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

Average Error: 20.6 → 17.8

Time: 11.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999429473585:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right) + \log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r975609 = 2.0;
        double r975610 = x;
        double r975611 = sqrt(r975610);
        double r975612 = r975609 * r975611;
        double r975613 = y;
        double r975614 = z;
        double r975615 = t;
        double r975616 = r975614 * r975615;
        double r975617 = 3.0;
        double r975618 = r975616 / r975617;
        double r975619 = r975613 - r975618;
        double r975620 = cos(r975619);
        double r975621 = r975612 * r975620;
        double r975622 = a;
        double r975623 = b;
        double r975624 = r975623 * r975617;
        double r975625 = r975622 / r975624;
        double r975626 = r975621 - r975625;
        return r975626;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r975627 = y;
        double r975628 = z;
        double r975629 = t;
        double r975630 = r975628 * r975629;
        double r975631 = 3.0;
        double r975632 = r975630 / r975631;
        double r975633 = r975627 - r975632;
        double r975634 = cos(r975633);
        double r975635 = 0.9999999429473585;
        bool r975636 = r975634 <= r975635;
        double r975637 = 2.0;
        double r975638 = x;
        double r975639 = sqrt(r975638);
        double r975640 = r975637 * r975639;
        double r975641 = cos(r975627);
        double r975642 = cos(r975632);
        double r975643 = exp(r975642);
        double r975644 = sqrt(r975643);
        double r975645 = log(r975644);
        double r975646 = r975645 + r975645;
        double r975647 = r975641 * r975646;
        double r975648 = r975640 * r975647;
        double r975649 = sin(r975627);
        double r975650 = 0.3333333333333333;
        double r975651 = r975629 * r975628;
        double r975652 = r975650 * r975651;
        double r975653 = sin(r975652);
        double r975654 = r975649 * r975653;
        double r975655 = r975640 * r975654;
        double r975656 = r975648 + r975655;
        double r975657 = a;
        double r975658 = b;
        double r975659 = r975658 * r975631;
        double r975660 = r975657 / r975659;
        double r975661 = r975656 - r975660;
        double r975662 = 1.0;
        double r975663 = 0.5;
        double r975664 = 2.0;
        double r975665 = pow(r975627, r975664);
        double r975666 = r975663 * r975665;
        double r975667 = r975662 - r975666;
        double r975668 = r975640 * r975667;
        double r975669 = r975668 - r975660;
        double r975670 = r975636 ? r975661 : r975669;
        return r975670;
}

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	20.6
Target	18.7
Herbie	17.8

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999429473585

   1. Initial program 19.9

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

   3. Applied cos-diff 19.2

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

   4. Applied distribute-lft-in 19.2

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

   5. Taylor expanded around inf 19.2

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

   7. Applied add-log-exp 19.2

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

   9. Applied add-sqr-sqrt 19.2

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

   10. Applied log-prod 19.2

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

   if 0.9999999429473585 < (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 15.5

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999429473585:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right) + \log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))))