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

Average Error: 20.6 → 12.3

Time: 25.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9999073125858173:\\ \;\;\;\;\begin{array}{l} t_3 := -\frac{t}{3}\\ t_4 := z \cdot \frac{t}{3}\\ t_5 := \mathsf{fma}\left(t_3, z, t_4\right)\\ t_6 := \cos t_4\\ t_7 := \sin \left(z \cdot t_3\right)\\ t_1 \cdot \left(\left(\cos y \cdot t_6 - \sin y \cdot \sqrt[3]{t_7 \cdot \left(t_7 \cdot t_7\right)}\right) \cdot \cos t_5 - \left(t_6 \cdot \sin y + \cos y \cdot t_7\right) \cdot \sin t_5\right) - t_2 \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \cos y - t_2\\ \end{array} \]

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
 (let* ((t_1 (* 2.0 (sqrt x))) (t_2 (/ a (* 3.0 b))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 0.9999073125858173)
     (let* ((t_3 (- (/ t 3.0)))
            (t_4 (* z (/ t 3.0)))
            (t_5 (fma t_3 z t_4))
            (t_6 (cos t_4))
            (t_7 (sin (* z t_3))))
       (-
        (*
         t_1
         (-
          (*
           (- (* (cos y) t_6) (* (sin y) (cbrt (* t_7 (* t_7 t_7)))))
           (cos t_5))
          (* (+ (* t_6 (sin y)) (* (cos y) t_7)) (sin t_5))))
        t_2))
     (- (* t_1 (cos y)) t_2))))

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 t_1 = 2.0 * sqrt(x);
	double t_2 = a / (3.0 * b);
	double tmp;
	if (cos(y - ((z * t) / 3.0)) <= 0.9999073125858173) {
		double t_3_1 = -(t / 3.0);
		double t_4_2 = z * (t / 3.0);
		double t_5_3 = fma(t_3_1, z, t_4_2);
		double t_6_4 = cos(t_4_2);
		double t_7_5 = sin(z * t_3_1);
		tmp = (t_1 * ((((cos(y) * t_6_4) - (sin(y) * cbrt(t_7_5 * (t_7_5 * t_7_5)))) * cos(t_5_3)) - (((t_6_4 * sin(y)) + (cos(y) * t_7_5)) * sin(t_5_3)))) - t_2;
	} else {
		tmp = (t_1 * cos(y)) - t_2;
	}
	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	20.6
Target	18.6
Herbie	12.3

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))) < 0.99990731258581733

   1. Initial program 20.2

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

   2. Applied *-un-lft-identity_binary64 20.2

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

   3. Applied times-frac_binary64 20.2

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

   4. Applied *-un-lft-identity_binary64 20.2

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

   5. Applied prod-diff_binary64 20.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right) + \mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)} - \frac{a}{b \cdot 3} \]

   6. Applied cos-sum_binary64 17.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right)} - \frac{a}{b \cdot 3} \]

   7. Applied fma-udef_binary64 17.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \color{blue}{\left(1 \cdot y + \left(-\frac{t}{3} \cdot \frac{z}{1}\right)\right)} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3} \]

   8. Applied cos-sum_binary64 16.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(\cos \left(1 \cdot y\right) \cdot \cos \left(-\frac{t}{3} \cdot \frac{z}{1}\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{3} \cdot \frac{z}{1}\right)\right)} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3} \]

   9. Simplified 16.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\color{blue}{\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right)} - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3} \]

   10. Simplified 16.3

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) - \color{blue}{\sin y \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)}\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3} \]

   11. Applied fma-udef_binary64 16.3

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) - \sin y \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \color{blue}{\left(1 \cdot y + \left(-\frac{t}{3} \cdot \frac{z}{1}\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3} \]

   12. Applied sin-sum_binary64 13.0

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) - \sin y \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \color{blue}{\left(\sin \left(1 \cdot y\right) \cdot \cos \left(-\frac{t}{3} \cdot \frac{z}{1}\right) + \cos \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{3} \cdot \frac{z}{1}\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3} \]

   13. Simplified 13.0

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) - \sin y \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \left(\color{blue}{\sin y \cdot \cos \left(z \cdot \frac{t}{3}\right)} + \cos \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3} \]

   14. Simplified 13.0

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) - \sin y \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \left(\sin y \cdot \cos \left(z \cdot \frac{t}{3}\right) + \color{blue}{\cos y \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)}\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3} \]

   15. Applied add-cbrt-cube_binary64 13.0

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) - \sin y \cdot \color{blue}{\sqrt[3]{\left(\sin \left(z \cdot \left(-\frac{t}{3}\right)\right) \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right) \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)}}\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \left(\sin y \cdot \cos \left(z \cdot \frac{t}{3}\right) + \cos y \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3} \]

   if 0.99990731258581733 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))

   1. Initial program 21.4

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

   2. Taylor expanded in z around 0 11.1

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

4. Final simplification 12.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9999073125858173:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) - \sin y \cdot \sqrt[3]{\sin \left(z \cdot \left(-\frac{t}{3}\right)\right) \cdot \left(\sin \left(z \cdot \left(-\frac{t}{3}\right)\right) \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right)}\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \left(\cos \left(z \cdot \frac{t}{3}\right) \cdot \sin y + \cos y \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{3 \cdot b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022020 
(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))))