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

Average Error: 4.7 → 3.4

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 1.4864422671369188 \cdot 10^{277}:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(x \cdot \left(-\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{1 - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.7
Target	4.0
Herbie	3.4

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

Derivation

1. Split input into 2 regimes

2. if (* x (- (/ y z) (/ t (- 1.0 z)))) < 1.4864422671369188e+277

   1. Initial program 3.3

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

   3. Applied sub-neg 3.3

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

   4. Applied distribute-lft-in 3.3

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

   6. Applied add-cube-cbrt 3.6

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

   7. Applied add-cube-cbrt 3.8

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

   8. Applied times-frac 3.8

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

   9. Applied distribute-lft-neg-in 3.8

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

   10. Applied associate-*r* 3.3

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

   if 1.4864422671369188e+277 < (* x (- (/ y z) (/ t (- 1.0 z))))

   1. Initial program 37.4

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

   3. Applied sub-neg 37.4

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

   4. Applied distribute-lft-in 37.4

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

   6. Applied div-inv 37.4

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

   7. Applied associate-*r* 6.0

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 1.4864422671369188 \cdot 10^{277}:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(x \cdot \left(-\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{1 - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020124 
(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) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

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