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

Average Error: 20.8 → 17.9

Time: 25.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 2.7474126037333123 \cdot 10^{+305}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt[3]{3}}}\right)\right) + \sin y \cdot \sin \left(\left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt[3]{3}}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt[3]{3}}}\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 2.7474126037333123e+305)))
   (- (* (* 2.0 (sqrt x)) (- 1.0 (* 0.5 (* y y)))) (/ a (* b 3.0)))
   (-
    (*
     (* 2.0 (sqrt x))
     (+
      (*
       (cos y)
       (cos
        (*
         (* (cbrt (/ (* z t) 3.0)) (cbrt (/ (* z t) 3.0)))
         (* (cbrt (/ z (* (cbrt 3.0) (cbrt 3.0)))) (cbrt (/ t (cbrt 3.0)))))))
      (*
       (sin y)
       (sin
        (*
         (* (cbrt (/ z (* (cbrt 3.0) (cbrt 3.0)))) (cbrt (/ t (cbrt 3.0))))
         (*
          (cbrt (/ (* z t) 3.0))
          (*
           (cbrt (/ z (* (cbrt 3.0) (cbrt 3.0))))
           (cbrt (/ t (cbrt 3.0))))))))))
    (/ a (* b 3.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos(y - ((z * t) / 3.0))) - (a / (b * 3.0));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 2.7474126037333123e+305)) {
		tmp = ((2.0 * sqrt(x)) * (1.0 - (0.5 * (y * y)))) - (a / (b * 3.0));
	} else {
		tmp = ((2.0 * sqrt(x)) * ((cos(y) * cos((cbrt((z * t) / 3.0) * cbrt((z * t) / 3.0)) * (cbrt(z / (cbrt(3.0) * cbrt(3.0))) * cbrt(t / cbrt(3.0))))) + (sin(y) * sin((cbrt(z / (cbrt(3.0) * cbrt(3.0))) * cbrt(t / cbrt(3.0))) * (cbrt((z * t) / 3.0) * (cbrt(z / (cbrt(3.0) * cbrt(3.0))) * cbrt(t / cbrt(3.0)))))))) - (a / (b * 3.0));
	}
	return tmp;
}

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	20.8
Target	18.7
Herbie	17.9

\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 2.74741260373331225e305 < (*.f64 z t)

   1. Initial program 63.7

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

   2. Taylor expanded around 0 45.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - 0.5 \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]

   3. Simplified 45.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - 0.5 \cdot \left(y \cdot y\right)\right)} - \frac{a}{b \cdot 3}\]

   if -inf.0 < (*.f64 z t) < 2.74741260373331225e305

   1. Initial program 14.4

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

   3. Applied add-cube-cbrt_binary64_18527 14.4

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

   5. Applied add-cube-cbrt_binary64_18527 14.4

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

   6. Applied times-frac_binary64_18498 14.4

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

   7. Applied cbrt-prod_binary64_18523 14.3

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

   9. Applied cos-diff_binary64_18629 13.9

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

   11. Applied add-cube-cbrt_binary64_18527 13.9

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

   12. Applied times-frac_binary64_18498 13.9

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

   13. Applied cbrt-prod_binary64_18523 13.9

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 2.7474126037333123 \cdot 10^{+305}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt[3]{3}}}\right)\right) + \sin y \cdot \sin \left(\left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt[3]{3}}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt[3]{3}}}\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.3793337487235141e+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))))