Average Error: 20.1 → 16.7
Time: 40.7s
Precision: 64
\[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -2.0242052511188462 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{b \cdot 3.0}\\ \mathbf{elif}\;z \cdot t \le -9.018122894565441 \cdot 10^{+61}:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{-t}{3.0} \cdot z\right)\right) \cdot \cos \left(\mathsf{fma}\left(\frac{-t}{3.0}, z, \frac{t}{3.0} \cdot z\right)\right) - \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{-t}{3.0} \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{-t}{3.0}, z, \frac{t}{3.0} \cdot z\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{b \cdot 3.0}\\ \mathbf{elif}\;z \cdot t \le 7.38439639698813 \cdot 10^{+306}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \left(\cos \left(\frac{z \cdot t}{3.0}\right) \cdot \cos y + \sin y \cdot \sin \left(\frac{t}{\sqrt[3]{3.0}} \cdot \frac{z}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}\right)\right) - \frac{a}{b \cdot 3.0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{b \cdot 3.0}\\ \end{array}\]
\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}
↓
\begin{array}{l}
\mathbf{if}\;z \cdot t \le -2.0242052511188462 \cdot 10^{+300}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{b \cdot 3.0}\\

\mathbf{elif}\;z \cdot t \le -9.018122894565441 \cdot 10^{+61}:\\
\;\;\;\;\left(\cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{-t}{3.0} \cdot z\right)\right) \cdot \cos \left(\mathsf{fma}\left(\frac{-t}{3.0}, z, \frac{t}{3.0} \cdot z\right)\right) - \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{-t}{3.0} \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{-t}{3.0}, z, \frac{t}{3.0} \cdot z\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{b \cdot 3.0}\\

\mathbf{elif}\;z \cdot t \le 7.38439639698813 \cdot 10^{+306}:\\
\;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \left(\cos \left(\frac{z \cdot t}{3.0}\right) \cdot \cos y + \sin y \cdot \sin \left(\frac{t}{\sqrt[3]{3.0}} \cdot \frac{z}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}\right)\right) - \frac{a}{b \cdot 3.0}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{b \cdot 3.0}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r33195052 = 2.0;
        double r33195053 = x;
        double r33195054 = sqrt(r33195053);
        double r33195055 = r33195052 * r33195054;
        double r33195056 = y;
        double r33195057 = z;
        double r33195058 = t;
        double r33195059 = r33195057 * r33195058;
        double r33195060 = 3.0;
        double r33195061 = r33195059 / r33195060;
        double r33195062 = r33195056 - r33195061;
        double r33195063 = cos(r33195062);
        double r33195064 = r33195055 * r33195063;
        double r33195065 = a;
        double r33195066 = b;
        double r33195067 = r33195066 * r33195060;
        double r33195068 = r33195065 / r33195067;
        double r33195069 = r33195064 - r33195068;
        return r33195069;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r33195070 = z;
        double r33195071 = t;
        double r33195072 = r33195070 * r33195071;
        double r33195073 = -2.0242052511188462e+300;
        bool r33195074 = r33195072 <= r33195073;
        double r33195075 = y;
        double r33195076 = r33195075 * r33195075;
        double r33195077 = -0.5;
        double r33195078 = 1.0;
        double r33195079 = fma(r33195076, r33195077, r33195078);
        double r33195080 = x;
        double r33195081 = sqrt(r33195080);
        double r33195082 = 2.0;
        double r33195083 = r33195081 * r33195082;
        double r33195084 = r33195079 * r33195083;
        double r33195085 = a;
        double r33195086 = b;
        double r33195087 = 3.0;
        double r33195088 = r33195086 * r33195087;
        double r33195089 = r33195085 / r33195088;
        double r33195090 = r33195084 - r33195089;
        double r33195091 = -9.018122894565441e+61;
        bool r33195092 = r33195072 <= r33195091;
        double r33195093 = cbrt(r33195075);
        double r33195094 = r33195093 * r33195093;
        double r33195095 = -r33195071;
        double r33195096 = r33195095 / r33195087;
        double r33195097 = r33195096 * r33195070;
        double r33195098 = fma(r33195094, r33195093, r33195097);
        double r33195099 = cos(r33195098);
        double r33195100 = r33195071 / r33195087;
        double r33195101 = r33195100 * r33195070;
        double r33195102 = fma(r33195096, r33195070, r33195101);
        double r33195103 = cos(r33195102);
        double r33195104 = r33195099 * r33195103;
        double r33195105 = sin(r33195098);
        double r33195106 = sin(r33195102);
        double r33195107 = r33195105 * r33195106;
        double r33195108 = r33195104 - r33195107;
        double r33195109 = r33195108 * r33195083;
        double r33195110 = r33195109 - r33195089;
        double r33195111 = 7.38439639698813e+306;
        bool r33195112 = r33195072 <= r33195111;
        double r33195113 = r33195072 / r33195087;
        double r33195114 = cos(r33195113);
        double r33195115 = cos(r33195075);
        double r33195116 = r33195114 * r33195115;
        double r33195117 = sin(r33195075);
        double r33195118 = cbrt(r33195087);
        double r33195119 = r33195071 / r33195118;
        double r33195120 = r33195118 * r33195118;
        double r33195121 = r33195070 / r33195120;
        double r33195122 = r33195119 * r33195121;
        double r33195123 = sin(r33195122);
        double r33195124 = r33195117 * r33195123;
        double r33195125 = r33195116 + r33195124;
        double r33195126 = r33195083 * r33195125;
        double r33195127 = r33195126 - r33195089;
        double r33195128 = r33195112 ? r33195127 : r33195090;
        double r33195129 = r33195092 ? r33195110 : r33195128;
        double r33195130 = r33195074 ? r33195090 : r33195129;
        return r33195130;
}

