Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, K

Average Error: 21.0 → 16.7

Time: 41.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9997361592973614818902206025086343288422:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \left(\cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \cos \left(\mathsf{fma}\left(1, y, \left(-z\right) \cdot \frac{t}{3}\right)\right) - \left(\cos \left(\left(-z\right) \cdot \frac{t}{3}\right) \cdot \sin y + \sin \left(\left(-z\right) \cdot \frac{t}{3}\right) \cdot \cos y\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r33396546 = 2.0;
        double r33396547 = x;
        double r33396548 = sqrt(r33396547);
        double r33396549 = r33396546 * r33396548;
        double r33396550 = y;
        double r33396551 = z;
        double r33396552 = t;
        double r33396553 = r33396551 * r33396552;
        double r33396554 = 3.0;
        double r33396555 = r33396553 / r33396554;
        double r33396556 = r33396550 - r33396555;
        double r33396557 = cos(r33396556);
        double r33396558 = r33396549 * r33396557;
        double r33396559 = a;
        double r33396560 = b;
        double r33396561 = r33396560 * r33396554;
        double r33396562 = r33396559 / r33396561;
        double r33396563 = r33396558 - r33396562;
        return r33396563;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r33396564 = y;
        double r33396565 = t;
        double r33396566 = z;
        double r33396567 = r33396565 * r33396566;
        double r33396568 = 3.0;
        double r33396569 = r33396567 / r33396568;
        double r33396570 = r33396564 - r33396569;
        double r33396571 = cos(r33396570);
        double r33396572 = 0.9997361592973615;
        bool r33396573 = r33396571 <= r33396572;
        double r33396574 = x;
        double r33396575 = sqrt(r33396574);
        double r33396576 = 2.0;
        double r33396577 = r33396575 * r33396576;
        double r33396578 = r33396565 / r33396568;
        double r33396579 = -r33396578;
        double r33396580 = r33396578 * r33396566;
        double r33396581 = fma(r33396579, r33396566, r33396580);
        double r33396582 = cos(r33396581);
        double r33396583 = 1.0;
        double r33396584 = -r33396566;
        double r33396585 = r33396584 * r33396578;
        double r33396586 = fma(r33396583, r33396564, r33396585);
        double r33396587 = cos(r33396586);
        double r33396588 = r33396582 * r33396587;
        double r33396589 = cos(r33396585);
        double r33396590 = sin(r33396564);
        double r33396591 = r33396589 * r33396590;
        double r33396592 = sin(r33396585);
        double r33396593 = cos(r33396564);
        double r33396594 = r33396592 * r33396593;
        double r33396595 = r33396591 + r33396594;
        double r33396596 = sin(r33396581);
        double r33396597 = r33396595 * r33396596;
        double r33396598 = r33396588 - r33396597;
        double r33396599 = r33396577 * r33396598;
        double r33396600 = a;
        double r33396601 = b;
        double r33396602 = r33396601 * r33396568;
        double r33396603 = r33396600 / r33396602;
        double r33396604 = r33396599 - r33396603;
        double r33396605 = r33396564 * r33396564;
        double r33396606 = -0.5;
        double r33396607 = fma(r33396605, r33396606, r33396583);
        double r33396608 = r33396577 * r33396607;
        double r33396609 = r33396608 - r33396603;
        double r33396610 = r33396573 ? r33396604 : r33396609;
        return r33396610;
}

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

Original	21.0
Target	18.9
Herbie	16.7

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9997361592973615

   1. Initial program 20.6

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

   3. Applied *-un-lft-identity 20.6

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

   4. Applied times-frac 20.6

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

   5. Applied *-un-lft-identity 20.6

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

   6. Applied prod-diff 20.6

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

   7. Applied cos-sum 17.3

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

   9. Applied fma-udef 17.3

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

   10. Applied sin-sum 17.2

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

   if 0.9997361592973615 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.8

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

   2. Taylor expanded around 0 16.1

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

   3. Simplified 16.1

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

4. Final simplification 16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9997361592973614818902206025086343288422:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \left(\cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \cos \left(\mathsf{fma}\left(1, y, \left(-z\right) \cdot \frac{t}{3}\right)\right) - \left(\cos \left(\left(-z\right) \cdot \frac{t}{3}\right) \cdot \sin y + \sin \left(\left(-z\right) \cdot \frac{t}{3}\right) \cdot \cos y\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

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))))