Average Error: 20.9 → 18.4
Time: 13.8s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999929107336669:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) + \sin y \cdot \sin \left(\frac{z}{\frac{3}{t}}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999929107336669:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) + \sin y \cdot \sin \left(\frac{z}{\frac{3}{t}}\right)\right) - \frac{\frac{a}{b}}{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 r741075 = 2.0;
        double r741076 = x;
        double r741077 = sqrt(r741076);
        double r741078 = r741075 * r741077;
        double r741079 = y;
        double r741080 = z;
        double r741081 = t;
        double r741082 = r741080 * r741081;
        double r741083 = 3.0;
        double r741084 = r741082 / r741083;
        double r741085 = r741079 - r741084;
        double r741086 = cos(r741085);
        double r741087 = r741078 * r741086;
        double r741088 = a;
        double r741089 = b;
        double r741090 = r741089 * r741083;
        double r741091 = r741088 / r741090;
        double r741092 = r741087 - r741091;
        return r741092;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r741093 = y;
        double r741094 = z;
        double r741095 = t;
        double r741096 = r741094 * r741095;
        double r741097 = 3.0;
        double r741098 = r741096 / r741097;
        double r741099 = r741093 - r741098;
        double r741100 = cos(r741099);
        double r741101 = 0.9999929107336669;
        bool r741102 = r741100 <= r741101;
        double r741103 = 2.0;
        double r741104 = x;
        double r741105 = sqrt(r741104);
        double r741106 = r741103 * r741105;
        double r741107 = cos(r741093);
        double r741108 = cbrt(r741097);
        double r741109 = r741108 * r741108;
        double r741110 = r741096 / r741109;
        double r741111 = r741110 / r741108;
        double r741112 = cos(r741111);
        double r741113 = r741107 * r741112;
        double r741114 = sin(r741093);
        double r741115 = r741097 / r741095;
        double r741116 = r741094 / r741115;
        double r741117 = sin(r741116);
        double r741118 = r741114 * r741117;
        double r741119 = r741113 + r741118;
        double r741120 = r741106 * r741119;
        double r741121 = a;
        double r741122 = b;
        double r741123 = r741121 / r741122;
        double r741124 = r741123 / r741097;
        double r741125 = r741120 - r741124;
        double r741126 = 1.0;
        double r741127 = 0.5;
        double r741128 = 2.0;
        double r741129 = pow(r741093, r741128);
        double r741130 = r741127 * r741129;
        double r741131 = r741126 - r741130;
        double r741132 = r741106 * r741131;
        double r741133 = r741122 * r741097;
        double r741134 = r741121 / r741133;
        double r741135 = r741132 - r741134;
        double r741136 = r741102 ? r741125 : r741135;
        return r741136;
}

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

Original20.9
Target19.1
Herbie18.4
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos (- y (/ (* z t) 3.0))) < 0.9999929107336669

    1. Initial program 20.3

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

    3. Applied cos-diff19.7

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

    5. Applied associate-/l*19.7

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

    7. Applied add-cube-cbrt19.7

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

    8. Applied associate-/r*19.7

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

    10. Applied associate-/r*19.7

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

    if 0.9999929107336669 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 21.9

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

    2. Taylor expanded around 0 16.2

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

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999929107336669:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) + \sin y \cdot \sin \left(\frac{z}{\frac{3}{t}}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))))