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

Original20.1
Target18.4
Herbie16.7
\[\begin{array}{l} \mathbf{if}\;z \lt -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \cos \left(y - \frac{t}{3.0} \cdot z\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2.0 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3.0}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* z t) < -2.0242052511188462e+300 or 7.38439639698813e+306 < (* z t)

    1. Initial program 61.6

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

    2. Taylor expanded around 0 44.8

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3.0}\]

    3. Simplified44.8

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right)} - \frac{a}{b \cdot 3.0}\]

    if -2.0242052511188462e+300 < (* z t) < -9.018122894565441e+61

    1. Initial program 33.5

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity33.5

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{\color{blue}{1 \cdot 3.0}}\right) - \frac{a}{b \cdot 3.0}\]

    4. Applied times-frac33.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\frac{z}{1} \cdot \frac{t}{3.0}}\right) - \frac{a}{b \cdot 3.0}\]

    5. Applied add-cube-cbrt33.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} - \frac{z}{1} \cdot \frac{t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

    6. Applied prod-diff33.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, -\frac{t}{3.0} \cdot \frac{z}{1}\right) + \mathsf{fma}\left(-\frac{t}{3.0}, \frac{z}{1}, \frac{t}{3.0} \cdot \frac{z}{1}\right)\right)} - \frac{a}{b \cdot 3.0}\]

    7. Applied cos-sum27.6

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, -\frac{t}{3.0} \cdot \frac{z}{1}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3.0}, \frac{z}{1}, \frac{t}{3.0} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, -\frac{t}{3.0} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3.0}, \frac{z}{1}, \frac{t}{3.0} \cdot \frac{z}{1}\right)\right)\right)} - \frac{a}{b \cdot 3.0}\]

    if -9.018122894565441e+61 < (* z t) < 7.38439639698813e+306

    1. Initial program 10.6

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
    2. Using strategy rm

    3. Applied cos-diff10.0

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3.0}\right)\right)} - \frac{a}{b \cdot 3.0}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt10.0

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{\color{blue}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}}}\right)\right) - \frac{a}{b \cdot 3.0}\]

    6. Applied times-frac10.0

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \color{blue}{\left(\frac{z}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \frac{t}{\sqrt[3]{3.0}}\right)}\right) - \frac{a}{b \cdot 3.0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -2.0242052511188462 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{b \cdot 3.0}\\ \mathbf{elif}\;z \cdot t \le -9.018122894565441 \cdot 10^{+61}:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{-t}{3.0} \cdot z\right)\right) \cdot \cos \left(\mathsf{fma}\left(\frac{-t}{3.0}, z, \frac{t}{3.0} \cdot z\right)\right) - \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{-t}{3.0} \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{-t}{3.0}, z, \frac{t}{3.0} \cdot z\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{b \cdot 3.0}\\ \mathbf{elif}\;z \cdot t \le 7.38439639698813 \cdot 10^{+306}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \left(\cos \left(\frac{z \cdot t}{3.0}\right) \cdot \cos y + \sin y \cdot \sin \left(\frac{t}{\sqrt[3]{3.0}} \cdot \frac{z}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}\right)\right) - \frac{a}{b \cdot 3.0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{b \cdot 3.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))