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

Average Error: 11.1 → 1.3

Time: 8.6s

Precision: binary64

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

Math

\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.73141670297118115 \cdot 10^{72}:\\ \;\;\;\;\sqrt{e^{-y}} \cdot \frac{\sqrt{e^{-y}}}{x}\\ \mathbf{elif}\;x \le 2.105386851920235:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original	11.1
Target	8.2
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.73141670297118115e72

   1. Initial program 13.0

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

   2. Simplified 13.0

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\]
   3. Using strategy rm

   4. Applied add-exp-log 64.0

      \[\leadsto \frac{{\left(\frac{x}{\color{blue}{e^{\log \left(x + y\right)}}}\right)}^{x}}{x}\]

   5. Applied add-exp-log 64.0

      \[\leadsto \frac{{\left(\frac{\color{blue}{e^{\log x}}}{e^{\log \left(x + y\right)}}\right)}^{x}}{x}\]

   6. Applied div-exp 64.0

      \[\leadsto \frac{{\color{blue}{\left(e^{\log x - \log \left(x + y\right)}\right)}}^{x}}{x}\]

   7. Applied pow-exp 64.0

      \[\leadsto \frac{\color{blue}{e^{\left(\log x - \log \left(x + y\right)\right) \cdot x}}}{x}\]

   8. Taylor expanded around inf 0.0

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x}\]

   9. Simplified 0.0

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]
   10. Using strategy rm

   11. Applied *-un-lft-identity 0.0

       \[\leadsto \frac{e^{-y}}{\color{blue}{1 \cdot x}}\]

   12. Applied add-sqr-sqrt 0.0

       \[\leadsto \frac{\color{blue}{\sqrt{e^{-y}} \cdot \sqrt{e^{-y}}}}{1 \cdot x}\]

   13. Applied times-frac 0.0

       \[\leadsto \color{blue}{\frac{\sqrt{e^{-y}}}{1} \cdot \frac{\sqrt{e^{-y}}}{x}}\]

   14. Simplified 0.0

       \[\leadsto \color{blue}{\sqrt{e^{-y}}} \cdot \frac{\sqrt{e^{-y}}}{x}\]

   if -1.73141670297118115e72 < x < 2.105386851920235

   1. Initial program 11.2

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

   2. Simplified 11.2

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 15.0

      \[\leadsto \frac{{\left(\frac{x}{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}\right)}^{x}}{x}\]

   5. Applied add-cube-cbrt 11.2

      \[\leadsto \frac{{\left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}\right)}^{x}}{x}\]

   6. Applied times-frac 11.2

      \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}^{x}}{x}\]

   7. Applied unpow-prod-down 2.6

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}}{x}\]

   if 2.105386851920235 < x

   1. Initial program 9.6

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

   2. Simplified 9.6

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\]
   3. Using strategy rm

   4. Applied add-exp-log 45.0

      \[\leadsto \frac{{\left(\frac{x}{\color{blue}{e^{\log \left(x + y\right)}}}\right)}^{x}}{x}\]

   5. Applied add-exp-log 9.8

      \[\leadsto \frac{{\left(\frac{\color{blue}{e^{\log x}}}{e^{\log \left(x + y\right)}}\right)}^{x}}{x}\]

   6. Applied div-exp 9.8

      \[\leadsto \frac{{\color{blue}{\left(e^{\log x - \log \left(x + y\right)}\right)}}^{x}}{x}\]

   7. Applied pow-exp 9.8

      \[\leadsto \frac{\color{blue}{e^{\left(\log x - \log \left(x + y\right)\right) \cdot x}}}{x}\]

   8. Taylor expanded around inf 0.1

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x}\]

   9. Simplified 0.1

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]
   10. Using strategy rm

   11. Applied clear-num 0.1

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{e^{-y}}}}\]

   12. Simplified 0.1

       \[\leadsto \frac{1}{\color{blue}{x \cdot e^{y}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.73141670297118115 \cdot 10^{72}:\\ \;\;\;\;\sqrt{e^{-y}} \cdot \frac{\sqrt{e^{-y}}}{x}\\ \mathbf{elif}\;x \le 2.105386851920235:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1.0 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))