Average Error: 20.0 → 15.9
Time: 15.2s
Precision: binary64
Cost: 34498
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.178476436106435 \cdot 10^{+192}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;z \cdot t \leq 1.0377641230388289 \cdot 10^{+126}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\cos y} \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \sqrt[3]{\cos y}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 -1.178476436106435 \cdot 10^{+192}:\\
\;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\

\mathbf{elif}\;z \cdot t \leq 1.0377641230388289 \cdot 10^{+126}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\cos y} \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \sqrt[3]{\cos y}\right) - \frac{a}{b \cdot 3}\\

\end{array}
(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 (<= (* z t) -1.178476436106435e+192)
   (- (* 2.0 (sqrt x)) (/ (/ a b) 3.0))
   (if (<= (* z t) 1.0377641230388289e+126)
     (-
      (*
       (* 2.0 (sqrt x))
       (+ (* (cos y) (cos (/ (* z t) 3.0))) (* (sin y) (sin (/ (* z t) 3.0)))))
      (/ a (* b 3.0)))
     (-
      (* (cbrt (cos y)) (* (* 2.0 (sqrt x)) (cbrt (cos y))))
      (/ a (* b 3.0))))))
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) <= -1.178476436106435e+192) {
		tmp = (2.0 * sqrt(x)) - ((a / b) / 3.0);
	} else if ((z * t) <= 1.0377641230388289e+126) {
		tmp = ((2.0 * sqrt(x)) * ((cos(y) * cos((z * t) / 3.0)) + (sin(y) * sin((z * t) / 3.0)))) - (a / (b * 3.0));
	} else {
		tmp = (cbrt(cos(y)) * ((2.0 * sqrt(x)) * cbrt(cos(y)))) - (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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target18.0
Herbie15.9
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Alternatives

Alternative 1
Error15.9
Cost34498
\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.178476436106435 \cdot 10^{+192}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;z \cdot t \leq 1.5307139395118401 \cdot 10^{+54}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) - \sin y \cdot \sin \left(z \cdot \left(t \cdot -0.3333333333333333\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\cos y} \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \sqrt[3]{\cos y}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
Alternative 2
Error16.5
Cost20993
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9999134170749396:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \end{array}\]
Alternative 3
Error16.6
Cost13504
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3}\]
Alternative 4
Error26.4
Cost32969
\[\begin{array}{l} \mathbf{if}\;\sqrt{x} \leq 6040219.589669799:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;\sqrt{x} \leq 66989658255580.19 \lor \neg \left(\sqrt{x} \leq 9.41012478605164 \cdot 10^{+68}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \end{array}\]
Alternative 5
Error24.6
Cost6976
\[2 \cdot \sqrt{x} - \frac{a}{b \cdot 3}\]
Alternative 6
Error24.6
Cost6976
\[2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\]
Alternative 7
Error35.5
Cost320
\[a \cdot \frac{-0.3333333333333333}{b}\]
Alternative 8
Error61.3
Cost64
\[1\]

Error

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 z t) < -1.178476436106435e192

    1. Initial program 47.6

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

    2. Taylor expanded around 0 31.7

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

    4. Applied associate-/r*_binary64_2014131.7

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

    5. Taylor expanded around 0 31.8

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{\frac{a}{b}}{3}\]

    6. Simplified31.8

      \[\leadsto \color{blue}{2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}}\]

    if -1.178476436106435e192 < (*.f64 z t) < 1.0377641230388289e126

    1. Initial program 10.7

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

    3. Applied cos-diff_binary64_2033410.1

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

    4. Simplified10.1

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

    if 1.0377641230388289e126 < (*.f64 z t)

    1. Initial program 45.0

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

    2. Taylor expanded around 0 32.7

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

    4. Applied add-cube-cbrt_binary64_2023232.7

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

    5. Applied associate-*r*_binary64_2013732.7

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

    6. Taylor expanded around 0 32.6

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

    7. Simplified32.6

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

  4. Final simplification15.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.178476436106435 \cdot 10^{+192}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;z \cdot t \leq 1.0377641230388289 \cdot 10^{+126}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\cos y} \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \sqrt[3]{\cos y}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))