Average Error: 20.5 → 18.2
Time: 14.4s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty:\\ \;\;\;\;\log \left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;z \cdot t \le 2.13358673845001374 \cdot 10^{302}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \log \left(e^{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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:\\
\;\;\;\;\log \left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{\frac{a}{b}}{3}\\

\mathbf{elif}\;z \cdot t \le 2.13358673845001374 \cdot 10^{302}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \log \left(e^{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{\frac{a}{b}}{3}\\

\mathbf{else}:\\
\;\;\;\;\log \left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r773939 = 2.0;
        double r773940 = x;
        double r773941 = sqrt(r773940);
        double r773942 = r773939 * r773941;
        double r773943 = y;
        double r773944 = z;
        double r773945 = t;
        double r773946 = r773944 * r773945;
        double r773947 = 3.0;
        double r773948 = r773946 / r773947;
        double r773949 = r773943 - r773948;
        double r773950 = cos(r773949);
        double r773951 = r773942 * r773950;
        double r773952 = a;
        double r773953 = b;
        double r773954 = r773953 * r773947;
        double r773955 = r773952 / r773954;
        double r773956 = r773951 - r773955;
        return r773956;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r773957 = z;
        double r773958 = t;
        double r773959 = r773957 * r773958;
        double r773960 = -inf.0;
        bool r773961 = r773959 <= r773960;
        double r773962 = 2.0;
        double r773963 = x;
        double r773964 = sqrt(r773963);
        double r773965 = r773962 * r773964;
        double r773966 = exp(r773965);
        double r773967 = y;
        double r773968 = cos(r773967);
        double r773969 = 0.3333333333333333;
        double r773970 = r773958 * r773957;
        double r773971 = r773969 * r773970;
        double r773972 = cos(r773971);
        double r773973 = sin(r773967);
        double r773974 = 3.0;
        double r773975 = r773959 / r773974;
        double r773976 = sin(r773975);
        double r773977 = r773973 * r773976;
        double r773978 = fma(r773968, r773972, r773977);
        double r773979 = pow(r773966, r773978);
        double r773980 = log(r773979);
        double r773981 = a;
        double r773982 = b;
        double r773983 = r773981 / r773982;
        double r773984 = r773983 / r773974;
        double r773985 = r773980 - r773984;
        double r773986 = 2.1335867384500137e+302;
        bool r773987 = r773959 <= r773986;
        double r773988 = r773968 * r773972;
        double r773989 = exp(r773977);
        double r773990 = log(r773989);
        double r773991 = r773988 + r773990;
        double r773992 = r773965 * r773991;
        double r773993 = r773992 - r773984;
        double r773994 = cos(r773975);
        double r773995 = fma(r773968, r773994, r773977);
        double r773996 = pow(r773966, r773995);
        double r773997 = log(r773996);
        double r773998 = r773982 * r773974;
        double r773999 = r773981 / r773998;
        double r774000 = r773997 - r773999;
        double r774001 = r773987 ? r773993 : r774000;
        double r774002 = r773961 ? r773985 : r774001;
        return r774002;
}

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
Target18.5
Herbie18.2
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* z t) < -inf.0

    1. Initial program 64.0

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

    3. Applied cos-diff64.0

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

    4. Taylor expanded around inf 64.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\]
    5. Using strategy rm

    6. Applied associate-/r*64.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \color{blue}{\frac{\frac{a}{b}}{3}}\]
    7. Using strategy rm

    8. Applied add-log-exp64.0

      \[\leadsto \color{blue}{\log \left(e^{\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)}\right)} - \frac{\frac{a}{b}}{3}\]

    9. Simplified48.5

      \[\leadsto \log \color{blue}{\left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right)} - \frac{\frac{a}{b}}{3}\]

    if -inf.0 < (* z t) < 2.1335867384500137e+302

    1. Initial program 14.1

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

    3. Applied cos-diff13.7

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

    4. Taylor expanded around inf 13.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\]
    5. Using strategy rm

    6. Applied associate-/r*13.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \color{blue}{\frac{\frac{a}{b}}{3}}\]
    7. Using strategy rm

    8. Applied add-log-exp13.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \color{blue}{\log \left(e^{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right)}\right) - \frac{\frac{a}{b}}{3}\]

    if 2.1335867384500137e+302 < (* z t)

    1. Initial program 63.0

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

    3. Applied cos-diff63.0

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

    5. Applied add-log-exp63.2

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

    6. Simplified47.7

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

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty:\\ \;\;\;\;\log \left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;z \cdot t \le 2.13358673845001374 \cdot 10^{302}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \log \left(e^{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))