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

Average Error: 11.1 → 0.7

Time: 22.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.901247255096526472927968321689508282152 \cdot 10^{93} \lor \neg \left(x \le 3.427314966810909420478563204128016522016 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right|\right)}^{x} \cdot {\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right|\right)}^{x}}{\frac{x}{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}}}\\ \end{array}\]

C

double f(double x, double y) {
        double r310237 = x;
        double r310238 = y;
        double r310239 = r310237 + r310238;
        double r310240 = r310237 / r310239;
        double r310241 = log(r310240);
        double r310242 = r310237 * r310241;
        double r310243 = exp(r310242);
        double r310244 = r310243 / r310237;
        return r310244;
}

↓

double f(double x, double y) {
        double r310245 = x;
        double r310246 = -1.9012472550965265e+93;
        bool r310247 = r310245 <= r310246;
        double r310248 = 3.4273149668109094e-18;
        bool r310249 = r310245 <= r310248;
        double r310250 = !r310249;
        bool r310251 = r310247 || r310250;
        double r310252 = y;
        double r310253 = -r310252;
        double r310254 = exp(r310253);
        double r310255 = r310254 / r310245;
        double r310256 = cbrt(r310245);
        double r310257 = r310252 + r310245;
        double r310258 = cbrt(r310257);
        double r310259 = r310256 / r310258;
        double r310260 = fabs(r310259);
        double r310261 = pow(r310260, r310245);
        double r310262 = r310261 * r310261;
        double r310263 = pow(r310259, r310245);
        double r310264 = r310245 / r310263;
        double r310265 = r310262 / r310264;
        double r310266 = r310251 ? r310255 : r310265;
        return r310266;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.1
Target	8.3
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -1.9012472550965265e+93 or 3.4273149668109094e-18 < x

   1. Initial program 11.2

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

   2. Simplified 11.2

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

   3. Taylor expanded around inf 0.8

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

   if -1.9012472550965265e+93 < x < 3.4273149668109094e-18

   1. Initial program 11.0

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

   2. Simplified 11.0

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

   4. Applied add-cube-cbrt 16.4

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

   5. Applied add-cube-cbrt 11.0

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

   6. Applied times-frac 11.0

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

   7. Applied unpow-prod-down 2.8

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

   8. Applied associate-/l* 2.8

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

   9. Simplified 2.8

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

   11. Applied add-sqr-sqrt 2.8

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

   12. Applied unpow-prod-down 2.8

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

   13. Simplified 2.8

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

   14. Simplified 0.7

       \[\leadsto \frac{{\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{x} \cdot \color{blue}{{\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{x}}}{\frac{x}{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.901247255096526472927968321689508282152 \cdot 10^{93} \lor \neg \left(x \le 3.427314966810909420478563204128016522016 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right|\right)}^{x} \cdot {\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right|\right)}^{x}}{\frac{x}{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"

  :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))