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

Average Error: 0.1 → 0.1

Time: 6.6s

Precision: binary64

Math

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

↓

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

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 sub-neg 0.1

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

4. Applied associate-+l+ 0.1

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

5. Simplified 0.1

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

7. Applied sub-neg 0.1

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

8. Applied distribute-lft-in 0.1

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

9. Applied associate--l+ 0.1

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

10. Applied associate-+r+ 0.1

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

11. Simplified 0.1

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

12. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020175 
(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) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

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