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

Average Error: 21.1 → 17.8

Time: 13.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99998911849150296:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) - \sin y \cdot \log \left(e^{\sin \left(-\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r1028015 = 2.0;
        double r1028016 = x;
        double r1028017 = sqrt(r1028016);
        double r1028018 = r1028015 * r1028017;
        double r1028019 = y;
        double r1028020 = z;
        double r1028021 = t;
        double r1028022 = r1028020 * r1028021;
        double r1028023 = 3.0;
        double r1028024 = r1028022 / r1028023;
        double r1028025 = r1028019 - r1028024;
        double r1028026 = cos(r1028025);
        double r1028027 = r1028018 * r1028026;
        double r1028028 = a;
        double r1028029 = b;
        double r1028030 = r1028029 * r1028023;
        double r1028031 = r1028028 / r1028030;
        double r1028032 = r1028027 - r1028031;
        return r1028032;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r1028033 = y;
        double r1028034 = z;
        double r1028035 = t;
        double r1028036 = r1028034 * r1028035;
        double r1028037 = 3.0;
        double r1028038 = r1028036 / r1028037;
        double r1028039 = r1028033 - r1028038;
        double r1028040 = cos(r1028039);
        double r1028041 = 0.999989118491503;
        bool r1028042 = r1028040 <= r1028041;
        double r1028043 = 2.0;
        double r1028044 = x;
        double r1028045 = sqrt(r1028044);
        double r1028046 = r1028043 * r1028045;
        double r1028047 = cos(r1028033);
        double r1028048 = cos(r1028038);
        double r1028049 = cbrt(r1028048);
        double r1028050 = r1028049 * r1028049;
        double r1028051 = r1028050 * r1028049;
        double r1028052 = cbrt(r1028051);
        double r1028053 = r1028049 * r1028052;
        double r1028054 = r1028053 * r1028049;
        double r1028055 = r1028047 * r1028054;
        double r1028056 = sin(r1028033);
        double r1028057 = -r1028038;
        double r1028058 = sin(r1028057);
        double r1028059 = exp(r1028058);
        double r1028060 = log(r1028059);
        double r1028061 = r1028056 * r1028060;
        double r1028062 = r1028055 - r1028061;
        double r1028063 = r1028046 * r1028062;
        double r1028064 = a;
        double r1028065 = b;
        double r1028066 = r1028065 * r1028037;
        double r1028067 = r1028064 / r1028066;
        double r1028068 = r1028063 - r1028067;
        double r1028069 = 1.0;
        double r1028070 = 0.5;
        double r1028071 = 2.0;
        double r1028072 = pow(r1028033, r1028071);
        double r1028073 = r1028070 * r1028072;
        double r1028074 = r1028069 - r1028073;
        double r1028075 = r1028046 * r1028074;
        double r1028076 = r1028075 - r1028067;
        double r1028077 = r1028042 ? r1028068 : r1028076;
        return r1028077;
}

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	21.1
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))) < 0.999989118491503

   1. Initial program 20.0

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

   3. Applied sub-neg 20.0

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

   4. Applied cos-sum 19.3

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

   5. Simplified 19.3

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

   7. Applied add-log-exp 19.3

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

   9. Applied add-cube-cbrt 19.3

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

   11. Applied add-cube-cbrt 19.3

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

   if 0.999989118491503 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.8

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

   2. Taylor expanded around 0 15.4

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99998911849150296:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) - \sin y \cdot \log \left(e^{\sin \left(-\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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))))