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

Average Error: 4.4 → 1.8

Time: 5.4s

Precision: binary64

Math

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.4
Target	4.0
Herbie	1.8

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

1. Initial program 4.4

   \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
2. Using strategy rm

3. Applied sub-neg 4.4

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

4. Applied distribute-lft-in 4.4

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

5. Simplified 4.4

   \[\leadsto x \cdot \frac{y}{z} + \color{blue}{x \cdot \frac{-t}{1 - z}}\]
6. Using strategy rm

7. Applied add-cube-cbrt 4.9

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

8. Applied add-cube-cbrt 5.0

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

9. Applied times-frac 5.0

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

10. Applied associate-*r* 1.8

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

11. Simplified 1.8

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

13. Applied add-cube-cbrt 2.1

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

14. Applied add-cube-cbrt 2.3

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

15. Applied times-frac 2.3

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

16. Applied associate-*r* 1.8

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

17. Simplified 1.8

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

18. Final simplification 1.8

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

Reproduce

herbie shell --seed 2020184 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (neg (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))