Average Error: 20.7 → 18.3
Time: 30.2s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999603707223627679923083633184433:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{t}{\sqrt[3]{3}}\right) - \frac{a}{b} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - a \cdot \frac{\frac{1}{b}}{3}\\ \end{array}\]
\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}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999603707223627679923083633184433:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{t}{\sqrt[3]{3}}\right) - \frac{a}{b} \cdot \frac{1}{3}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - a \cdot \frac{\frac{1}{b}}{3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r507214 = 2.0;
        double r507215 = x;
        double r507216 = sqrt(r507215);
        double r507217 = r507214 * r507216;
        double r507218 = y;
        double r507219 = z;
        double r507220 = t;
        double r507221 = r507219 * r507220;
        double r507222 = 3.0;
        double r507223 = r507221 / r507222;
        double r507224 = r507218 - r507223;
        double r507225 = cos(r507224);
        double r507226 = r507217 * r507225;
        double r507227 = a;
        double r507228 = b;
        double r507229 = r507228 * r507222;
        double r507230 = r507227 / r507229;
        double r507231 = r507226 - r507230;
        return r507231;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r507232 = y;
        double r507233 = z;
        double r507234 = t;
        double r507235 = r507233 * r507234;
        double r507236 = 3.0;
        double r507237 = r507235 / r507236;
        double r507238 = r507232 - r507237;
        double r507239 = cos(r507238);
        double r507240 = 0.9999999996037072;
        bool r507241 = r507239 <= r507240;
        double r507242 = 2.0;
        double r507243 = x;
        double r507244 = sqrt(r507243);
        double r507245 = r507242 * r507244;
        double r507246 = cbrt(r507236);
        double r507247 = r507246 * r507246;
        double r507248 = r507233 / r507247;
        double r507249 = r507234 / r507246;
        double r507250 = r507248 * r507249;
        double r507251 = r507232 - r507250;
        double r507252 = cos(r507251);
        double r507253 = r507245 * r507252;
        double r507254 = a;
        double r507255 = b;
        double r507256 = r507254 / r507255;
        double r507257 = 1.0;
        double r507258 = r507257 / r507236;
        double r507259 = r507256 * r507258;
        double r507260 = r507253 - r507259;
        double r507261 = -0.5;
        double r507262 = 2.0;
        double r507263 = pow(r507232, r507262);
        double r507264 = fma(r507261, r507263, r507257);
        double r507265 = r507245 * r507264;
        double r507266 = r507257 / r507255;
        double r507267 = r507266 / r507236;
        double r507268 = r507254 * r507267;
        double r507269 = r507265 - r507268;
        double r507270 = r507241 ? r507260 : r507269;
        return r507270;
}

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

Original20.7
Target18.6
Herbie18.3
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999996037072

    1. Initial program 19.9

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

    3. Applied add-cube-cbrt19.9

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

    4. Applied times-frac19.8

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

    6. Applied associate-/r*19.9

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

    8. Applied div-inv19.9

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

    if 0.9999999996037072 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 22.1

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

    3. Applied add-cube-cbrt22.1

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

    4. Applied times-frac22.1

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

    6. Applied associate-/r*22.1

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

    8. Applied *-un-lft-identity22.1

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

    9. Applied div-inv22.1

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

    10. Applied times-frac22.1

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

    11. Simplified22.1

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

    12. Taylor expanded around 0 15.6

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

    13. Simplified15.6

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

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999603707223627679923083633184433:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{t}{\sqrt[3]{3}}\right) - \frac{a}{b} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - a \cdot \frac{\frac{1}{b}}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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))))