Average Error: 18.9 → 0.1
Time: 1.1m
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -224500362.7019620835781097412109375:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 51921085.7710403501987457275390625:\\ \;\;\;\;\log \left(\frac{1}{\sqrt{1 - \frac{x - y}{1 - y}}} \cdot \frac{e^{1}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{x - y}{1 \cdot 1 - y \cdot y} \cdot \left(-\left(1 + y\right)\right)\right) + \mathsf{fma}\left(-\left(1 + y\right), \frac{x - y}{1 \cdot 1 - y \cdot y}, \frac{x - y}{1 \cdot 1 - y \cdot y} \cdot \left(1 + y\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -224500362.7019620835781097412109375:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\

\mathbf{elif}\;y \le 51921085.7710403501987457275390625:\\
\;\;\;\;\log \left(\frac{1}{\sqrt{1 - \frac{x - y}{1 - y}}} \cdot \frac{e^{1}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{x - y}{1 \cdot 1 - y \cdot y} \cdot \left(-\left(1 + y\right)\right)\right) + \mathsf{fma}\left(-\left(1 + y\right), \frac{x - y}{1 \cdot 1 - y \cdot y}, \frac{x - y}{1 \cdot 1 - y \cdot y} \cdot \left(1 + y\right)\right)}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r14570006 = 1.0;
        double r14570007 = x;
        double r14570008 = y;
        double r14570009 = r14570007 - r14570008;
        double r14570010 = r14570006 - r14570008;
        double r14570011 = r14570009 / r14570010;
        double r14570012 = r14570006 - r14570011;
        double r14570013 = log(r14570012);
        double r14570014 = r14570006 - r14570013;
        return r14570014;
}
↓
double f(double x, double y) {
        double r14570015 = y;
        double r14570016 = -224500362.70196208;
        bool r14570017 = r14570015 <= r14570016;
        double r14570018 = 1.0;
        double r14570019 = x;
        double r14570020 = r14570019 / r14570015;
        double r14570021 = r14570018 / r14570015;
        double r14570022 = r14570020 - r14570021;
        double r14570023 = fma(r14570020, r14570021, r14570022);
        double r14570024 = log(r14570023);
        double r14570025 = r14570018 - r14570024;
        double r14570026 = 51921085.77104035;
        bool r14570027 = r14570015 <= r14570026;
        double r14570028 = 1.0;
        double r14570029 = r14570019 - r14570015;
        double r14570030 = r14570018 - r14570015;
        double r14570031 = r14570029 / r14570030;
        double r14570032 = r14570018 - r14570031;
        double r14570033 = sqrt(r14570032);
        double r14570034 = r14570028 / r14570033;
        double r14570035 = exp(r14570018);
        double r14570036 = r14570018 * r14570018;
        double r14570037 = r14570015 * r14570015;
        double r14570038 = r14570036 - r14570037;
        double r14570039 = r14570029 / r14570038;
        double r14570040 = r14570018 + r14570015;
        double r14570041 = -r14570040;
        double r14570042 = r14570039 * r14570041;
        double r14570043 = fma(r14570028, r14570018, r14570042);
        double r14570044 = r14570039 * r14570040;
        double r14570045 = fma(r14570041, r14570039, r14570044);
        double r14570046 = r14570043 + r14570045;
        double r14570047 = sqrt(r14570046);
        double r14570048 = r14570035 / r14570047;
        double r14570049 = r14570034 * r14570048;
        double r14570050 = log(r14570049);
        double r14570051 = r14570027 ? r14570050 : r14570025;
        double r14570052 = r14570017 ? r14570025 : r14570051;
        return r14570052;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.9
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 < -224500362.70196208 or 51921085.77104035 < y

    1. Initial program 47.4

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

    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

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

    if -224500362.70196208 < y < 51921085.77104035

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied diff-log0.1

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied *-un-lft-identity0.1

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

    8. Applied times-frac0.1

      \[\leadsto \log \color{blue}{\left(\frac{1}{\sqrt{1 - \frac{x - y}{1 - y}}} \cdot \frac{e^{1}}{\sqrt{1 - \frac{x - y}{1 - y}}}\right)}\]
    9. Using strategy rm

    10. Applied flip--0.1

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

    11. Applied associate-/r/0.1

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

    12. Applied *-un-lft-identity0.1

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

    13. Applied prod-diff0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -224500362.7019620835781097412109375:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 51921085.7710403501987457275390625:\\ \;\;\;\;\log \left(\frac{1}{\sqrt{1 - \frac{x - y}{1 - y}}} \cdot \frac{e^{1}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{x - y}{1 \cdot 1 - y \cdot y} \cdot \left(-\left(1 + y\right)\right)\right) + \mathsf{fma}\left(-\left(1 + y\right), \frac{x - y}{1 \cdot 1 - y \cdot y}, \frac{x - y}{1 \cdot 1 - y \cdot y} \cdot \left(1 + y\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))