Average Error: 29.0 → 1.2
Time: 5.4s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -9.1364780761282189 \cdot 10^{32}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{1}{\left|z\right|} \cdot \frac{t}{\left|z\right|}, x\right)\\ \mathbf{elif}\;z \le 1.54616507477103977 \cdot 10^{25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{\frac{t}{z}}{z}, x\right)\\ \end{array}\]
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}\;z \le -9.1364780761282189 \cdot 10^{32}:\\
\;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{1}{\left|z\right|} \cdot \frac{t}{\left|z\right|}, x\right)\\

\mathbf{elif}\;z \le 1.54616507477103977 \cdot 10^{25}:\\
\;\;\;\;\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)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r319399 = x;
        double r319400 = y;
        double r319401 = z;
        double r319402 = 3.13060547623;
        double r319403 = r319401 * r319402;
        double r319404 = 11.1667541262;
        double r319405 = r319403 + r319404;
        double r319406 = r319405 * r319401;
        double r319407 = t;
        double r319408 = r319406 + r319407;
        double r319409 = r319408 * r319401;
        double r319410 = a;
        double r319411 = r319409 + r319410;
        double r319412 = r319411 * r319401;
        double r319413 = b;
        double r319414 = r319412 + r319413;
        double r319415 = r319400 * r319414;
        double r319416 = 15.234687407;
        double r319417 = r319401 + r319416;
        double r319418 = r319417 * r319401;
        double r319419 = 31.4690115749;
        double r319420 = r319418 + r319419;
        double r319421 = r319420 * r319401;
        double r319422 = 11.9400905721;
        double r319423 = r319421 + r319422;
        double r319424 = r319423 * r319401;
        double r319425 = 0.607771387771;
        double r319426 = r319424 + r319425;
        double r319427 = r319415 / r319426;
        double r319428 = r319399 + r319427;
        return r319428;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r319429 = z;
        double r319430 = -9.136478076128219e+32;
        bool r319431 = r319429 <= r319430;
        double r319432 = y;
        double r319433 = 3.13060547623;
        double r319434 = 1.0;
        double r319435 = fabs(r319429);
        double r319436 = r319434 / r319435;
        double r319437 = t;
        double r319438 = r319437 / r319435;
        double r319439 = r319436 * r319438;
        double r319440 = r319433 + r319439;
        double r319441 = x;
        double r319442 = fma(r319432, r319440, r319441);
        double r319443 = 1.5461650747710398e+25;
        bool r319444 = r319429 <= r319443;
        double r319445 = 15.234687407;
        double r319446 = r319429 + r319445;
        double r319447 = 31.4690115749;
        double r319448 = fma(r319446, r319429, r319447);
        double r319449 = 11.9400905721;
        double r319450 = fma(r319448, r319429, r319449);
        double r319451 = 0.607771387771;
        double r319452 = fma(r319450, r319429, r319451);
        double r319453 = r319432 / r319452;
        double r319454 = 11.1667541262;
        double r319455 = fma(r319429, r319433, r319454);
        double r319456 = fma(r319455, r319429, r319437);
        double r319457 = a;
        double r319458 = fma(r319456, r319429, r319457);
        double r319459 = b;
        double r319460 = fma(r319458, r319429, r319459);
        double r319461 = fma(r319453, r319460, r319441);
        double r319462 = r319437 / r319429;
        double r319463 = r319462 / r319429;
        double r319464 = r319433 + r319463;
        double r319465 = fma(r319432, r319464, r319441);
        double r319466 = r319444 ? r319461 : r319465;
        double r319467 = r319431 ? r319442 : r319466;
        return r319467;
}

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

Original29.0
Target1.0
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -9.136478076128219e+32

    1. Initial program 59.2

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

    2. Simplified56.9

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

    3. Taylor expanded around inf 8.7

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

    4. Simplified1.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt1.7

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

    7. Applied *-un-lft-identity1.7

      \[\leadsto \mathsf{fma}\left(y, 3.13060547622999996 + \frac{\color{blue}{1 \cdot t}}{\sqrt{{z}^{2}} \cdot \sqrt{{z}^{2}}}, x\right)\]

    8. Applied times-frac1.7

      \[\leadsto \mathsf{fma}\left(y, 3.13060547622999996 + \color{blue}{\frac{1}{\sqrt{{z}^{2}}} \cdot \frac{t}{\sqrt{{z}^{2}}}}, x\right)\]

    9. Simplified1.7

      \[\leadsto \mathsf{fma}\left(y, 3.13060547622999996 + \color{blue}{\frac{1}{\left|z\right|}} \cdot \frac{t}{\sqrt{{z}^{2}}}, x\right)\]

    10. Simplified1.7

      \[\leadsto \mathsf{fma}\left(y, 3.13060547622999996 + \frac{1}{\left|z\right|} \cdot \color{blue}{\frac{t}{\left|z\right|}}, x\right)\]

    if -9.136478076128219e+32 < z < 1.5461650747710398e+25

    1. Initial program 0.8

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

    2. Simplified0.5

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

    if 1.5461650747710398e+25 < z

    1. Initial program 58.1

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

    2. Simplified55.4

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

    3. Taylor expanded around inf 9.1

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

    4. Simplified2.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)}\]
    5. Using strategy rm

    6. Applied unpow22.2

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

    7. Applied associate-/r*2.3

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.1364780761282189 \cdot 10^{32}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{1}{\left|z\right|} \cdot \frac{t}{\left|z\right|}, x\right)\\ \mathbf{elif}\;z \le 1.54616507477103977 \cdot 10^{25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{\frac{t}{z}}{z}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))))