Average Error: 20.2 → 15.1
Time: 31.8s
Precision: binary64
Cost: 48328
\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
↓
\[\begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := \left(z \cdot t\right) \cdot 0.3333333333333333\\ t_3 := \mathsf{fma}\left(-t, z \cdot 0.3333333333333333, t_2\right)\\ t_4 := y + \left(z \cdot t\right) \cdot -0.3333333333333333\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+248}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(1 - e^{t_2}\right)\right) - t_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+228}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos t_4 \cdot \cos t_3 - \sin t_4 \cdot \sin t_3\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right) - {\left(3 \cdot \frac{b}{a}\right)}^{-1}\\ \end{array} \]
(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
 (let* ((t_1 (/ a (* 3.0 b)))
        (t_2 (* (* z t) 0.3333333333333333))
        (t_3 (fma (- t) (* z 0.3333333333333333) t_2))
        (t_4 (+ y (* (* z t) -0.3333333333333333))))
   (if (<= (* z t) -2e+248)
     (- (* 2.0 (* (sqrt x) (cos (- 1.0 (exp t_2))))) t_1)
     (if (<= (* z t) 1e+228)
       (-
        (*
         2.0
         (* (sqrt x) (- (* (cos t_4) (cos t_3)) (* (sin t_4) (sin t_3)))))
        t_1)
       (- (* 2.0 (* (sqrt x) (cos y))) (pow (* 3.0 (/ b a)) -1.0))))))
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 t_1 = a / (3.0 * b);
	double t_2 = (z * t) * 0.3333333333333333;
	double t_3 = fma(-t, (z * 0.3333333333333333), t_2);
	double t_4 = y + ((z * t) * -0.3333333333333333);
	double tmp;
	if ((z * t) <= -2e+248) {
		tmp = (2.0 * (sqrt(x) * cos((1.0 - exp(t_2))))) - t_1;
	} else if ((z * t) <= 1e+228) {
		tmp = (2.0 * (sqrt(x) * ((cos(t_4) * cos(t_3)) - (sin(t_4) * sin(t_3))))) - t_1;
	} else {
		tmp = (2.0 * (sqrt(x) * cos(y))) - pow((3.0 * (b / a)), -1.0);
	}
	return tmp;
}

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end
↓
function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(Float64(z * t) * 0.3333333333333333)
	t_3 = fma(Float64(-t), Float64(z * 0.3333333333333333), t_2)
	t_4 = Float64(y + Float64(Float64(z * t) * -0.3333333333333333))
	tmp = 0.0
	if (Float64(z * t) <= -2e+248)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * cos(Float64(1.0 - exp(t_2))))) - t_1);
	elseif (Float64(z * t) <= 1e+228)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(cos(t_4) * cos(t_3)) - Float64(sin(t_4) * sin(t_3))))) - t_1);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * cos(y))) - (Float64(3.0 * Float64(b / a)) ^ -1.0));
	end
	return tmp
end

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$3 = N[((-t) * N[(z * 0.3333333333333333), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(y + N[(N[(z * t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+248], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[N[(1.0 - N[Exp[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+228], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Cos[t$95$4], $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$4], $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[(3.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]]]

\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 := \left(z \cdot t\right) \cdot 0.3333333333333333\\
t_3 := \mathsf{fma}\left(-t, z \cdot 0.3333333333333333, t_2\right)\\
t_4 := y + \left(z \cdot t\right) \cdot -0.3333333333333333\\
\mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+248}:\\
\;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(1 - e^{t_2}\right)\right) - t_1\\

\mathbf{elif}\;z \cdot t \leq 10^{+228}:\\
\;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos t_4 \cdot \cos t_3 - \sin t_4 \cdot \sin t_3\right)\right) - t_1\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right) - {\left(3 \cdot \frac{b}{a}\right)}^{-1}\\


\end{array}

Error

Target

Original20.2
Target18.5
Herbie15.1
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 z t) < -2.00000000000000009e248

    1. Initial program 52.8

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

    2. Simplified52.9

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{3} \cdot t\right)\right) - \frac{a}{3 \cdot b}} \]
      Proof
      (-.f64 (*.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (*.f64 (/.f64 z 3) t))))) (/.f64 a (*.f64 3 b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 z t) 3)))))) (/.f64 a (*.f64 3 b))): 7 points increase in error, 6 points decrease in error
      (-.f64 (*.f64 2 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))))))) (/.f64 a (*.f64 3 b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (neg.f64 (neg.f64 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))))) (/.f64 a (Rewrite<= *-commutative_binary64 (*.f64 b 3)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> fma-neg_binary64 (fma.f64 2 (neg.f64 (neg.f64 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))))) (neg.f64 (/.f64 a (*.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (Rewrite=> remove-double-neg_binary64 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (neg.f64 (/.f64 a (*.f64 b 3)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (/.f64 a (*.f64 b 3)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (/.f64 a (*.f64 b 3))): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr64.0

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(z \cdot 0.3333333333333333\right) \cdot t\right)\right)}\right)\right) - \frac{a}{3 \cdot b} \]

    4. Taylor expanded in z around 0 32.5

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \mathsf{expm1}\left(\color{blue}{0.3333333333333333 \cdot \left(t \cdot z\right)}\right)\right)\right) - \frac{a}{3 \cdot b} \]

