Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Average Error: 9.3 → 0.5

Time: 23.9s

Precision: 64

Math

\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

↓

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

C

double f(double x, double y, double z, double t) {
        double r18112007 = x;
        double r18112008 = y;
        double r18112009 = log(r18112008);
        double r18112010 = r18112007 * r18112009;
        double r18112011 = z;
        double r18112012 = 1.0;
        double r18112013 = r18112012 - r18112008;
        double r18112014 = log(r18112013);
        double r18112015 = r18112011 * r18112014;
        double r18112016 = r18112010 + r18112015;
        double r18112017 = t;
        double r18112018 = r18112016 - r18112017;
        return r18112018;
}

↓

double f(double x, double y, double z, double t) {
        double r18112019 = z;
        double r18112020 = 1.0;
        double r18112021 = log(r18112020);
        double r18112022 = y;
        double r18112023 = r18112022 / r18112020;
        double r18112024 = r18112023 * r18112023;
        double r18112025 = 0.5;
        double r18112026 = r18112022 * r18112020;
        double r18112027 = fma(r18112024, r18112025, r18112026);
        double r18112028 = r18112021 - r18112027;
        double r18112029 = log(r18112022);
        double r18112030 = cbrt(r18112022);
        double r18112031 = log(r18112030);
        double r18112032 = r18112029 * r18112031;
        double r18112033 = r18112030 * r18112030;
        double r18112034 = log(r18112033);
        double r18112035 = r18112029 * r18112034;
        double r18112036 = r18112032 + r18112035;
        double r18112037 = cbrt(r18112036);
        double r18112038 = cbrt(r18112029);
        double r18112039 = x;
        double r18112040 = r18112038 * r18112039;
        double r18112041 = r18112037 * r18112040;
        double r18112042 = fma(r18112019, r18112028, r18112041);
        double r18112043 = t;
        double r18112044 = r18112042 - r18112043;
        return r18112044;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.3
Target	0.2
Herbie	0.5

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.3

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

2. Simplified 9.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \log \left(1 - y\right), \log y \cdot x\right) - t}\]

3. Taylor expanded around 0 0.3

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

4. Simplified 0.3

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{\log 1 - \mathsf{fma}\left(\frac{y}{1} \cdot \frac{y}{1}, \frac{1}{2}, y \cdot 1\right)}, \log y \cdot x\right) - t\]
5. Using strategy rm

6. Applied add-cube-cbrt 0.7

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

7. Applied associate-*l* 0.7

   \[\leadsto \mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(\frac{y}{1} \cdot \frac{y}{1}, \frac{1}{2}, y \cdot 1\right), \color{blue}{\left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right) \cdot \left(\sqrt[3]{\log y} \cdot x\right)}\right) - t\]
8. Using strategy rm

9. Applied cbrt-unprod 0.6

   \[\leadsto \mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(\frac{y}{1} \cdot \frac{y}{1}, \frac{1}{2}, y \cdot 1\right), \color{blue}{\sqrt[3]{\log y \cdot \log y}} \cdot \left(\sqrt[3]{\log y} \cdot x\right)\right) - t\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.6

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

12. Applied log-prod 0.6

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

13. Applied distribute-lft-in 0.5

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

14. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))