Average Error: 21.0 → 20.7
Time: 31.1s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(z \cdot 0.3333333333333333148296162562473909929395\right) \cdot t\right)\right)\right) \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) + \sin y \cdot \left(\sin \left(\frac{t}{\frac{3}{z}}\right) \cdot \left(2 \cdot \sqrt{x}\right)\right)\right) - \frac{\frac{a}{b}}{3}\]
\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}
↓
\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(z \cdot 0.3333333333333333148296162562473909929395\right) \cdot t\right)\right)\right) \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) + \sin y \cdot \left(\sin \left(\frac{t}{\frac{3}{z}}\right) \cdot \left(2 \cdot \sqrt{x}\right)\right)\right) - \frac{\frac{a}{b}}{3}
double f(double x, double y, double z, double t, double a, double b) {
        double r554397 = 2.0;
        double r554398 = x;
        double r554399 = sqrt(r554398);
        double r554400 = r554397 * r554399;
        double r554401 = y;
        double r554402 = z;
        double r554403 = t;
        double r554404 = r554402 * r554403;
        double r554405 = 3.0;
        double r554406 = r554404 / r554405;
        double r554407 = r554401 - r554406;
        double r554408 = cos(r554407);
        double r554409 = r554400 * r554408;
        double r554410 = a;
        double r554411 = b;
        double r554412 = r554411 * r554405;
        double r554413 = r554410 / r554412;
        double r554414 = r554409 - r554413;
        return r554414;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r554415 = z;
        double r554416 = 0.3333333333333333;
        double r554417 = r554415 * r554416;
        double r554418 = t;
        double r554419 = r554417 * r554418;
        double r554420 = cos(r554419);
        double r554421 = log1p(r554420);
        double r554422 = expm1(r554421);
        double r554423 = 2.0;
        double r554424 = x;
        double r554425 = sqrt(r554424);
        double r554426 = r554423 * r554425;
        double r554427 = y;
        double r554428 = cos(r554427);
        double r554429 = r554426 * r554428;
        double r554430 = r554422 * r554429;
        double r554431 = sin(r554427);
        double r554432 = 3.0;
        double r554433 = r554432 / r554415;
        double r554434 = r554418 / r554433;
        double r554435 = sin(r554434);
        double r554436 = r554435 * r554426;
        double r554437 = r554431 * r554436;
        double r554438 = r554430 + r554437;
        double r554439 = a;
        double r554440 = b;
        double r554441 = r554439 / r554440;
        double r554442 = r554441 / r554432;
        double r554443 = r554438 - r554442;
        return r554443;
}

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

Original21.0
Target18.9
Herbie20.7
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Initial program 21.0

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

  3. Applied cos-diff20.7

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

  4. Applied distribute-lft-in20.7

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

  5. Simplified20.6

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

  6. Simplified20.6

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

  7. Taylor expanded around inf 20.7

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

  8. Simplified20.7

    \[\leadsto \left(\left(\left(\sqrt{x} \cdot 2\right) \cdot \cos y\right) \cdot \color{blue}{\cos \left(t \cdot \left(0.3333333333333333148296162562473909929395 \cdot z\right)\right)} + \sin y \cdot \left(\sin \left(\frac{t}{\frac{3}{z}}\right) \cdot \left(\sqrt{x} \cdot 2\right)\right)\right) - \frac{a}{b \cdot 3}\]
  9. Using strategy rm

  10. Applied associate-/r*20.7

    \[\leadsto \left(\left(\left(\sqrt{x} \cdot 2\right) \cdot \cos y\right) \cdot \cos \left(t \cdot \left(0.3333333333333333148296162562473909929395 \cdot z\right)\right) + \sin y \cdot \left(\sin \left(\frac{t}{\frac{3}{z}}\right) \cdot \left(\sqrt{x} \cdot 2\right)\right)\right) - \color{blue}{\frac{\frac{a}{b}}{3}}\]
  11. Using strategy rm

  12. Applied expm1-log1p-u20.7

    \[\leadsto \left(\left(\left(\sqrt{x} \cdot 2\right) \cdot \cos y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(t \cdot \left(0.3333333333333333148296162562473909929395 \cdot z\right)\right)\right)\right)} + \sin y \cdot \left(\sin \left(\frac{t}{\frac{3}{z}}\right) \cdot \left(\sqrt{x} \cdot 2\right)\right)\right) - \frac{\frac{a}{b}}{3}\]

  13. Simplified20.7

    \[\leadsto \left(\left(\left(\sqrt{x} \cdot 2\right) \cdot \cos y\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos \left(\left(z \cdot 0.3333333333333333148296162562473909929395\right) \cdot t\right)\right)}\right) + \sin y \cdot \left(\sin \left(\frac{t}{\frac{3}{z}}\right) \cdot \left(\sqrt{x} \cdot 2\right)\right)\right) - \frac{\frac{a}{b}}{3}\]

  14. Final simplification20.7

    \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(z \cdot 0.3333333333333333148296162562473909929395\right) \cdot t\right)\right)\right) \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) + \sin y \cdot \left(\sin \left(\frac{t}{\frac{3}{z}}\right) \cdot \left(2 \cdot \sqrt{x}\right)\right)\right) - \frac{\frac{a}{b}}{3}\]

Reproduce

herbie shell --seed 2019179 +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))))