Average Error: 20.7 → 16.5
Time: 42.9s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.999999999991291077527932884549954906106:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(\frac{-t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) - \sin y \cdot \sin \left(\frac{t}{3} \cdot \left(-z\right)\right)\right) - \sin \left(\mathsf{fma}\left(\frac{-t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(1, y, \frac{t}{3} \cdot \left(-z\right)\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) - \frac{a}{3 \cdot b}\\ \end{array}\]
\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{t \cdot z}{3}\right) \le 0.999999999991291077527932884549954906106:\\
\;\;\;\;\left(\cos \left(\mathsf{fma}\left(\frac{-t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) - \sin y \cdot \sin \left(\frac{t}{3} \cdot \left(-z\right)\right)\right) - \sin \left(\mathsf{fma}\left(\frac{-t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(1, y, \frac{t}{3} \cdot \left(-z\right)\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r30695990 = 2.0;
        double r30695991 = x;
        double r30695992 = sqrt(r30695991);
        double r30695993 = r30695990 * r30695992;
        double r30695994 = y;
        double r30695995 = z;
        double r30695996 = t;
        double r30695997 = r30695995 * r30695996;
        double r30695998 = 3.0;
        double r30695999 = r30695997 / r30695998;
        double r30696000 = r30695994 - r30695999;
        double r30696001 = cos(r30696000);
        double r30696002 = r30695993 * r30696001;
        double r30696003 = a;
        double r30696004 = b;
        double r30696005 = r30696004 * r30695998;
        double r30696006 = r30696003 / r30696005;
        double r30696007 = r30696002 - r30696006;
        return r30696007;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r30696008 = y;
        double r30696009 = t;
        double r30696010 = z;
        double r30696011 = r30696009 * r30696010;
        double r30696012 = 3.0;
        double r30696013 = r30696011 / r30696012;
        double r30696014 = r30696008 - r30696013;
        double r30696015 = cos(r30696014);
        double r30696016 = 0.9999999999912911;
        bool r30696017 = r30696015 <= r30696016;
        double r30696018 = -r30696009;
        double r30696019 = r30696018 / r30696012;
        double r30696020 = r30696009 / r30696012;
        double r30696021 = r30696010 * r30696020;
        double r30696022 = fma(r30696019, r30696010, r30696021);
        double r30696023 = cos(r30696022);
        double r30696024 = cos(r30696008);
        double r30696025 = 0.3333333333333333;
        double r30696026 = r30696011 * r30696025;
        double r30696027 = cos(r30696026);
        double r30696028 = r30696024 * r30696027;
        double r30696029 = sin(r30696008);
        double r30696030 = -r30696010;
        double r30696031 = r30696020 * r30696030;
        double r30696032 = sin(r30696031);
        double r30696033 = r30696029 * r30696032;
        double r30696034 = r30696028 - r30696033;
        double r30696035 = r30696023 * r30696034;
        double r30696036 = sin(r30696022);
        double r30696037 = 1.0;
        double r30696038 = fma(r30696037, r30696008, r30696031);
        double r30696039 = sin(r30696038);
        double r30696040 = r30696036 * r30696039;
        double r30696041 = r30696035 - r30696040;
        double r30696042 = x;
        double r30696043 = sqrt(r30696042);
        double r30696044 = 2.0;
        double r30696045 = r30696043 * r30696044;
        double r30696046 = r30696041 * r30696045;
        double r30696047 = a;
        double r30696048 = b;
        double r30696049 = r30696012 * r30696048;
        double r30696050 = r30696047 / r30696049;
        double r30696051 = r30696046 - r30696050;
        double r30696052 = -0.5;
        double r30696053 = r30696008 * r30696008;
        double r30696054 = fma(r30696052, r30696053, r30696037);
        double r30696055 = r30696045 * r30696054;
        double r30696056 = r30696055 - r30696050;
        double r30696057 = r30696017 ? r30696051 : r30696056;
        return r30696057;
}

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
Target18.7
Herbie16.5
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999912911

    1. Initial program 19.7

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

    3. Applied *-un-lft-identity19.7

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

    4. Applied times-frac19.7

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

    5. Applied *-un-lft-identity19.7

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

    6. Applied prod-diff19.7

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

    7. Applied cos-sum16.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right)} - \frac{a}{b \cdot 3}\]
    8. Using strategy rm

    9. Applied fma-udef16.6

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

    10. Applied cos-sum15.7

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

    11. Taylor expanded around inf 16.9

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(1 \cdot y\right) \cdot \color{blue}{\cos \left(-0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3}\]

    12. Simplified16.9

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(1 \cdot y\right) \cdot \color{blue}{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right)} - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3}\]

    if 0.9999999999912911 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 22.3

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

    2. Taylor expanded around 0 15.6

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

    3. Simplified15.6

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.999999999991291077527932884549954906106:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(\frac{-t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) - \sin y \cdot \sin \left(\frac{t}{3} \cdot \left(-z\right)\right)\right) - \sin \left(\mathsf{fma}\left(\frac{-t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(1, y, \frac{t}{3} \cdot \left(-z\right)\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +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 (- (/ 1.0 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))))