Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Average Error: 9.3 → 0.4

Time: 4.5s

Precision: binary64

Math

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

↓

\[\left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{1.6666666666666667} \cdot \sqrt[3]{{y}^{0.3333333333333333}}\right) + \left(x \cdot \log \left(\sqrt[3]{y}\right) + z \cdot \left(\log 1 - \left(y \cdot 1 + 0.5 \cdot \frac{y \cdot y}{1 \cdot 1}\right)\right)\right)\right) - t\]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))

↓

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (*
    x
    (log
     (* (pow (cbrt y) 1.6666666666666667) (cbrt (pow y 0.3333333333333333)))))
   (+
    (* x (log (cbrt y)))
    (* z (- (log 1.0) (+ (* y 1.0) (* 0.5 (/ (* y y) (* 1.0 1.0))))))))
  t))

C

double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * ((double) log(y)))) + ((double) (z * ((double) log(((double) (1.0 - y)))))))) - t));
}

↓

double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * ((double) log(((double) (((double) pow(((double) cbrt(y)), 1.6666666666666667)) * ((double) cbrt(((double) pow(y, 0.3333333333333333)))))))))) + ((double) (((double) (x * ((double) log(((double) cbrt(y)))))) + ((double) (z * ((double) (((double) log(1.0)) - ((double) (((double) (y * 1.0)) + ((double) (0.5 * (((double) (y * y)) / ((double) (1.0 * 1.0))))))))))))))) - t));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.3
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.3

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

2. Taylor expanded around 0 0.3

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

3. Simplified 0.3

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1 - \left(1 \cdot y + 0.5 \cdot \frac{y \cdot y}{1 \cdot 1}\right)\right)}\right) - t\]
4. Using strategy rm

5. Applied add-cube-cbrt_binary64 0.3

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

6. Applied log-prod_binary64 0.4

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

7. Applied distribute-lft-in_binary64 0.4

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

8. Applied associate-+l+_binary64 0.4

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

9. Simplified 0.4

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

11. Applied pow1/3_binary64 0.4

    \[\leadsto \left(x \cdot \log \left(\sqrt[3]{y} \cdot \color{blue}{{y}^{0.3333333333333333}}\right) + \left(x \cdot \log \left(\sqrt[3]{y}\right) + z \cdot \left(\log 1 - \left(y \cdot 1 + 0.5 \cdot \frac{y \cdot y}{1 \cdot 1}\right)\right)\right)\right) - t\]
12. Using strategy rm

13. Applied add-cube-cbrt_binary64 0.4

    \[\leadsto \left(x \cdot \log \left(\sqrt[3]{y} \cdot \color{blue}{\left(\left(\sqrt[3]{{y}^{0.3333333333333333}} \cdot \sqrt[3]{{y}^{0.3333333333333333}}\right) \cdot \sqrt[3]{{y}^{0.3333333333333333}}\right)}\right) + \left(x \cdot \log \left(\sqrt[3]{y}\right) + z \cdot \left(\log 1 - \left(y \cdot 1 + 0.5 \cdot \frac{y \cdot y}{1 \cdot 1}\right)\right)\right)\right) - t\]

14. Applied associate-*r*_binary64 0.4

    \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \left(\sqrt[3]{{y}^{0.3333333333333333}} \cdot \sqrt[3]{{y}^{0.3333333333333333}}\right)\right) \cdot \sqrt[3]{{y}^{0.3333333333333333}}\right)} + \left(x \cdot \log \left(\sqrt[3]{y}\right) + z \cdot \left(\log 1 - \left(y \cdot 1 + 0.5 \cdot \frac{y \cdot y}{1 \cdot 1}\right)\right)\right)\right) - t\]

15. Simplified 0.4

    \[\leadsto \left(x \cdot \log \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{1.6666666666666667}} \cdot \sqrt[3]{{y}^{0.3333333333333333}}\right) + \left(x \cdot \log \left(\sqrt[3]{y}\right) + z \cdot \left(\log 1 - \left(y \cdot 1 + 0.5 \cdot \frac{y \cdot y}{1 \cdot 1}\right)\right)\right)\right) - t\]

16. Final simplification 0.4

    \[\leadsto \left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{1.6666666666666667} \cdot \sqrt[3]{{y}^{0.3333333333333333}}\right) + \left(x \cdot \log \left(\sqrt[3]{y}\right) + z \cdot \left(\log 1 - \left(y \cdot 1 + 0.5 \cdot \frac{y \cdot y}{1 \cdot 1}\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))