Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 25.9s

Precision: 64

Math

\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]

↓

\[\left(\left(y + \left(\left(z - \log \left({t}^{\frac{2}{3}}\right) \cdot z\right) - z \cdot \log \left(\sqrt[3]{t}\right)\right)\right) + x\right) + b \cdot \left(a - 0.5\right)\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r21487129 = x;
        double r21487130 = y;
        double r21487131 = r21487129 + r21487130;
        double r21487132 = z;
        double r21487133 = r21487131 + r21487132;
        double r21487134 = t;
        double r21487135 = log(r21487134);
        double r21487136 = r21487132 * r21487135;
        double r21487137 = r21487133 - r21487136;
        double r21487138 = a;
        double r21487139 = 0.5;
        double r21487140 = r21487138 - r21487139;
        double r21487141 = b;
        double r21487142 = r21487140 * r21487141;
        double r21487143 = r21487137 + r21487142;
        return r21487143;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r21487144 = y;
        double r21487145 = z;
        double r21487146 = t;
        double r21487147 = 0.6666666666666666;
        double r21487148 = pow(r21487146, r21487147);
        double r21487149 = log(r21487148);
        double r21487150 = r21487149 * r21487145;
        double r21487151 = r21487145 - r21487150;
        double r21487152 = cbrt(r21487146);
        double r21487153 = log(r21487152);
        double r21487154 = r21487145 * r21487153;
        double r21487155 = r21487151 - r21487154;
        double r21487156 = r21487144 + r21487155;
        double r21487157 = x;
        double r21487158 = r21487156 + r21487157;
        double r21487159 = b;
        double r21487160 = a;
        double r21487161 = 0.5;
        double r21487162 = r21487160 - r21487161;
        double r21487163 = r21487159 * r21487162;
        double r21487164 = r21487158 + r21487163;
        return r21487164;
}

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

Target

Original	0.1
Target	0.3
Herbie	0.1

\[\left(\left(x + y\right) + \frac{\left(1 - {\left(\log t\right)}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b\]

Derivation

1. Initial program 0.1

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]

2. Taylor expanded around inf 0.1

   \[\leadsto \color{blue}{\left(x + \left(z + \left(z \cdot \log \left(\frac{1}{t}\right) + y\right)\right)\right)} + \left(a - 0.5\right) \cdot b\]

3. Simplified 0.1

   \[\leadsto \color{blue}{\left(x + \left(\left(z - z \cdot \log t\right) + y\right)\right)} + \left(a - 0.5\right) \cdot b\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.1

   \[\leadsto \left(x + \left(\left(z - z \cdot \log \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}\right) + y\right)\right) + \left(a - 0.5\right) \cdot b\]

6. Applied log-prod 0.1

   \[\leadsto \left(x + \left(\left(z - z \cdot \color{blue}{\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)}\right) + y\right)\right) + \left(a - 0.5\right) \cdot b\]

7. Applied distribute-lft-in 0.1

   \[\leadsto \left(x + \left(\left(z - \color{blue}{\left(z \cdot \log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + z \cdot \log \left(\sqrt[3]{t}\right)\right)}\right) + y\right)\right) + \left(a - 0.5\right) \cdot b\]

8. Applied associate--r+ 0.1

   \[\leadsto \left(x + \left(\color{blue}{\left(\left(z - z \cdot \log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) - z \cdot \log \left(\sqrt[3]{t}\right)\right)} + y\right)\right) + \left(a - 0.5\right) \cdot b\]
9. Using strategy rm

10. Applied pow1/3 0.1

    \[\leadsto \left(x + \left(\left(\left(z - z \cdot \log \left(\sqrt[3]{t} \cdot \color{blue}{{t}^{\frac{1}{3}}}\right)\right) - z \cdot \log \left(\sqrt[3]{t}\right)\right) + y\right)\right) + \left(a - 0.5\right) \cdot b\]

11. Applied pow1/3 0.1

    \[\leadsto \left(x + \left(\left(\left(z - z \cdot \log \left(\color{blue}{{t}^{\frac{1}{3}}} \cdot {t}^{\frac{1}{3}}\right)\right) - z \cdot \log \left(\sqrt[3]{t}\right)\right) + y\right)\right) + \left(a - 0.5\right) \cdot b\]

12. Applied pow-prod-up 0.1

    \[\leadsto \left(x + \left(\left(\left(z - z \cdot \log \color{blue}{\left({t}^{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)}\right) - z \cdot \log \left(\sqrt[3]{t}\right)\right) + y\right)\right) + \left(a - 0.5\right) \cdot b\]

13. Simplified 0.1

    \[\leadsto \left(x + \left(\left(\left(z - z \cdot \log \left({t}^{\color{blue}{\frac{2}{3}}}\right)\right) - z \cdot \log \left(\sqrt[3]{t}\right)\right) + y\right)\right) + \left(a - 0.5\right) \cdot b\]

14. Final simplification 0.1

    \[\leadsto \left(\left(y + \left(\left(z - \log \left({t}^{\frac{2}{3}}\right) \cdot z\right) - z \cdot \log \left(\sqrt[3]{t}\right)\right)\right) + x\right) + b \cdot \left(a - 0.5\right)\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))