Average Error: 1.9 → 0.2
Time: 24.6s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\mathsf{fma}\left(y, \log z - t, \mathsf{fma}\left(\sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, -b \cdot 1\right) \cdot a\right) + a \cdot \mathsf{fma}\left(-b, 1, b \cdot 1\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\mathsf{fma}\left(y, \log z - t, \mathsf{fma}\left(\sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, -b \cdot 1\right) \cdot a\right) + a \cdot \mathsf{fma}\left(-b, 1, b \cdot 1\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r389 = x;
        double r390 = y;
        double r391 = z;
        double r392 = log(r391);
        double r393 = t;
        double r394 = r392 - r393;
        double r395 = r390 * r394;
        double r396 = a;
        double r397 = 1.0;
        double r398 = r397 - r391;
        double r399 = log(r398);
        double r400 = b;
        double r401 = r399 - r400;
        double r402 = r396 * r401;
        double r403 = r395 + r402;
        double r404 = exp(r403);
        double r405 = r389 * r404;
        return r405;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r406 = x;
        double r407 = y;
        double r408 = z;
        double r409 = log(r408);
        double r410 = t;
        double r411 = r409 - r410;
        double r412 = 1.0;
        double r413 = log(r412);
        double r414 = 0.5;
        double r415 = 2.0;
        double r416 = pow(r408, r415);
        double r417 = pow(r412, r415);
        double r418 = r416 / r417;
        double r419 = r414 * r418;
        double r420 = r412 * r408;
        double r421 = r419 + r420;
        double r422 = r413 - r421;
        double r423 = cbrt(r422);
        double r424 = r423 * r423;
        double r425 = b;
        double r426 = 1.0;
        double r427 = r425 * r426;
        double r428 = -r427;
        double r429 = fma(r424, r423, r428);
        double r430 = a;
        double r431 = r429 * r430;
        double r432 = fma(r407, r411, r431);
        double r433 = -r425;
        double r434 = fma(r433, r426, r427);
        double r435 = r430 * r434;
        double r436 = r432 + r435;
        double r437 = exp(r436);
        double r438 = r406 * r437;
        return r438;
}

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

Derivation

  1. Initial program 1.9

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]

  2. Taylor expanded around 0 0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right)} - b\right)}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - \color{blue}{1 \cdot b}\right)}\]

  5. Applied add-cube-cbrt0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}\right) \cdot \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}} - 1 \cdot b\right)}\]

  6. Applied prod-diff0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, -b \cdot 1\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)\right)}}\]

  7. Applied distribute-lft-in0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(a \cdot \mathsf{fma}\left(\sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, -b \cdot 1\right) + a \cdot \mathsf{fma}\left(-b, 1, b \cdot 1\right)\right)}}\]

  8. Applied associate-+r+0.5

    \[\leadsto x \cdot e^{\color{blue}{\left(y \cdot \left(\log z - t\right) + a \cdot \mathsf{fma}\left(\sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, -b \cdot 1\right)\right) + a \cdot \mathsf{fma}\left(-b, 1, b \cdot 1\right)}}\]

  9. Simplified0.2

    \[\leadsto x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z - t, \mathsf{fma}\left(\sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, -b \cdot 1\right) \cdot a\right)} + a \cdot \mathsf{fma}\left(-b, 1, b \cdot 1\right)}\]

  10. Final simplification0.2

    \[\leadsto x \cdot e^{\mathsf{fma}\left(y, \log z - t, \mathsf{fma}\left(\sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, \sqrt[3]{\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)}, -b \cdot 1\right) \cdot a\right) + a \cdot \mathsf{fma}\left(-b, 1, b \cdot 1\right)}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1 z)) b))))))