Average Error: 17.8 → 0.5
Time: 26.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 9.496174125024586353937400273750313317578 \cdot 10^{-5}:\\ \;\;\;\;1 - \log \left(\left({\left(\sqrt[3]{1}\right)}^{3} - \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 9.496174125024586353937400273750313317578 \cdot 10^{-5}:\\
\;\;\;\;1 - \log \left(\left({\left(\sqrt[3]{1}\right)}^{3} - \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r291385 = 1.0;
        double r291386 = x;
        double r291387 = y;
        double r291388 = r291386 - r291387;
        double r291389 = r291385 - r291387;
        double r291390 = r291388 / r291389;
        double r291391 = r291385 - r291390;
        double r291392 = log(r291391);
        double r291393 = r291385 - r291392;
        return r291393;
}
↓
double f(double x, double y) {
        double r291394 = x;
        double r291395 = y;
        double r291396 = r291394 - r291395;
        double r291397 = 1.0;
        double r291398 = r291397 - r291395;
        double r291399 = r291396 / r291398;
        double r291400 = 9.496174125024586e-05;
        bool r291401 = r291399 <= r291400;
        double r291402 = cbrt(r291397);
        double r291403 = 3.0;
        double r291404 = pow(r291402, r291403);
        double r291405 = cbrt(r291398);
        double r291406 = pow(r291405, r291403);
        double r291407 = r291396 / r291406;
        double r291408 = r291404 - r291407;
        double r291409 = -r291407;
        double r291410 = r291407 + r291409;
        double r291411 = r291408 + r291410;
        double r291412 = log(r291411);
        double r291413 = r291397 - r291412;
        double r291414 = 2.0;
        double r291415 = pow(r291395, r291414);
        double r291416 = r291394 / r291415;
        double r291417 = r291394 / r291395;
        double r291418 = fma(r291416, r291397, r291417);
        double r291419 = r291397 / r291395;
        double r291420 = r291418 - r291419;
        double r291421 = log(r291420);
        double r291422 = r291397 - r291421;
        double r291423 = r291401 ? r291413 : r291422;
        return r291423;
}

Error

Bits error versus x

Bits error versus y

Target

Original17.8
Target0.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 9.496174125024586e-05

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    4. Applied add-cube-cbrt0.0

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

    5. Applied times-frac0.0

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

    6. Applied add-cube-cbrt0.0

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

    7. Applied prod-diff0.0

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

    8. Simplified0.0

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

    9. Simplified0.0

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

    if 9.496174125024586e-05 < (/ (- x y) (- 1.0 y))

    1. Initial program 59.8

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

    2. Taylor expanded around inf 1.6

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

    3. Simplified1.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 9.496174125024586353937400273750313317578 \cdot 10^{-5}:\\ \;\;\;\;1 - \log \left(\left({\left(\sqrt[3]{1}\right)}^{3} - \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.619472414) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

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