464;
        return r227465;
}
↓
double f(double x, double y, double z) {
        double r227466 = z;
        double r227467 = -220354992.6475145;
        bool r227468 = r227466 <= r227467;
        double r227469 = 7.61910361890589e-05;
        bool r227470 = r227466 <= r227469;
        double r227471 = !r227470;
        bool r227472 = r227468 || r227471;
        double r227473 = y;
        double r227474 = 0.0692910599291889;
        double r227475 = r227473 / r227466;
        double r227476 = 0.07512208616047561;
        double r227477 = x;
        double r227478 = fma(r227475, r227476, r227477);
        double r227479 = fma(r227473, r227474, r227478);
        double r227480 = r227466 * r227474;
        double r227481 = 0.4917317610505968;
        double r227482 = r227480 + r227481;
        double r227483 = r227482 * r227466;
        double r227484 = 0.279195317918525;
        double r227485 = r227483 + r227484;
        double r227486 = r227473 * r227485;
        double r227487 = 6.012459259764103;
        double r227488 = r227466 + r227487;
        double r227489 = r227488 * r227466;
        double r227490 = 3.350343815022304;
        double r227491 = r227489 + r227490;
        double r227492 = r227486 / r227491;
        double r227493 = r227477 + r227492;
        double r227494 = r227472 ? r227479 : r227493;
        return r227494;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.8
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -220354992.6475145 or 7.61910361890589e-05 < z

    1. Initial program 39.9

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

    2. Simplified33.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)}\]

    3. Taylor expanded around 0 33.5

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \color{blue}{0.4917317610505967939715787906607147306204 \cdot z + \left(0.06929105992918889456166908757950295694172 \cdot {z}^{2} + 0.2791953179185249767080279070796677842736\right)}, x\right)\]

    4. Simplified33.5

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), 0.2791953179185249767080279070796677842736\right)}, x\right)\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt33.5

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \color{blue}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), 0.2791953179185249767080279070796677842736\right)} \cdot \sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), 0.2791953179185249767080279070796677842736\right)}}, x\right)\]

    7. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{x + \left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right)}\]

    8. Simplified0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)}\]

    if -220354992.6475145 < z < 7.61910361890589e-05

    1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -220354992.6475144922733306884765625 \lor \neg \left(z \le 7.619103618905890033200650757194694051577 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))