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

Average Error: 20.8 → 16.3

Time: 15.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ t_3 := \cos y \cdot t_2\\ t_4 := \frac{z \cdot t}{3}\\ \mathbf{if}\;z \cdot t \leq -6.460990352425691 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\left(x \cdot 4\right) \cdot {\cos y}^{2}}, \sqrt[3]{t_3}, -t_1\right)\\ \mathbf{elif}\;z \cdot t \leq 4.0205360772180477 \cdot 10^{+155}:\\ \;\;\;\;\left(t_2 \cdot \left(\cos y \cdot \cos t_4\right) + t_2 \cdot \left(\sin y \cdot \sin t_4\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_3 - \frac{\frac{a}{b}}{3}\\ \end{array} \]

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 (* b 3.0)))
        (t_2 (* 2.0 (sqrt x)))
        (t_3 (* (cos y) t_2))
        (t_4 (/ (* z t) 3.0)))
   (if (<= (* z t) -6.460990352425691e+204)
     (fma (cbrt (* (* x 4.0) (pow (cos y) 2.0))) (cbrt t_3) (- t_1))
     (if (<= (* z t) 4.0205360772180477e+155)
       (- (+ (* t_2 (* (cos y) (cos t_4))) (* t_2 (* (sin y) (sin t_4)))) t_1)
       (- t_3 (/ (/ 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 t_1 = a / (b * 3.0);
	double t_2 = 2.0 * sqrt(x);
	double t_3 = cos(y) * t_2;
	double t_4 = (z * t) / 3.0;
	double tmp;
	if ((z * t) <= -6.460990352425691e+204) {
		tmp = fma(cbrt(((x * 4.0) * pow(cos(y), 2.0))), cbrt(t_3), -t_1);
	} else if ((z * t) <= 4.0205360772180477e+155) {
		tmp = ((t_2 * (cos(y) * cos(t_4))) + (t_2 * (sin(y) * sin(t_4)))) - t_1;
	} else {
		tmp = t_3 - ((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	18.6
Herbie	16.3

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -6.4609903524256911e204

   1. Initial program 51.1

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

   2. Taylor expanded in z around 0 32.6

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

   3. Applied egg-rr 32.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\left(x \cdot 4\right) \cdot {\cos y}^{2}}, \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y}, -\frac{a}{b \cdot 3}\right)} \]

   if -6.4609903524256911e204 < (*.f64 z t) < 4.0205360772180477e155

   1. Initial program 11.6

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

   2. Applied egg-rr 11.0

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

   if 4.0205360772180477e155 < (*.f64 z t)

   1. Initial program 48.0

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

   2. Taylor expanded in z around 0 33.0

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

   3. Applied egg-rr 33.1

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

   4. Applied egg-rr 33.0

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

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -6.460990352425691 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\left(x \cdot 4\right) \cdot {\cos y}^{2}}, \sqrt[3]{\cos y \cdot \left(2 \cdot \sqrt{x}\right)}, -\frac{a}{b \cdot 3}\right)\\ \mathbf{elif}\;z \cdot t \leq 4.0205360772180477 \cdot 10^{+155}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))))