Average Error: 19.8 → 0.2
Time: 16.6s
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 r233461 = x;
        double r233462 = y;
        double r233463 = z;
        double r233464 = 0.0692910599291889;
        double r233465 = r233463 * r233464;
        double r233466 = 0.4917317610505968;
        double r233467 = r233465 + r233466;
        double r233468 = r233467 * r233463;
        double r233469 = 0.279195317918525;
        double r233470 = r233468 + r233469;
        double r233471 = r233462 * r233470;
        double r233472 = 6.012459259764103;
        double r233473 = r233463 + r233472;
        double r233474 = r233473 * r233463;
        double r233475 = 3.350343815022304;
        double r233476 = r233474 + r233475;
        double r233477 = r233471 / r233476;
        double r233478 = r233461 + r233477;
        return r233478;
}
↓
double f(double x, double y, double z) {
        double r233479 = z;
        double r233480 = -220354992.6475145;
        bool r233481 = r233479 <= r233480;
        double r233482 = 7.61910361890589e-05;
        bool r233483 = r233479 <= r233482;
        double r233484 = !r233483;
        bool r233485 = r233481 || r233484;
        double r233486 = y;
        double r233487 = 0.0692910599291889;
        double r233488 = r233486 / r233479;
        double r233489 = 0.07512208616047561;
        double r233490 = x;
        double r233491 = fma(r233488, r233489, r233490);
        double r233492 = fma(r233486, r233487, r233491);
        double r233493 = r233479 * r233487;
        double r233494 = 0.4917317610505968;
        double r233495 = r233493 + r233494;
        double r233496 = r233495 * r233479;
        double r233497 = 0.279195317918525;
        double r233498 = r233496 + r233497;
        double r233499 = r233486 * r233498;
        double r233500 = 6.012459259764103;
        double r233501 = r233479 + r233500;
        double r233502 = r233501 * r233479;
        double r233503 = 3.350343815022304;
        double r233504 = r233502 + r233503;
        double r233505 = r233499 / r233504;
        double r233506 = r233490 + r233505;
        double r233507 = r233485 ? r233492 : r233506;
        return r233507;
}

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))))