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

Average Error: 20.5 → 15.1

Time: 16.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z \cdot t \leq -3.0732563653436038 \cdot 10^{+193}:\\ \;\;\;\;t_1 \cdot \cos y - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;z \cdot t \leq 1.545117997407016 \cdot 10^{+295}:\\ \;\;\;\;\begin{array}{l} t_3 := z \cdot \frac{t}{3}\\ t_4 := \mathsf{fma}\left(-\frac{t}{3}, z, t_3\right)\\ t_5 := \mathsf{fma}\left(1, y, -t_3\right)\\ t_1 \cdot \left(\cos t_5 \cdot \cos t_4 - \sin t_5 \cdot \sin t_4\right) - t_2 \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_6 := \sqrt[3]{2 \cdot \left(\sqrt{x} \cdot \cos y\right)}\\ t_6 \cdot \left(t_6 \cdot t_6\right) - t_2 \end{array}\\ \end{array}\\ \end{array} \]

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))))
   (if (<= (* z t) -3.0732563653436038e+193)
     (- (* t_1 (cos y)) (/ (/ a b) 3.0))
     (let* ((t_2 (/ a (* b 3.0))))
       (if (<= (* z t) 1.545117997407016e+295)
         (let* ((t_3 (* z (/ t 3.0)))
                (t_4 (fma (- (/ t 3.0)) z t_3))
                (t_5 (fma 1.0 y (- t_3))))
           (- (* t_1 (- (* (cos t_5) (cos t_4)) (* (sin t_5) (sin t_4)))) t_2))
         (let* ((t_6 (cbrt (* 2.0 (* (sqrt x) (cos y))))))
           (- (* t_6 (* t_6 t_6)) 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 tmp;
	if ((z * t) <= -3.0732563653436038e+193) {
		tmp = (t_1 * cos(y)) - ((a / b) / 3.0);
	} else {
		double t_2 = a / (b * 3.0);
		double tmp_1;
		if ((z * t) <= 1.545117997407016e+295) {
			double t_3_2 = z * (t / 3.0);
			double t_4_3 = fma(-(t / 3.0), z, t_3_2);
			double t_5_4 = fma(1.0, y, -t_3_2);
			tmp_1 = (t_1 * ((cos(t_5_4) * cos(t_4_3)) - (sin(t_5_4) * sin(t_4_3)))) - t_2;
		} else {
			double t_6 = cbrt(2.0 * (sqrt(x) * cos(y)));
			tmp_1 = (t_6 * (t_6 * t_6)) - t_2;
		}
		tmp = tmp_1;
	}
	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	20.5
Target	18.6
Herbie	15.1

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -3.073256365343604e193

   1. Initial program 48.7

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

   2. Taylor expanded in z around 0 32.5

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

   3. Applied associate-/r*_binary64 32.6

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

   if -3.073256365343604e193 < (*.f64 z t) < 1.5451179974070159e295

   1. Initial program 13.2

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

   2. Applied *-un-lft-identity_binary64 13.2

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

   3. Applied times-frac_binary64 13.2

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

   4. Applied *-un-lft-identity_binary64 13.2

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

   5. Applied prod-diff_binary64 13.2

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

   6. Applied cos-sum_binary64 11.2

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

   if 1.5451179974070159e295 < (*.f64 z t)

   1. Initial program 61.2

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

   2. Taylor expanded in z around 0 33.2

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

   3. Applied add-cube-cbrt_binary64 33.2

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

   4. Simplified 33.2

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

   5. Simplified 33.2

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

4. Final simplification 15.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -3.0732563653436038 \cdot 10^{+193}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;z \cdot t \leq 1.545117997407016 \cdot 10^{+295}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} \cdot \left(\sqrt[3]{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} \cdot \sqrt[3]{2 \cdot \left(\sqrt{x} \cdot \cos y\right)}\right) - \frac{a}{b \cdot 3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(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))))