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

Average Error: 0.1 → 0.1

Time: 7.6s

Precision: 64

Math

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

↓

\[\left(\left(\left(\left(x + y\right) + z\right) - \log \left({t}^{\frac{1}{3}} \cdot \sqrt[3]{t}\right) \cdot z\right) - z \cdot \log \left({\left({t}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{t}\right)}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) + z) - (z * log(t))) + ((a - 0.5) * b));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	return (((((x + y) + z) - (log((pow(t, 0.3333333333333333) * cbrt(t))) * z)) - (z * log((pow(pow(t, 0.3333333333333333), 0.6666666666666666) * pow(cbrt(t), 0.3333333333333333))))) + ((a - 0.5) * b));
}

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. Applied associate--r+ 0.1

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

7. Simplified 0.1

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

9. Applied pow1/3 0.1

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

11. Applied add-cube-cbrt 0.1

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

12. Simplified 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left({t}^{\frac{1}{3}} \cdot \sqrt[3]{t}\right) \cdot z\right) - z \cdot \log \left(\color{blue}{{\left({t}^{\frac{1}{3}}\right)}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)\right) + \left(a - 0.5\right) \cdot b\]

13. Simplified 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left({t}^{\frac{1}{3}} \cdot \sqrt[3]{t}\right) \cdot z\right) - z \cdot \log \left({\left({t}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot \color{blue}{{\left(\sqrt[3]{t}\right)}^{\frac{1}{3}}}\right)\right) + \left(a - 0.5\right) \cdot b\]

14. Final simplification 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left({t}^{\frac{1}{3}} \cdot \sqrt[3]{t}\right) \cdot z\right) - z \cdot \log \left({\left({t}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{t}\right)}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

Reproduce

herbie shell --seed 2020053 
(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)))