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

Average Error: 4.9 → 0.6

Time: 8.3s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) = -inf.0:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(-\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot x\right) \cdot \frac{\sqrt[3]{t}}{1 - z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le -3.5523710845838534 \cdot 10^{-263}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 0.0:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 1.5580996770493964 \cdot 10^{299}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.9
Target	4.5
Herbie	0.6

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

Derivation

1. Split input into 4 regimes

2. if (* x (- (/ y z) (/ t (- 1.0 z)))) < -inf.0

   1. Initial program 64.0

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

   3. Applied sub-neg 64.0

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

   4. Applied distribute-lft-in 64.0

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

   5. Simplified 0.3

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

   7. Applied *-un-lft-identity 0.3

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

   8. Applied add-cube-cbrt 0.3

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

   9. Applied times-frac 0.3

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

   10. Applied distribute-lft-neg-in 0.3

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

   11. Applied associate-*r* 0.3

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

   12. Simplified 0.3

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

   if -inf.0 < (* x (- (/ y z) (/ t (- 1.0 z)))) < -3.5523710845838534e-263 or 0.0 < (* x (- (/ y z) (/ t (- 1.0 z)))) < 1.5580996770493964e299

   1. Initial program 0.3

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

   3. Applied sub-neg 0.3

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

   4. Applied distribute-lft-in 0.3

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

   5. Simplified 5.9

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

   7. Applied associate-/l* 0.4

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

   if -3.5523710845838534e-263 < (* x (- (/ y z) (/ t (- 1.0 z)))) < 0.0

   1. Initial program 8.1

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

   3. Applied sub-neg 8.1

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

   4. Applied distribute-lft-in 8.1

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

   5. Simplified 5.8

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

   7. Applied distribute-neg-frac 5.8

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

   8. Applied associate-*r/ 1.1

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

   if 1.5580996770493964e299 < (* x (- (/ y z) (/ t (- 1.0 z))))

   1. Initial program 55.3

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

   3. Applied sub-neg 55.3

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

   4. Applied distribute-lft-in 55.3

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

   5. Simplified 2.7

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

   7. Applied div-inv 2.9

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) = -inf.0:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(-\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot x\right) \cdot \frac{\sqrt[3]{t}}{1 - z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le -3.5523710845838534 \cdot 10^{-263}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 0.0:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 1.5580996770493964 \cdot 10^{299}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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)))))