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

Average Error: 21.1 → 12.5

Time: 25.7s

Precision: binary64

\[[z, t] = \mathsf{sort}([z, t]) \\]

Math

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

↓

\[\begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := 2 \cdot \sqrt{x}\\ t_3 := t_2 \cdot \cos y - t_1\\ t_4 := y - \frac{z \cdot t}{3}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_4 \leq 4.4498753054159406 \cdot 10^{+287}:\\ \;\;\;\;\begin{array}{l} t_5 := z \cdot \frac{t}{3}\\ t_6 := \cos t_5\\ t_7 := -\frac{t}{3}\\ t_8 := \sin \left(z \cdot t_7\right)\\ t_9 := \mathsf{fma}\left(t_7, z, t_5\right)\\ t_2 \cdot \left(\left(\cos y \cdot t_6 - \sin y \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_8\right)\right)\right) \cdot \cos t_9 - \left(t_6 \cdot \sin y + \cos y \cdot t_8\right) \cdot \sin t_9\right) - t_1 \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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
 (let* ((t_1 (/ a (* 3.0 b)))
        (t_2 (* 2.0 (sqrt x)))
        (t_3 (- (* t_2 (cos y)) t_1))
        (t_4 (- y (/ (* z t) 3.0))))
   (if (<= t_4 (- INFINITY))
     t_3
     (if (<= t_4 4.4498753054159406e+287)
       (let* ((t_5 (* z (/ t 3.0)))
              (t_6 (cos t_5))
              (t_7 (- (/ t 3.0)))
              (t_8 (sin (* z t_7)))
              (t_9 (fma t_7 z t_5)))
         (-
          (*
           t_2
           (-
            (* (- (* (cos y) t_6) (* (sin y) (expm1 (log1p t_8)))) (cos t_9))
            (* (+ (* t_6 (sin y)) (* (cos y) t_8)) (sin t_9))))
          t_1))
       t_3))))

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 = a / (3.0 * b);
	double t_2 = 2.0 * sqrt(x);
	double t_3 = (t_2 * cos(y)) - t_1;
	double t_4 = y - ((z * t) / 3.0);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_4 <= 4.4498753054159406e+287) {
		double t_5 = z * (t / 3.0);
		double t_6 = cos(t_5);
		double t_7 = -(t / 3.0);
		double t_8 = sin(z * t_7);
		double t_9 = fma(t_7, z, t_5);
		tmp = (t_2 * ((((cos(y) * t_6) - (sin(y) * expm1(log1p(t_8)))) * cos(t_9)) - (((t_6 * sin(y)) + (cos(y) * t_8)) * sin(t_9)))) - t_1;
	} else {
		tmp = t_3;
	}
	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	21.1
Target	19.1
Herbie	12.5

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 y (/.f64 (*.f64 z t) 3)) < -inf.0 or 4.44987530541594057e287 < (-.f64 y (/.f64 (*.f64 z t) 3))

   1. Initial program 57.9

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

   2. Taylor expanded in z around 0 31.3

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

   if -inf.0 < (-.f64 y (/.f64 (*.f64 z t) 3)) < 4.44987530541594057e287

   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. Applied *-un-lft-identity_binary64 14.4

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

   3. Applied times-frac_binary64 14.4

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

   4. Applied *-un-lft-identity_binary64 14.4

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

   5. Applied prod-diff_binary64 14.4

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

   6. Applied cos-sum_binary64 12.1

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

   7. Applied fma-udef_binary64 12.1

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

   8. Applied cos-sum_binary64 11.5

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

   9. Simplified 11.5

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

   10. Simplified 11.5

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

   11. Applied fma-udef_binary64 11.5

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

   12. Applied sin-sum_binary64 9.1

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

   13. Simplified 9.1

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

   14. Simplified 9.1

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

   15. Applied expm1-log1p-u_binary64 9.1

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) - \sin y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \left(\sin y \cdot \cos \left(z \cdot \frac{t}{3}\right) + \cos y \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -\infty:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 4.4498753054159406 \cdot 10^{+287}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) - \sin y \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \left(\cos \left(z \cdot \frac{t}{3}\right) \cdot \sin y + \cos y \cdot \sin \left(z \cdot \left(-\frac{t}{3}\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{3 \cdot b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022096 
(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))))