Average Error: 2.0 → 1.3
Time: 38.4s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x}{\frac{y \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}\right)}{\frac{{a}^{\left(-1\right)}}{\sqrt[3]{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x}{\frac{y \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}\right)}{\frac{{a}^{\left(-1\right)}}{\sqrt[3]{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r81548 = x;
        double r81549 = y;
        double r81550 = z;
        double r81551 = log(r81550);
        double r81552 = r81549 * r81551;
        double r81553 = t;
        double r81554 = 1.0;
        double r81555 = r81553 - r81554;
        double r81556 = a;
        double r81557 = log(r81556);
        double r81558 = r81555 * r81557;
        double r81559 = r81552 + r81558;
        double r81560 = b;
        double r81561 = r81559 - r81560;
        double r81562 = exp(r81561);
        double r81563 = r81548 * r81562;
        double r81564 = r81563 / r81549;
        return r81564;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r81565 = x;
        double r81566 = y;
        double r81567 = z;
        double r81568 = log(r81567);
        double r81569 = -r81568;
        double r81570 = a;
        double r81571 = log(r81570);
        double r81572 = -r81571;
        double r81573 = t;
        double r81574 = b;
        double r81575 = fma(r81572, r81573, r81574);
        double r81576 = fma(r81566, r81569, r81575);
        double r81577 = exp(r81576);
        double r81578 = cbrt(r81577);
        double r81579 = r81578 * r81578;
        double r81580 = r81566 * r81579;
        double r81581 = 1.0;
        double r81582 = -r81581;
        double r81583 = pow(r81570, r81582);
        double r81584 = r81583 / r81578;
        double r81585 = r81580 / r81584;
        double r81586 = r81565 / r81585;
        return r81586;
}

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

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

  2. Taylor expanded around inf 2.0

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

  3. Simplified1.3

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

  5. Applied associate-/l*1.3

    \[\leadsto \color{blue}{\frac{x}{\frac{y}{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt1.3

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

  8. Applied *-un-lft-identity1.3

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

  9. Applied unpow-prod-down1.3

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

  10. Applied times-frac1.3

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

  11. Applied associate-/r*1.3

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

  12. Simplified1.3

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

  13. Final simplification1.3

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))