Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Average Error: 22.7 → 7.9

Time: 14.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3610893527941651.5 \lor \neg \left(y \le 131485688765959957771190272\right):\\ \;\;\;\;x + 1 \cdot \left(\frac{x}{y \cdot y} - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt[3]{y + 1}} \cdot \frac{1}{\frac{\sqrt[3]{\left(1 - y\right) \cdot \left(y + 1\right)} \cdot \sqrt[3]{\left(1 - y\right) \cdot \left(y + 1\right)}}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}, x - 1, 1\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r424284 = 1.0;
        double r424285 = x;
        double r424286 = r424284 - r424285;
        double r424287 = y;
        double r424288 = r424286 * r424287;
        double r424289 = r424287 + r424284;
        double r424290 = r424288 / r424289;
        double r424291 = r424284 - r424290;
        return r424291;
}

↓

double f(double x, double y) {
        double r424292 = y;
        double r424293 = -3610893527941651.5;
        bool r424294 = r424292 <= r424293;
        double r424295 = 1.3148568876595996e+26;
        bool r424296 = r424292 <= r424295;
        double r424297 = !r424296;
        bool r424298 = r424294 || r424297;
        double r424299 = x;
        double r424300 = 1.0;
        double r424301 = r424292 * r424292;
        double r424302 = r424299 / r424301;
        double r424303 = r424299 / r424292;
        double r424304 = r424302 - r424303;
        double r424305 = r424300 * r424304;
        double r424306 = r424299 + r424305;
        double r424307 = r424292 + r424300;
        double r424308 = cbrt(r424307);
        double r424309 = r424292 / r424308;
        double r424310 = 1.0;
        double r424311 = r424300 - r424292;
        double r424312 = r424311 * r424307;
        double r424313 = cbrt(r424312);
        double r424314 = r424313 * r424313;
        double r424315 = cbrt(r424311);
        double r424316 = r424315 * r424315;
        double r424317 = r424314 / r424316;
        double r424318 = r424310 / r424317;
        double r424319 = r424309 * r424318;
        double r424320 = r424299 - r424300;
        double r424321 = fma(r424319, r424320, r424300);
        double r424322 = r424298 ? r424306 : r424321;
        return r424322;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.7
Target	0.2
Herbie	7.9

\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -3610893527941651.5 or 1.3148568876595996e+26 < y

   1. Initial program 47.0

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

   2. Simplified 29.6

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

   3. Taylor expanded around inf 15.3

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

   4. Simplified 15.3

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

   if -3610893527941651.5 < y < 1.3148568876595996e+26

   1. Initial program 1.3

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

   2. Simplified 1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 1.3

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

   5. Applied *-un-lft-identity 1.3

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

   6. Applied times-frac 1.3

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

   7. Simplified 1.3

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

   8. Simplified 1.3

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

   10. Applied flip-+ 1.3

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

   11. Applied cbrt-div 1.3

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

   12. Applied flip-+ 1.3

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

   13. Applied cbrt-div 1.3

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

   14. Applied frac-times 1.3

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

   15. Simplified 1.3

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

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3610893527941651.5 \lor \neg \left(y \le 131485688765959957771190272\right):\\ \;\;\;\;x + 1 \cdot \left(\frac{x}{y \cdot y} - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt[3]{y + 1}} \cdot \frac{1}{\frac{\sqrt[3]{\left(1 - y\right) \cdot \left(y + 1\right)} \cdot \sqrt[3]{\left(1 - y\right) \cdot \left(y + 1\right)}}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))