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

Average Error: 0.1 → 0.1

Time: 6.3s

Precision: 64

Math

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

↓

\[\left(\left(x + y\right) + \left(1 - \log t\right) \cdot z\right) + \left(a - 0.5\right) \cdot b\]

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) + ((1.0 - log(t)) * z)) + ((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 associate--l+ 0.1

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

4. Taylor expanded around 0 0.1

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

5. Simplified 0.1

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

6. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020100 
(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)))