Average Error: 20.8 → 17.9
Time: 9.8s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 6.6600521486915968 \cdot 10^{153}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 6.6600521486915968 \cdot 10^{153}:\\
\;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r750050 = 2.0;
        double r750051 = x;
        double r750052 = sqrt(r750051);
        double r750053 = r750050 * r750052;
        double r750054 = y;
        double r750055 = z;
        double r750056 = t;
        double r750057 = r750055 * r750056;
        double r750058 = 3.0;
        double r750059 = r750057 / r750058;
        double r750060 = r750054 - r750059;
        double r750061 = cos(r750060);
        double r750062 = r750053 * r750061;
        double r750063 = a;
        double r750064 = b;
        double r750065 = r750064 * r750058;
        double r750066 = r750063 / r750065;
        double r750067 = r750062 - r750066;
        return r750067;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r750068 = 2.0;
        double r750069 = x;
        double r750070 = sqrt(r750069);
        double r750071 = r750068 * r750070;
        double r750072 = y;
        double r750073 = z;
        double r750074 = t;
        double r750075 = r750073 * r750074;
        double r750076 = 3.0;
        double r750077 = r750075 / r750076;
        double r750078 = r750072 - r750077;
        double r750079 = cos(r750078);
        double r750080 = r750071 * r750079;
        double r750081 = 6.660052148691597e+153;
        bool r750082 = r750080 <= r750081;
        double r750083 = cos(r750072);
        double r750084 = cos(r750077);
        double r750085 = r750083 * r750084;
        double r750086 = r750071 * r750085;
        double r750087 = sin(r750072);
        double r750088 = sin(r750077);
        double r750089 = r750087 * r750088;
        double r750090 = cbrt(r750089);
        double r750091 = r750090 * r750090;
        double r750092 = r750091 * r750090;
        double r750093 = r750071 * r750092;
        double r750094 = r750086 + r750093;
        double r750095 = a;
        double r750096 = b;
        double r750097 = r750096 * r750076;
        double r750098 = r750095 / r750097;
        double r750099 = r750094 - r750098;
        double r750100 = 1.0;
        double r750101 = 0.5;
        double r750102 = 2.0;
        double r750103 = pow(r750072, r750102);
        double r750104 = r750101 * r750103;
        double r750105 = r750100 - r750104;
        double r750106 = r750071 * r750105;
        double r750107 = r750106 - r750098;
        double r750108 = r750082 ? r750099 : r750107;
        return r750108;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.8
Target18.6
Herbie17.9
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) < 6.660052148691597e+153

    1. Initial program 14.2

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

    3. Applied cos-diff13.8

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

    4. Applied distribute-lft-in13.8

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

    6. Applied add-cube-cbrt13.8

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

    if 6.660052148691597e+153 < (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))

    1. Initial program 63.8

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

    2. Taylor expanded around 0 44.9

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 6.6600521486915968 \cdot 10^{153}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))))