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

Average Error: 20.4 → 17.6

Time: 14.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 1.0112129461852768 \cdot 10^{290}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) + \sin y \cdot \log \left(e^{\sin \left(\frac{z}{\sqrt{3}} \cdot \left(\left(\sqrt[3]{\frac{t}{\sqrt{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right) \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right)\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r885545 = 2.0;
        double r885546 = x;
        double r885547 = sqrt(r885546);
        double r885548 = r885545 * r885547;
        double r885549 = y;
        double r885550 = z;
        double r885551 = t;
        double r885552 = r885550 * r885551;
        double r885553 = 3.0;
        double r885554 = r885552 / r885553;
        double r885555 = r885549 - r885554;
        double r885556 = cos(r885555);
        double r885557 = r885548 * r885556;
        double r885558 = a;
        double r885559 = b;
        double r885560 = r885559 * r885553;
        double r885561 = r885558 / r885560;
        double r885562 = r885557 - r885561;
        return r885562;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r885563 = z;
        double r885564 = t;
        double r885565 = r885563 * r885564;
        double r885566 = -inf.0;
        bool r885567 = r885565 <= r885566;
        double r885568 = 1.0112129461852768e+290;
        bool r885569 = r885565 <= r885568;
        double r885570 = !r885569;
        bool r885571 = r885567 || r885570;
        double r885572 = 2.0;
        double r885573 = x;
        double r885574 = sqrt(r885573);
        double r885575 = r885572 * r885574;
        double r885576 = 1.0;
        double r885577 = 0.5;
        double r885578 = y;
        double r885579 = 2.0;
        double r885580 = pow(r885578, r885579);
        double r885581 = r885577 * r885580;
        double r885582 = r885576 - r885581;
        double r885583 = r885575 * r885582;
        double r885584 = a;
        double r885585 = b;
        double r885586 = 3.0;
        double r885587 = r885585 * r885586;
        double r885588 = r885584 / r885587;
        double r885589 = r885583 - r885588;
        double r885590 = cos(r885578);
        double r885591 = sqrt(r885586);
        double r885592 = r885563 / r885591;
        double r885593 = r885564 / r885591;
        double r885594 = r885592 * r885593;
        double r885595 = cos(r885594);
        double r885596 = r885590 * r885595;
        double r885597 = sin(r885578);
        double r885598 = cbrt(r885593);
        double r885599 = r885598 * r885598;
        double r885600 = r885599 * r885598;
        double r885601 = r885592 * r885600;
        double r885602 = sin(r885601);
        double r885603 = exp(r885602);
        double r885604 = log(r885603);
        double r885605 = r885597 * r885604;
        double r885606 = r885596 + r885605;
        double r885607 = r885575 * r885606;
        double r885608 = r885607 - r885588;
        double r885609 = r885571 ? r885589 : r885608;
        return r885609;
}

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	20.4
Target	18.3
Herbie	17.6

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

Derivation

1. Split input into 2 regimes

2. if (* z t) < -inf.0 or 1.0112129461852768e+290 < (* z t)

   1. Initial program 62.4

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

   2. Taylor expanded around 0 44.1

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

   if -inf.0 < (* z t) < 1.0112129461852768e+290

   1. Initial program 13.9

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

   3. Applied add-sqr-sqrt 14.0

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

   4. Applied times-frac 13.9

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

   6. Applied cos-diff 13.5

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

   8. Applied add-cube-cbrt 13.5

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

   10. Applied add-log-exp 13.5

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 1.0112129461852768 \cdot 10^{290}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) + \sin y \cdot \log \left(e^{\sin \left(\frac{z}{\sqrt{3}} \cdot \left(\left(\sqrt[3]{\frac{t}{\sqrt{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right) \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right)\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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))))