    5. Simplified32.5

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \mathsf{expm1}\left(\color{blue}{t \cdot \left(0.3333333333333333 \cdot z\right)}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      Proof
      (*.f64 t (*.f64 1/3 z)): 0 points increase in error, 0 points decrease in error
      (*.f64 t (Rewrite<= *-commutative_binary64 (*.f64 z 1/3))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t z) 1/3)): 22 points increase in error, 32 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 1/3 (*.f64 t z))): 0 points increase in error, 0 points decrease in error

    6. Taylor expanded in y around 0 32.5

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\cos \left(1 - e^{0.3333333333333333 \cdot \left(t \cdot z\right)}\right)}\right) - \frac{a}{3 \cdot b} \]

    if -2.00000000000000009e248 < (*.f64 z t) < 9.9999999999999992e227

    1. Initial program 12.8

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

    2. Simplified12.8

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{3} \cdot t\right)\right) - \frac{a}{3 \cdot b}} \]
      Proof
      (-.f64 (*.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (*.f64 (/.f64 z 3) t))))) (/.f64 a (*.f64 3 b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 z t) 3)))))) (/.f64 a (*.f64 3 b))): 7 points increase in error, 6 points decrease in error
      (-.f64 (*.f64 2 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))))))) (/.f64 a (*.f64 3 b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (neg.f64 (neg.f64 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))))) (/.f64 a (Rewrite<= *-commutative_binary64 (*.f64 b 3)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> fma-neg_binary64 (fma.f64 2 (neg.f64 (neg.f64 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))))) (neg.f64 (/.f64 a (*.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (Rewrite=> remove-double-neg_binary64 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (neg.f64 (/.f64 a (*.f64 b 3)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (/.f64 a (*.f64 b 3)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (/.f64 a (*.f64 b 3))): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr19.2

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(z \cdot 0.3333333333333333\right) \cdot t\right)\right)}\right)\right) - \frac{a}{3 \cdot b} \]

    4. Applied egg-rr11.1

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(\cos \left(y - 0.3333333333333333 \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(-t, z \cdot 0.3333333333333333, 0.3333333333333333 \cdot \left(z \cdot t\right)\right)\right) - \sin \left(y - 0.3333333333333333 \cdot \left(z \cdot t\right)\right) \cdot \sin \left(\mathsf{fma}\left(-t, z \cdot 0.3333333333333333, 0.3333333333333333 \cdot \left(z \cdot t\right)\right)\right)\right)}\right) - \frac{a}{3 \cdot b} \]

    if 9.9999999999999992e227 < (*.f64 z t)

    1. Initial program 51.3

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

    2. Simplified51.3

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right) - \frac{\frac{a}{b}}{3}} \]
      Proof
      (-.f64 (*.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (/.f64 (/.f64 a b) 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (/.f64 (/.f64 a b) 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (Rewrite<= associate-/r*_binary64 (/.f64 a (*.f64 b 3)))): 20 points increase in error, 16 points decrease in error

    3. Taylor expanded in z around 0 32.7

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

    4. Applied egg-rr32.7

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

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+248}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(1 - e^{\left(z \cdot t\right) \cdot 0.3333333333333333}\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;z \cdot t \leq 10^{+228}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos \left(y + \left(z \cdot t\right) \cdot -0.3333333333333333\right) \cdot \cos \left(\mathsf{fma}\left(-t, z \cdot 0.3333333333333333, \left(z \cdot t\right) \cdot 0.3333333333333333\right)\right) - \sin \left(y + \left(z \cdot t\right) \cdot -0.3333333333333333\right) \cdot \sin \left(\mathsf{fma}\left(-t, z \cdot 0.3333333333333333, \left(z \cdot t\right) \cdot 0.3333333333333333\right)\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right) - {\left(3 \cdot \frac{b}{a}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Error15.0
Cost48712
\[\begin{array}{l} t_1 := y + t \cdot \left(z \cdot -0.3333333333333333\right)\\ t_2 := y - \frac{z \cdot t}{3}\\ t_3 := \mathsf{fma}\left(t \cdot -0.3333333333333333, z, t \cdot \left(z \cdot 0.3333333333333333\right)\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+291}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right) - {\left(3 \cdot \frac{b}{a}\right)}^{-1}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+281}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos t_1 \cdot \cos t_3 - \sin t_1 \cdot \sin t_3\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} + -0.3333333333333333 \cdot \frac{a}{b}\\ \end{array} \]
Alternative 2
Error16.9
Cost19968
\[2 \cdot \left(\sqrt{x} \cdot \cos y\right) - {\left(3 \cdot \frac{b}{a}\right)}^{-1} \]
Alternative 3
Error16.8
Cost13504
\[2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{a}{3 \cdot b} \]
Alternative 4
Error16.8
Cost13504
\[2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{\frac{a}{b}}{3} \]
Alternative 5
Error24.2
Cost13384
\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{-226}:\\ \;\;\;\;t_1 - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-200}:\\ \;\;\;\;\cos y \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 + -0.3333333333333333 \cdot \frac{a}{b}\\ \end{array} \]
Alternative 6
Error25.3
Cost6976
\[2 \cdot \sqrt{x} + -0.3333333333333333 \cdot \frac{a}{b} \]
Alternative 7
Error25.2
Cost6976
\[2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3} \]
Alternative 8
Error36.3
Cost6784
\[{\left(\frac{b}{a} \cdot -3\right)}^{-1} \]
Alternative 9
Error36.3
Cost320
\[-0.3333333333333333 \cdot \frac{a}{b} \]

Error

Reproduce

herbie shell --seed 2022329 
(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))))