Average Error: 2.0 → 0.4
Time: 17.1s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \mathsf{fma}\left(\log \left(\sqrt[3]{z}\right) - t, y, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{y \cdot \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \mathsf{fma}\left(\log \left(\sqrt[3]{z}\right) - t, y, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r223495 = x;
        double r223496 = y;
        double r223497 = z;
        double r223498 = log(r223497);
        double r223499 = t;
        double r223500 = r223498 - r223499;
        double r223501 = r223496 * r223500;
        double r223502 = a;
        double r223503 = 1.0;
        double r223504 = r223503 - r223497;
        double r223505 = log(r223504);
        double r223506 = b;
        double r223507 = r223505 - r223506;
        double r223508 = r223502 * r223507;
        double r223509 = r223501 + r223508;
        double r223510 = exp(r223509);
        double r223511 = r223495 * r223510;
        return r223511;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r223512 = x;
        double r223513 = y;
        double r223514 = z;
        double r223515 = cbrt(r223514);
        double r223516 = r223515 * r223515;
        double r223517 = log(r223516);
        double r223518 = r223513 * r223517;
        double r223519 = log(r223515);
        double r223520 = t;
        double r223521 = r223519 - r223520;
        double r223522 = 1.0;
        double r223523 = log(r223522);
        double r223524 = 0.5;
        double r223525 = 2.0;
        double r223526 = pow(r223514, r223525);
        double r223527 = pow(r223522, r223525);
        double r223528 = r223526 / r223527;
        double r223529 = r223522 * r223514;
        double r223530 = fma(r223524, r223528, r223529);
        double r223531 = b;
        double r223532 = r223530 + r223531;
        double r223533 = r223523 - r223532;
        double r223534 = a;
        double r223535 = r223533 * r223534;
        double r223536 = fma(r223521, r223513, r223535);
        double r223537 = r223518 + r223536;
        double r223538 = exp(r223537);
        double r223539 = r223512 * r223538;
        return r223539;
}

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 2.0

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

  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 add-cube-cbrt0.5

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

  5. Applied log-prod0.5

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

  6. Applied associate--l+0.5

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

  7. Applied distribute-lft-in0.6

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

  8. Applied associate-+l+0.6

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020100 +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))))))