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

Average Error: 9.7 → 0.4

Time: 20.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r338086 = x;
        double r338087 = y;
        double r338088 = log(r338087);
        double r338089 = r338086 * r338088;
        double r338090 = z;
        double r338091 = 1.0;
        double r338092 = r338091 - r338087;
        double r338093 = log(r338092);
        double r338094 = r338090 * r338093;
        double r338095 = r338089 + r338094;
        double r338096 = t;
        double r338097 = r338095 - r338096;
        return r338097;
}

↓

double f(double x, double y, double z, double t) {
        double r338098 = y;
        double r338099 = cbrt(r338098);
        double r338100 = r338099 * r338099;
        double r338101 = log(r338100);
        double r338102 = x;
        double r338103 = r338101 * r338102;
        double r338104 = log(r338099);
        double r338105 = r338102 * r338104;
        double r338106 = 1.0;
        double r338107 = log(r338106);
        double r338108 = r338106 * r338098;
        double r338109 = r338107 - r338108;
        double r338110 = z;
        double r338111 = r338109 * r338110;
        double r338112 = r338105 + r338111;
        double r338113 = r338103 + r338112;
        double r338114 = r338098 * r338098;
        double r338115 = r338114 / r338106;
        double r338116 = -0.5;
        double r338117 = r338116 / r338106;
        double r338118 = r338115 * r338117;
        double r338119 = r338110 * r338118;
        double r338120 = r338113 + r338119;
        double r338121 = t;
        double r338122 = r338120 - r338121;
        return r338122;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.7
Target	0.3
Herbie	0.4

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

Derivation

1. Initial program 9.7

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

2. Taylor expanded around 0 0.4

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

3. Simplified 0.4

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

5. Applied sub-neg 0.4

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

6. Applied distribute-rgt-in 0.4

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

7. Applied associate-+r+ 0.4

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

8. Simplified 0.4

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

10. Applied add-cube-cbrt 0.4

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

11. Applied log-prod 0.4

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

12. Applied distribute-rgt-in 0.4

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

13. Applied associate-+l+ 0.4

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

14. Simplified 0.4

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

15. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019195 
(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))