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

Average Error: 0.1 → 0.1

Time: 6.2s

Precision: 64

Math

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

↓

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

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 fma(b, (a - 0.5), (fma(z, ((1.0 - (2.0 * log(cbrt(t)))) - log(cbrt(t))), (x + y)) + (log(t) * (-z + z))));
}

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. Simplified 0.1

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

4. Applied add-sqr-sqrt 32.1

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

5. Applied prod-diff 32.1

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

6. Simplified 0.1

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

7. Simplified 0.1

   \[\leadsto \mathsf{fma}\left(b, a - 0.5, \mathsf{fma}\left(z, 1 - \log t, x + y\right) + \color{blue}{\log t \cdot \left(\left(-z\right) + z\right)}\right)\]
8. Using strategy rm

9. Applied add-cube-cbrt 0.1

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

10. Applied log-prod 0.1

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

11. Applied associate--r+ 0.1

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

12. Simplified 0.1

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

13. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020056 +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)))