Average Error: 18.6 → 0.1
Time: 15.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -125230047.9964575469493865966796875 \lor \neg \left(y \le 3758135.54599315486848354339599609375\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{\left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1} - \frac{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(y, y, 1 \cdot y\right)\right) \cdot \left(x - y\right)}{{1}^{3} - {y}^{3}}\right) + \mathsf{fma}\left(1, 1, \mathsf{fma}\left(y, y, 1 \cdot y\right)\right) \cdot \left(\left(-\frac{x - y}{{1}^{3} - {y}^{3}}\right) + \frac{x - y}{{1}^{3} - {y}^{3}}\right)}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -125230047.9964575469493865966796875 \lor \neg \left(y \le 3758135.54599315486848354339599609375\right):\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{\left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1} - \frac{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(y, y, 1 \cdot y\right)\right) \cdot \left(x - y\right)}{{1}^{3} - {y}^{3}}\right) + \mathsf{fma}\left(1, 1, \mathsf{fma}\left(y, y, 1 \cdot y\right)\right) \cdot \left(\left(-\frac{x - y}{{1}^{3} - {y}^{3}}\right) + \frac{x - y}{{1}^{3} - {y}^{3}}\right)}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r327899 = 1.0;
        double r327900 = x;
        double r327901 = y;
        double r327902 = r327900 - r327901;
        double r327903 = r327899 - r327901;
        double r327904 = r327902 / r327903;
        double r327905 = r327899 - r327904;
        double r327906 = log(r327905);
        double r327907 = r327899 - r327906;
        return r327907;
}
↓
double f(double x, double y) {
        double r327908 = y;
        double r327909 = -125230047.99645755;
        bool r327910 = r327908 <= r327909;
        double r327911 = 3758135.545993155;
        bool r327912 = r327908 <= r327911;
        double r327913 = !r327912;
        bool r327914 = r327910 || r327913;
        double r327915 = 1.0;
        double r327916 = x;
        double r327917 = 2.0;
        double r327918 = pow(r327908, r327917);
        double r327919 = r327916 / r327918;
        double r327920 = 1.0;
        double r327921 = r327920 / r327908;
        double r327922 = r327919 - r327921;
        double r327923 = r327916 / r327908;
        double r327924 = fma(r327915, r327922, r327923);
        double r327925 = log(r327924);
        double r327926 = r327915 - r327925;
        double r327927 = r327916 - r327908;
        double r327928 = r327915 - r327908;
        double r327929 = r327927 / r327928;
        double r327930 = r327915 - r327929;
        double r327931 = sqrt(r327930);
        double r327932 = log(r327931);
        double r327933 = cbrt(r327915);
        double r327934 = r327933 * r327933;
        double r327935 = r327934 * r327933;
        double r327936 = r327915 * r327908;
        double r327937 = fma(r327908, r327908, r327936);
        double r327938 = fma(r327915, r327915, r327937);
        double r327939 = r327938 * r327927;
        double r327940 = 3.0;
        double r327941 = pow(r327915, r327940);
        double r327942 = pow(r327908, r327940);
        double r327943 = r327941 - r327942;
        double r327944 = r327939 / r327943;
        double r327945 = r327935 - r327944;
        double r327946 = r327927 / r327943;
        double r327947 = -r327946;
        double r327948 = r327947 + r327946;
        double r327949 = r327938 * r327948;
        double r327950 = r327945 + r327949;
        double r327951 = sqrt(r327950);
        double r327952 = log(r327951);
        double r327953 = r327932 + r327952;
        double r327954 = r327915 - r327953;
        double r327955 = r327914 ? r327926 : r327954;
        return r327955;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.6
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -125230047.99645755 or 3758135.545993155 < y

    1. Initial program 47.9

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -125230047.99645755 < y < 3758135.545993155

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied log-prod0.1

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

    6. Applied flip3--0.1

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

    7. Applied associate-/r/0.1

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

    8. Applied add-cube-cbrt0.1

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

    9. Applied prod-diff0.1

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

    10. Simplified0.1

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

    11. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -125230047.9964575469493865966796875 \lor \neg \left(y \le 3758135.54599315486848354339599609375\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{\left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1} - \frac{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(y, y, 1 \cdot y\right)\right) \cdot \left(x - y\right)}{{1}^{3} - {y}^{3}}\right) + \mathsf{fma}\left(1, 1, \mathsf{fma}\left(y, y, 1 \cdot y\right)\right) \cdot \left(\left(-\frac{x - y}{{1}^{3} - {y}^{3}}\right) + \frac{x - y}{{1}^{3} - {y}^{3}}\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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))))))