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

Average Error: 0.1 → 0.1

Time: 20.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r468048 = x;
        double r468049 = y;
        double r468050 = r468048 + r468049;
        double r468051 = z;
        double r468052 = r468050 + r468051;
        double r468053 = t;
        double r468054 = log(r468053);
        double r468055 = r468051 * r468054;
        double r468056 = r468052 - r468055;
        double r468057 = a;
        double r468058 = 0.5;
        double r468059 = r468057 - r468058;
        double r468060 = b;
        double r468061 = r468059 * r468060;
        double r468062 = r468056 + r468061;
        return r468062;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r468063 = x;
        double r468064 = y;
        double r468065 = r468063 + r468064;
        double r468066 = z;
        double r468067 = r468065 + r468066;
        double r468068 = 2.0;
        double r468069 = t;
        double r468070 = cbrt(r468069);
        double r468071 = log(r468070);
        double r468072 = r468068 * r468071;
        double r468073 = r468066 * r468072;
        double r468074 = 0.6666666666666666;
        double r468075 = pow(r468069, r468074);
        double r468076 = cbrt(r468075);
        double r468077 = log(r468076);
        double r468078 = r468066 * r468077;
        double r468079 = cbrt(r468070);
        double r468080 = log(r468079);
        double r468081 = r468066 * r468080;
        double r468082 = r468078 + r468081;
        double r468083 = r468073 + r468082;
        double r468084 = r468067 - r468083;
        double r468085 = a;
        double r468086 = 0.5;
        double r468087 = r468085 - r468086;
        double r468088 = b;
        double r468089 = r468087 * r468088;
        double r468090 = r468084 + r468089;
        return r468090;
}

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	0.4
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. Using strategy rm

3. Applied add-cube-cbrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Simplified 0.1

   \[\leadsto \left(\left(\left(x + y\right) + z\right) - \left(\color{blue}{z \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right)} + z \cdot \log \left(\sqrt[3]{t}\right)\right)\right) + \left(a - 0.5\right) \cdot b\]
7. Using strategy rm

8. Applied add-cube-cbrt 0.1

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

9. Applied cbrt-prod 0.1

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

10. Applied log-prod 0.1

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

11. Applied distribute-lft-in 0.1

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

12. Simplified 0.1

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

13. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :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)))