Average Error: 18.9 → 1.5
Time: 7.1s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\
\;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r353221 = 1.0;
        double r353222 = x;
        double r353223 = y;
        double r353224 = r353222 - r353223;
        double r353225 = r353221 - r353223;
        double r353226 = r353224 / r353225;
        double r353227 = r353221 - r353226;
        double r353228 = log(r353227);
        double r353229 = r353221 - r353228;
        return r353229;
}
↓
double f(double x, double y) {
        double r353230 = x;
        double r353231 = y;
        double r353232 = r353230 - r353231;
        double r353233 = 1.0;
        double r353234 = r353233 - r353231;
        double r353235 = r353232 / r353234;
        double r353236 = 3.2716599026957865e-16;
        bool r353237 = r353235 <= r353236;
        double r353238 = r353233 - r353235;
        double r353239 = cbrt(r353238);
        double r353240 = r353239 * r353239;
        double r353241 = r353240 * r353239;
        double r353242 = sqrt(r353241);
        double r353243 = log(r353242);
        double r353244 = sqrt(r353238);
        double r353245 = log(r353244);
        double r353246 = r353243 + r353245;
        double r353247 = r353233 - r353246;
        double r353248 = 2.0;
        double r353249 = pow(r353231, r353248);
        double r353250 = r353230 / r353249;
        double r353251 = 1.0;
        double r353252 = r353251 / r353231;
        double r353253 = r353250 - r353252;
        double r353254 = r353230 / r353231;
        double r353255 = fma(r353233, r353253, r353254);
        double r353256 = log(r353255);
        double r353257 = r353233 - r353256;
        double r353258 = r353237 ? r353247 : r353257;
        return r353258;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.9
Target0.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 3.2716599026957865e-16

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

      \[\leadsto 1 - \log \color{blue}{\left(\sqrt{1 - \frac{x - y}{1 - y}} \cdot \sqrt{1 - \frac{x - y}{1 - y}}\right)}\]

    4. Applied log-prod0.0

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

    6. Applied add-cube-cbrt0.1

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

    if 3.2716599026957865e-16 < (/ (- x y) (- 1.0 y))

    1. Initial program 57.6

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]

    2. Taylor expanded around inf 4.4

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

    3. Simplified4.4

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))