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

Average Error: 9.5 → 0.4

Time: 1.2m

Precision: 64

Math

\[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

↓

\[\left(\left(x \cdot \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot x + \log \left(\sqrt[3]{y}\right) \cdot x\right)\right) + z \cdot \left(\left(\log 1.0 - \frac{y}{1.0} \cdot \left(\frac{1}{2} \cdot \frac{y}{1.0}\right)\right) - 1.0 \cdot y\right)\right) - t\]

C

double f(double x, double y, double z, double t) {
        double r18085912 = x;
        double r18085913 = y;
        double r18085914 = log(r18085913);
        double r18085915 = r18085912 * r18085914;
        double r18085916 = z;
        double r18085917 = 1.0;
        double r18085918 = r18085917 - r18085913;
        double r18085919 = log(r18085918);
        double r18085920 = r18085916 * r18085919;
        double r18085921 = r18085915 + r18085920;
        double r18085922 = t;
        double r18085923 = r18085921 - r18085922;
        return r18085923;
}

↓

double f(double x, double y, double z, double t) {
        double r18085924 = x;
        double r18085925 = y;
        double r18085926 = cbrt(r18085925);
        double r18085927 = cbrt(r18085926);
        double r18085928 = r18085927 * r18085927;
        double r18085929 = r18085928 * r18085927;
        double r18085930 = log(r18085929);
        double r18085931 = r18085924 * r18085930;
        double r18085932 = log(r18085926);
        double r18085933 = r18085932 * r18085924;
        double r18085934 = r18085933 + r18085933;
        double r18085935 = r18085931 + r18085934;
        double r18085936 = z;
        double r18085937 = 1.0;
        double r18085938 = log(r18085937);
        double r18085939 = r18085925 / r18085937;
        double r18085940 = 0.5;
        double r18085941 = r18085940 * r18085939;
        double r18085942 = r18085939 * r18085941;
        double r18085943 = r18085938 - r18085942;
        double r18085944 = r18085937 * r18085925;
        double r18085945 = r18085943 - r18085944;
        double r18085946 = r18085936 * r18085945;
        double r18085947 = r18085935 + r18085946;
        double r18085948 = t;
        double r18085949 = r18085947 - r18085948;
        return r18085949;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.5
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333}{1.0 \cdot \left(1.0 \cdot 1.0\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.5

   \[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

2. Taylor expanded around 0 0.4

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

3. Simplified 0.4

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\left(\log 1.0 - \left(\frac{1}{2} \cdot \frac{y}{1.0}\right) \cdot \frac{y}{1.0}\right) - 1.0 \cdot y\right)}\right) - t\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.4

   \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z \cdot \left(\left(\log 1.0 - \left(\frac{1}{2} \cdot \frac{y}{1.0}\right) \cdot \frac{y}{1.0}\right) - 1.0 \cdot y\right)\right) - t\]

6. Applied log-prod 0.4

   \[\leadsto \left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\left(\log 1.0 - \left(\frac{1}{2} \cdot \frac{y}{1.0}\right) \cdot \frac{y}{1.0}\right) - 1.0 \cdot y\right)\right) - t\]

7. Applied distribute-lft-in 0.4

   \[\leadsto \left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\left(\log 1.0 - \left(\frac{1}{2} \cdot \frac{y}{1.0}\right) \cdot \frac{y}{1.0}\right) - 1.0 \cdot y\right)\right) - t\]

8. Simplified 0.4

   \[\leadsto \left(\left(\color{blue}{\left(\log \left(\sqrt[3]{y}\right) \cdot x + \log \left(\sqrt[3]{y}\right) \cdot x\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\left(\log 1.0 - \left(\frac{1}{2} \cdot \frac{y}{1.0}\right) \cdot \frac{y}{1.0}\right) - 1.0 \cdot y\right)\right) - t\]

9. Simplified 0.4

   \[\leadsto \left(\left(\left(\log \left(\sqrt[3]{y}\right) \cdot x + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \color{blue}{\log \left(\sqrt[3]{y}\right) \cdot x}\right) + z \cdot \left(\left(\log 1.0 - \left(\frac{1}{2} \cdot \frac{y}{1.0}\right) \cdot \frac{y}{1.0}\right) - 1.0 \cdot y\right)\right) - t\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.4

    \[\leadsto \left(\left(\left(\log \left(\sqrt[3]{y}\right) \cdot x + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)} \cdot x\right) + z \cdot \left(\left(\log 1.0 - \left(\frac{1}{2} \cdot \frac{y}{1.0}\right) \cdot \frac{y}{1.0}\right) - 1.0 \cdot y\right)\right) - t\]

12. Final simplification 0.4

    \[\leadsto \left(\left(x \cdot \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot x + \log \left(\sqrt[3]{y}\right) \cdot x\right)\right) + z \cdot \left(\left(\log 1.0 - \frac{y}{1.0} \cdot \left(\frac{1}{2} \cdot \frac{y}{1.0}\right)\right) - 1.0 \cdot y\right)\right) - t\]

Reproduce

herbie shell --seed 2019165 
(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))