Average Error: 20.3 → 0.2
Time: 10.1s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.2730222216892933 \cdot 10^{26} \lor \neg \left(z \le 72489.2439122225187\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, 0.0692910599291888946 \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.2730222216892933 \cdot 10^{26} \lor \neg \left(z \le 72489.2439122225187\right):\\
\;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, 0.0692910599291888946 \cdot y\right) + x\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\

\end{array}
double f(double x, double y, double z) {
        double r449442 = x;
        double r449443 = y;
        double r449444 = z;
        double r449445 = 0.0692910599291889;
        double r449446 = r449444 * r449445;
        double r449447 = 0.4917317610505968;
        double r449448 = r449446 + r449447;
        double r449449 = r449448 * r449444;
        double r449450 = 0.279195317918525;
        double r449451 = r449449 + r449450;
        double r449452 = r449443 * r449451;
        double r449453 = 6.012459259764103;
        double r449454 = r449444 + r449453;
        double r449455 = r449454 * r449444;
        double r449456 = 3.350343815022304;
        double r449457 = r449455 + r449456;
        double r449458 = r449452 / r449457;
        double r449459 = r449442 + r449458;
        return r449459;
}
↓
double f(double x, double y, double z) {
        double r449460 = z;
        double r449461 = -1.2730222216892933e+26;
        bool r449462 = r449460 <= r449461;
        double r449463 = 72489.24391222252;
        bool r449464 = r449460 <= r449463;
        double r449465 = !r449464;
        bool r449466 = r449462 || r449465;
        double r449467 = 0.07512208616047561;
        double r449468 = y;
        double r449469 = r449468 / r449460;
        double r449470 = 0.0692910599291889;
        double r449471 = r449470 * r449468;
        double r449472 = fma(r449467, r449469, r449471);
        double r449473 = x;
        double r449474 = r449472 + r449473;
        double r449475 = r449460 * r449470;
        double r449476 = 0.4917317610505968;
        double r449477 = r449475 + r449476;
        double r449478 = r449477 * r449460;
        double r449479 = 0.279195317918525;
        double r449480 = r449478 + r449479;
        double r449481 = r449468 * r449480;
        double r449482 = 6.012459259764103;
        double r449483 = r449460 + r449482;
        double r449484 = r449483 * r449460;
        double r449485 = 3.350343815022304;
        double r449486 = r449484 + r449485;
        double r449487 = r449481 / r449486;
        double r449488 = r449473 + r449487;
        double r449489 = r449466 ? r449474 : r449488;
        return r449489;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.3
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.2730222216892933e+26 or 72489.24391222252 < z

    1. Initial program 42.3

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

    2. Simplified35.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt35.3

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{\sqrt{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)} \cdot \sqrt{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)\]

    5. Applied associate-/r*35.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{\sqrt{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}}}{\sqrt{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)\]

    6. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)}\]

    7. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, 0.0692910599291888946 \cdot y\right) + x}\]

    if -1.2730222216892933e+26 < z < 72489.24391222252

    1. Initial program 0.3

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.2730222216892933 \cdot 10^{26} \lor \neg \left(z \le 72489.2439122225187\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, 0.0692910599291888946 \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))))