Average Error: 20.5 → 18.1
Time: 33.0s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 3.982214551187210407512533577328782205679 \cdot 10^{305}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\left(\frac{z}{\sqrt{3}} \cdot t\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt{3}}} \cdot \sqrt[3]{\frac{1}{\sqrt{3}}}\right)\right) \cdot \sqrt[3]{\frac{1}{\sqrt{3}}}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}
↓
\begin{array}{l}
\mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 3.982214551187210407512533577328782205679 \cdot 10^{305}\right):\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\left(\frac{z}{\sqrt{3}} \cdot t\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt{3}}} \cdot \sqrt[3]{\frac{1}{\sqrt{3}}}\right)\right) \cdot \sqrt[3]{\frac{1}{\sqrt{3}}}\right) - \frac{a}{b \cdot 3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r532316 = 2.0;
        double r532317 = x;
        double r532318 = sqrt(r532317);
        double r532319 = r532316 * r532318;
        double r532320 = y;
        double r532321 = z;
        double r532322 = t;
        double r532323 = r532321 * r532322;
        double r532324 = 3.0;
        double r532325 = r532323 / r532324;
        double r532326 = r532320 - r532325;
        double r532327 = cos(r532326);
        double r532328 = r532319 * r532327;
        double r532329 = a;
        double r532330 = b;
        double r532331 = r532330 * r532324;
        double r532332 = r532329 / r532331;
        double r532333 = r532328 - r532332;
        return r532333;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r532334 = z;
        double r532335 = t;
        double r532336 = r532334 * r532335;
        double r532337 = -inf.0;
        bool r532338 = r532336 <= r532337;
        double r532339 = 3.9822145511872104e+305;
        bool r532340 = r532336 <= r532339;
        double r532341 = !r532340;
        bool r532342 = r532338 || r532341;
        double r532343 = 2.0;
        double r532344 = x;
        double r532345 = sqrt(r532344);
        double r532346 = r532343 * r532345;
        double r532347 = y;
        double r532348 = 2.0;
        double r532349 = pow(r532347, r532348);
        double r532350 = -0.5;
        double r532351 = 1.0;
        double r532352 = fma(r532349, r532350, r532351);
        double r532353 = r532346 * r532352;
        double r532354 = a;
        double r532355 = b;
        double r532356 = 3.0;
        double r532357 = r532355 * r532356;
        double r532358 = r532354 / r532357;
        double r532359 = r532353 - r532358;
        double r532360 = sqrt(r532356);
        double r532361 = r532334 / r532360;
        double r532362 = r532361 * r532335;
        double r532363 = r532351 / r532360;
        double r532364 = cbrt(r532363);
        double r532365 = r532364 * r532364;
        double r532366 = r532362 * r532365;
        double r532367 = r532366 * r532364;
        double r532368 = r532347 - r532367;
        double r532369 = cos(r532368);
        double r532370 = r532346 * r532369;
        double r532371 = r532370 - r532358;
        double r532372 = r532342 ? r532359 : r532371;
        return r532372;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original20.5
Target18.6
Herbie18.1
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z t) < -inf.0 or 3.9822145511872104e+305 < (* z t)

    1. Initial program 63.7

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt63.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}\right) - \frac{a}{b \cdot 3}\]

    4. Applied times-frac63.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}}\right) - \frac{a}{b \cdot 3}\]
    5. Using strategy rm

    6. Applied div-inv63.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt{3}} \cdot \color{blue}{\left(t \cdot \frac{1}{\sqrt{3}}\right)}\right) - \frac{a}{b \cdot 3}\]

    7. Applied associate-*r*63.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\left(\frac{z}{\sqrt{3}} \cdot t\right) \cdot \frac{1}{\sqrt{3}}}\right) - \frac{a}{b \cdot 3}\]

    8. Taylor expanded around 0 45.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]

    9. Simplified45.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right)} - \frac{a}{b \cdot 3}\]

    if -inf.0 < (* z t) < 3.9822145511872104e+305

    1. Initial program 14.3

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt14.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}\right) - \frac{a}{b \cdot 3}\]

    4. Applied times-frac14.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}}\right) - \frac{a}{b \cdot 3}\]
    5. Using strategy rm

    6. Applied div-inv14.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt{3}} \cdot \color{blue}{\left(t \cdot \frac{1}{\sqrt{3}}\right)}\right) - \frac{a}{b \cdot 3}\]

    7. Applied associate-*r*14.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\left(\frac{z}{\sqrt{3}} \cdot t\right) \cdot \frac{1}{\sqrt{3}}}\right) - \frac{a}{b \cdot 3}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt14.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\frac{z}{\sqrt{3}} \cdot t\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{\sqrt{3}}} \cdot \sqrt[3]{\frac{1}{\sqrt{3}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt{3}}}\right)}\right) - \frac{a}{b \cdot 3}\]

    10. Applied associate-*r*14.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\left(\left(\frac{z}{\sqrt{3}} \cdot t\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt{3}}} \cdot \sqrt[3]{\frac{1}{\sqrt{3}}}\right)\right) \cdot \sqrt[3]{\frac{1}{\sqrt{3}}}}\right) - \frac{a}{b \cdot 3}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 3.982214551187210407512533577328782205679 \cdot 10^{305}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\left(\frac{z}{\sqrt{3}} \cdot t\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt{3}}} \cdot \sqrt[3]{\frac{1}{\sqrt{3}}}\right)\right) \cdot \sqrt[3]{\frac{1}{\sqrt{3}}}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e+129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))