Average Error: 0.5 → 0.4
Time: 37.8s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \log \left(e^{\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \log \left(e^{\cos x - \cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}
↓
\frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \log \left(e^{\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \log \left(e^{\cos x - \cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}
double f(double x, double y) {
        double r189265 = 2.0;
        double r189266 = sqrt(r189265);
        double r189267 = x;
        double r189268 = sin(r189267);
        double r189269 = y;
        double r189270 = sin(r189269);
        double r189271 = 16.0;
        double r189272 = r189270 / r189271;
        double r189273 = r189268 - r189272;
        double r189274 = r189266 * r189273;
        double r189275 = r189268 / r189271;
        double r189276 = r189270 - r189275;
        double r189277 = r189274 * r189276;
        double r189278 = cos(r189267);
        double r189279 = cos(r189269);
        double r189280 = r189278 - r189279;
        double r189281 = r189277 * r189280;
        double r189282 = r189265 + r189281;
        double r189283 = 3.0;
        double r189284 = 1.0;
        double r189285 = 5.0;
        double r189286 = sqrt(r189285);
        double r189287 = r189286 - r189284;
        double r189288 = r189287 / r189265;
        double r189289 = r189288 * r189278;
        double r189290 = r189284 + r189289;
        double r189291 = r189283 - r189286;
        double r189292 = r189291 / r189265;
        double r189293 = r189292 * r189279;
        double r189294 = r189290 + r189293;
        double r189295 = r189283 * r189294;
        double r189296 = r189282 / r189295;
        return r189296;
}
↓
double f(double x, double y) {
        double r189297 = 2.0;
        double r189298 = sqrt(r189297);
        double r189299 = sqrt(r189298);
        double r189300 = x;
        double r189301 = sin(r189300);
        double r189302 = y;
        double r189303 = sin(r189302);
        double r189304 = 16.0;
        double r189305 = r189303 / r189304;
        double r189306 = r189301 - r189305;
        double r189307 = r189299 * r189306;
        double r189308 = exp(r189307);
        double r189309 = log(r189308);
        double r189310 = r189299 * r189309;
        double r189311 = r189301 / r189304;
        double r189312 = r189303 - r189311;
        double r189313 = r189310 * r189312;
        double r189314 = cos(r189300);
        double r189315 = cos(r189302);
        double r189316 = r189314 - r189315;
        double r189317 = exp(r189316);
        double r189318 = log(r189317);
        double r189319 = r189313 * r189318;
        double r189320 = r189297 + r189319;
        double r189321 = 3.0;
        double r189322 = 1.0;
        double r189323 = 5.0;
        double r189324 = sqrt(r189323);
        double r189325 = r189324 - r189322;
        double r189326 = r189325 / r189297;
        double r189327 = r189326 * r189314;
        double r189328 = r189322 + r189327;
        double r189329 = r189321 * r189321;
        double r189330 = r189329 - r189323;
        double r189331 = r189321 + r189324;
        double r189332 = r189330 / r189331;
        double r189333 = r189332 / r189297;
        double r189334 = r189333 * r189315;
        double r189335 = r189328 + r189334;
        double r189336 = r189321 * r189335;
        double r189337 = r189320 / r189336;
        return r189337;
}

Error

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Bits error versus y

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
  2. Using strategy rm

  3. Applied flip--0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)}\]

  4. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{3 \cdot 3 - 5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  7. Applied sqrt-prod0.4

    \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  8. Applied associate-*l*0.4

    \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
  9. Using strategy rm

  10. Applied add-log-exp0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\log \left(e^{\cos y}\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  11. Applied add-log-exp0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\log \left(e^{\cos x}\right)} - \log \left(e^{\cos y}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  12. Applied diff-log0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\log \left(\frac{e^{\cos x}}{e^{\cos y}}\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  13. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \log \color{blue}{\left(e^{\cos x - \cos y}\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
  14. Using strategy rm

  15. Applied add-log-exp0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \color{blue}{\log \left(e^{\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)}\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \log \left(e^{\cos x - \cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  16. Final simplification0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \log \left(e^{\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \log \left(e^{\cos x - \cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  (/ (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (- (sin y) (/ (sin x) 16.0))) (- (cos x) (cos y)))) (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))