Average Error: 20.4 → 15.0
Time: 18.1s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
↓
\[\begin{array}{l} t_1 := z \cdot \frac{t}{3}\\ t_2 := \mathsf{fma}\left(-\frac{t}{3}, z, t_1\right)\\ t_3 := \frac{a}{b \cdot 3}\\ t_4 := 2 \cdot \sqrt{x}\\ t_5 := \mathsf{fma}\left(1, y, -t_1\right)\\ \mathbf{if}\;z \cdot t \leq -9.068849681521214 \cdot 10^{+303}:\\ \;\;\;\;t_4 - a \cdot \frac{0.3333333333333333}{b}\\ \mathbf{elif}\;z \cdot t \leq 4.50876227333934 \cdot 10^{+210}:\\ \;\;\;\;t_4 \cdot \left(\cos t_5 \cdot \cos t_2 - \sin t_5 \cdot \sin t_2\right) - t_3\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \cos y - t_3\\ \end{array} \]
(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
 (let* ((t_1 (* z (/ t 3.0)))
        (t_2 (fma (- (/ t 3.0)) z t_1))
        (t_3 (/ a (* b 3.0)))
        (t_4 (* 2.0 (sqrt x)))
        (t_5 (fma 1.0 y (- t_1))))
   (if (<= (* z t) -9.068849681521214e+303)
     (- t_4 (* a (/ 0.3333333333333333 b)))
     (if (<= (* z t) 4.50876227333934e+210)
       (- (* t_4 (- (* (cos t_5) (cos t_2)) (* (sin t_5) (sin t_2)))) t_3)
       (- (* t_4 (cos y)) t_3)))))
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 = z * (t / 3.0);
	double t_2 = fma(-(t / 3.0), z, t_1);
	double t_3 = a / (b * 3.0);
	double t_4 = 2.0 * sqrt(x);
	double t_5 = fma(1.0, y, -t_1);
	double tmp;
	if ((z * t) <= -9.068849681521214e+303) {
		tmp = t_4 - (a * (0.3333333333333333 / b));
	} else if ((z * t) <= 4.50876227333934e+210) {
		tmp = (t_4 * ((cos(t_5) * cos(t_2)) - (sin(t_5) * sin(t_2)))) - t_3;
	} else {
		tmp = (t_4 * cos(y)) - t_3;
	}
	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(z * Float64(t / 3.0))
	t_2 = fma(Float64(-Float64(t / 3.0)), z, t_1)
	t_3 = Float64(a / Float64(b * 3.0))
	t_4 = Float64(2.0 * sqrt(x))
	t_5 = fma(1.0, y, Float64(-t_1))
	tmp = 0.0
	if (Float64(z * t) <= -9.068849681521214e+303)
		tmp = Float64(t_4 - Float64(a * Float64(0.3333333333333333 / b)));
	elseif (Float64(z * t) <= 4.50876227333934e+210)
		tmp = Float64(Float64(t_4 * Float64(Float64(cos(t_5) * cos(t_2)) - Float64(sin(t_5) * sin(t_2)))) - t_3);
	else
		tmp = Float64(Float64(t_4 * cos(y)) - t_3);
	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[(z * N[(t / 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(t / 3.0), $MachinePrecision]) * z + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 * y + (-t$95$1)), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -9.068849681521214e+303], N[(t$95$4 - N[(a * N[(0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4.50876227333934e+210], N[(N[(t$95$4 * N[(N[(N[Cos[t$95$5], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$5], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision], N[(N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$3), $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 := z \cdot \frac{t}{3}\\
t_2 := \mathsf{fma}\left(-\frac{t}{3}, z, t_1\right)\\
t_3 := \frac{a}{b \cdot 3}\\
t_4 := 2 \cdot \sqrt{x}\\
t_5 := \mathsf{fma}\left(1, y, -t_1\right)\\
\mathbf{if}\;z \cdot t \leq -9.068849681521214 \cdot 10^{+303}:\\
\;\;\;\;t_4 - a \cdot \frac{0.3333333333333333}{b}\\

\mathbf{elif}\;z \cdot t \leq 4.50876227333934 \cdot 10^{+210}:\\
\;\;\;\;t_4 \cdot \left(\cos t_5 \cdot \cos t_2 - \sin t_5 \cdot \sin t_2\right) - t_3\\

\mathbf{else}:\\
\;\;\;\;t_4 \cdot \cos y - t_3\\


\end{array}

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

Original20.4
Target18.5
Herbie15.0
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 z t) < -9.068849681521214e303

    1. Initial program 63.0

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

    2. Taylor expanded in z around 0 34.3

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

    3. Applied div-inv_binary6434.3

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

    4. Simplified34.3

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

    5. Taylor expanded in y around 0 34.1

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - a \cdot \frac{0.3333333333333333}{b} \]

    if -9.068849681521214e303 < (*.f64 z t) < 4.50876227333934e210

    1. Initial program 13.1

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

    2. Applied *-un-lft-identity_binary6413.1

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

    3. Applied times-frac_binary6413.2

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

    4. Applied *-un-lft-identity_binary6413.2

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

    5. Applied prod-diff_binary6413.2

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

    6. Applied cos-sum_binary6411.1

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

    if 4.50876227333934e210 < (*.f64 z t)

    1. Initial program 51.2

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

    2. Taylor expanded in z around 0 33.8

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

    3. Applied sub-neg_binary6433.8

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -9.068849681521214 \cdot 10^{+303}:\\ \;\;\;\;2 \cdot \sqrt{x} - a \cdot \frac{0.3333333333333333}{b}\\ \mathbf{elif}\;z \cdot t \leq 4.50876227333934 \cdot 10^{+210}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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))))