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

Average Error: 10.8 → 6.2

Time: 6.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.5737049212414563 \cdot 10^{60}:\\ \;\;\;\;\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{x} + \left(\frac{1}{2} \cdot \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y \cdot y} \cdot \left(x \cdot x + {x}^{3}\right)\right) - x \cdot \frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y}\right)\\ \mathbf{elif}\;y \le 7.2610625153204842 \cdot 10^{61}:\\ \;\;\;\;\frac{{\left(\sqrt{\frac{x}{y + x}}\right)}^{x} \cdot {\left(\sqrt{\frac{x}{y + x}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt{x}}{\sqrt{y + x}}\right)}^{x} \cdot {\left(\frac{\sqrt{x}}{\sqrt{y + x}}\right)}^{x}}{x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original	10.8
Target	7.6
Herbie	6.2

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

Derivation

1. Split input into 3 regimes

2. if y < -3.5737049212414563e60

   1. Initial program 34.2

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

   2. Simplified 34.2

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

   3. Taylor expanded around -inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{e^{x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)} \cdot {x}^{2}}{{y}^{2}} + \left(\frac{1}{2} \cdot \frac{e^{x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)} \cdot {x}^{3}}{{y}^{2}} + \frac{e^{x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)}}{x}\right)\right) - \frac{e^{x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)} \cdot x}{y}}\]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{x} + \left(\frac{1}{2} \cdot \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y \cdot y} \cdot \left(x \cdot x + {x}^{3}\right)\right) - \frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y} \cdot x\right)}\]

   if -3.5737049212414563e60 < y < 7.2610625153204842e61

   1. Initial program 3.6

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

   2. Simplified 3.6

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

   4. Applied add-sqr-sqrt 3.6

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

   5. Applied unpow-prod-down 3.6

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

   if 7.2610625153204842e61 < y

   1. Initial program 30.9

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

   2. Simplified 30.9

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

   4. Applied add-sqr-sqrt 30.9

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

   5. Applied add-sqr-sqrt 33.1

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

   6. Applied times-frac 33.1

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

   7. Applied unpow-prod-down 19.4

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt{x}}{\sqrt{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt{x}}{\sqrt{x + y}}\right)}^{x}}}{x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.5737049212414563 \cdot 10^{60}:\\ \;\;\;\;\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{x} + \left(\frac{1}{2} \cdot \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y \cdot y} \cdot \left(x \cdot x + {x}^{3}\right)\right) - x \cdot \frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y}\right)\\ \mathbf{elif}\;y \le 7.2610625153204842 \cdot 10^{61}:\\ \;\;\;\;\frac{{\left(\sqrt{\frac{x}{y + x}}\right)}^{x} \cdot {\left(\sqrt{\frac{x}{y + x}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt{x}}{\sqrt{y + x}}\right)}^{x} \cdot {\left(\frac{\sqrt{x}}{\sqrt{y + x}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))