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

Average Error: 20.4 → 16.5

Time: 56.6s

Precision: 64

Math

\[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3.0}\right) \le 0.9964954255751017:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \left(\cos \left(\mathsf{fma}\left(\frac{-t}{3.0}, z, z \cdot \frac{t}{3.0}\right)\right) \cdot \left(\cos y \cdot \cos \left(-z \cdot \frac{t}{3.0}\right) - \sin \left(-z \cdot \frac{t}{3.0}\right) \cdot \sin y\right) - \sin \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3.0}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{-t}{3.0}, z, z \cdot \frac{t}{3.0}\right)\right)\right) - \frac{a}{b \cdot 3.0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3.0}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r29467050 = 2.0;
        double r29467051 = x;
        double r29467052 = sqrt(r29467051);
        double r29467053 = r29467050 * r29467052;
        double r29467054 = y;
        double r29467055 = z;
        double r29467056 = t;
        double r29467057 = r29467055 * r29467056;
        double r29467058 = 3.0;
        double r29467059 = r29467057 / r29467058;
        double r29467060 = r29467054 - r29467059;
        double r29467061 = cos(r29467060);
        double r29467062 = r29467053 * r29467061;
        double r29467063 = a;
        double r29467064 = b;
        double r29467065 = r29467064 * r29467058;
        double r29467066 = r29467063 / r29467065;
        double r29467067 = r29467062 - r29467066;
        return r29467067;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r29467068 = y;
        double r29467069 = t;
        double r29467070 = z;
        double r29467071 = r29467069 * r29467070;
        double r29467072 = 3.0;
        double r29467073 = r29467071 / r29467072;
        double r29467074 = r29467068 - r29467073;
        double r29467075 = cos(r29467074);
        double r29467076 = 0.9964954255751017;
        bool r29467077 = r29467075 <= r29467076;
        double r29467078 = x;
        double r29467079 = sqrt(r29467078);
        double r29467080 = 2.0;
        double r29467081 = r29467079 * r29467080;
        double r29467082 = -r29467069;
        double r29467083 = r29467082 / r29467072;
        double r29467084 = r29467069 / r29467072;
        double r29467085 = r29467070 * r29467084;
        double r29467086 = fma(r29467083, r29467070, r29467085);
        double r29467087 = cos(r29467086);
        double r29467088 = cos(r29467068);
        double r29467089 = -r29467085;
        double r29467090 = cos(r29467089);
        double r29467091 = r29467088 * r29467090;
        double r29467092 = sin(r29467089);
        double r29467093 = sin(r29467068);
        double r29467094 = r29467092 * r29467093;
        double r29467095 = r29467091 - r29467094;
        double r29467096 = r29467087 * r29467095;
        double r29467097 = 1.0;
        double r29467098 = fma(r29467097, r29467068, r29467089);
        double r29467099 = sin(r29467098);
        double r29467100 = sin(r29467086);
        double r29467101 = r29467099 * r29467100;
        double r29467102 = r29467096 - r29467101;
        double r29467103 = r29467081 * r29467102;
        double r29467104 = a;
        double r29467105 = b;
        double r29467106 = r29467105 * r29467072;
        double r29467107 = r29467104 / r29467106;
        double r29467108 = r29467103 - r29467107;
        double r29467109 = r29467068 * r29467068;
        double r29467110 = -0.5;
        double r29467111 = fma(r29467109, r29467110, r29467097);
        double r29467112 = r29467081 * r29467111;
        double r29467113 = r29467112 - r29467107;
        double r29467114 = r29467077 ? r29467108 : r29467113;
        return r29467114;
}

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	18.7
Herbie	16.5

\[\begin{array}{l} \mathbf{if}\;z \lt -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \cos \left(y - \frac{t}{3.0} \cdot z\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2.0 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3.0}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9964954255751017

   1. Initial program 20.0

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 20.0

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{\color{blue}{1 \cdot 3.0}}\right) - \frac{a}{b \cdot 3.0}\]

   4. Applied times-frac 19.9

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\frac{z}{1} \cdot \frac{t}{3.0}}\right) - \frac{a}{b \cdot 3.0}\]

   5. Applied *-un-lft-identity 19.9

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(\color{blue}{1 \cdot y} - \frac{z}{1} \cdot \frac{t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

   6. Applied prod-diff 19.9

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(1, y, -\frac{t}{3.0} \cdot \frac{z}{1}\right) + \mathsf{fma}\left(-\frac{t}{3.0}, \frac{z}{1}, \frac{t}{3.0} \cdot \frac{z}{1}\right)\right)} - \frac{a}{b \cdot 3.0}\]

   7. Applied cos-sum 16.9

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{3.0} \cdot \frac{z}{1}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3.0}, \frac{z}{1}, \frac{t}{3.0} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3.0} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3.0}, \frac{z}{1}, \frac{t}{3.0} \cdot \frac{z}{1}\right)\right)\right)} - \frac{a}{b \cdot 3.0}\]
   8. Using strategy rm

   9. Applied fma-udef 16.9

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos \color{blue}{\left(1 \cdot y + \left(-\frac{t}{3.0} \cdot \frac{z}{1}\right)\right)} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3.0}, \frac{z}{1}, \frac{t}{3.0} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3.0} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3.0}, \frac{z}{1}, \frac{t}{3.0} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3.0}\]

   10. Applied cos-sum 16.0

       \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(\cos \left(1 \cdot y\right) \cdot \cos \left(-\frac{t}{3.0} \cdot \frac{z}{1}\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{3.0} \cdot \frac{z}{1}\right)\right)} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3.0}, \frac{z}{1}, \frac{t}{3.0} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3.0} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3.0}, \frac{z}{1}, \frac{t}{3.0} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3.0}\]

   if 0.9964954255751017 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.1

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

   2. Taylor expanded around 0 17.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3.0}\]

   3. Simplified 17.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right)} - \frac{a}{b \cdot 3.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3.0}\right) \le 0.9964954255751017:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \left(\cos \left(\mathsf{fma}\left(\frac{-t}{3.0}, z, z \cdot \frac{t}{3.0}\right)\right) \cdot \left(\cos y \cdot \cos \left(-z \cdot \frac{t}{3.0}\right) - \sin \left(-z \cdot \frac{t}{3.0}\right) \cdot \sin y\right) - \sin \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3.0}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{-t}{3.0}, z, z \cdot \frac{t}{3.0}\right)\right)\right) - \frac{a}{b \cdot 3.0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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 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))))