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

Average Error: 21.1 → 16.1

Time: 18.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := z \cdot \frac{t}{3}\\ t_3 := \mathsf{fma}\left(-\frac{t}{3}, z, t_2\right)\\ t_4 := 2 \cdot \sqrt{x}\\ t_5 := \sqrt[3]{t_4 \cdot \cos y}\\ t_6 := \mathsf{fma}\left(1, y, -t_2\right)\\ \mathbf{if}\;z \cdot t \leq -614615024502844300:\\ \;\;\;\;t_4 - t_1\\ \mathbf{elif}\;z \cdot t \leq 1.7974798995587906 \cdot 10^{+301}:\\ \;\;\;\;t_4 \cdot \left(\cos t_6 \cdot \cos t_3 - \sin t_6 \cdot \sin t_3\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \left(\left(\sqrt[3]{2} \cdot {x}^{0.16666666666666666}\right) \cdot t_5\right) - t_1\\ \end{array} \]

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 (* z (/ t 3.0)))
        (t_3 (fma (- (/ t 3.0)) z t_2))
        (t_4 (* 2.0 (sqrt x)))
        (t_5 (cbrt (* t_4 (cos y))))
        (t_6 (fma 1.0 y (- t_2))))
   (if (<= (* z t) -614615024502844300.0)
     (- t_4 t_1)
     (if (<= (* z t) 1.7974798995587906e+301)
       (- (* t_4 (- (* (cos t_6) (cos t_3)) (* (sin t_6) (sin t_3)))) t_1)
       (- (* t_5 (* (* (cbrt 2.0) (pow x 0.16666666666666666)) t_5)) t_1)))))

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 = z * (t / 3.0);
	double t_3 = fma(-(t / 3.0), z, t_2);
	double t_4 = 2.0 * sqrt(x);
	double t_5 = cbrt((t_4 * cos(y)));
	double t_6 = fma(1.0, y, -t_2);
	double tmp;
	if ((z * t) <= -614615024502844300.0) {
		tmp = t_4 - t_1;
	} else if ((z * t) <= 1.7974798995587906e+301) {
		tmp = (t_4 * ((cos(t_6) * cos(t_3)) - (sin(t_6) * sin(t_3)))) - t_1;
	} else {
		tmp = (t_5 * ((cbrt(2.0) * pow(x, 0.16666666666666666)) * t_5)) - t_1;
	}
	return tmp;
}

Julia

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(b * 3.0))
	t_2 = Float64(z * Float64(t / 3.0))
	t_3 = fma(Float64(-Float64(t / 3.0)), z, t_2)
	t_4 = Float64(2.0 * sqrt(x))
	t_5 = cbrt(Float64(t_4 * cos(y)))
	t_6 = fma(1.0, y, Float64(-t_2))
	tmp = 0.0
	if (Float64(z * t) <= -614615024502844300.0)
		tmp = Float64(t_4 - t_1);
	elseif (Float64(z * t) <= 1.7974798995587906e+301)
		tmp = Float64(Float64(t_4 * Float64(Float64(cos(t_6) * cos(t_3)) - Float64(sin(t_6) * sin(t_3)))) - t_1);
	else
		tmp = Float64(Float64(t_5 * Float64(Float64(cbrt(2.0) * (x ^ 0.16666666666666666)) * t_5)) - t_1);
	end
	return tmp
end

Wolfram

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[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(t / 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[(t / 3.0), $MachinePrecision]) * z + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$6 = N[(1.0 * y + (-t$95$2)), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -614615024502844300.0], N[(t$95$4 - t$95$1), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1.7974798995587906e+301], N[(N[(t$95$4 * N[(N[(N[Cos[t$95$6], $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$6], $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(t$95$5 * N[(N[(N[Power[2.0, 1/3], $MachinePrecision] * N[Power[x, 0.16666666666666666], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]

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	21.1
Target	19.0
Herbie	16.1

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -614615024502844288

   1. Initial program 42.4

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

   2. Taylor expanded in z around 0 34.1

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

   3. Taylor expanded in y around 0 34.1

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

   if -614615024502844288 < (*.f64 z t) < 1.79747989955879061e301

   1. Initial program 9.5

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

   2. Applied *-un-lft-identity_binary64 9.5

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

   3. Applied times-frac_binary64 9.5

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

   4. Applied *-un-lft-identity_binary64 9.5

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

   5. Applied prod-diff_binary64 9.5

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

   6. Applied cos-sum_binary64 8.1

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

   if 1.79747989955879061e301 < (*.f64 z t)

   1. Initial program 62.2

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

   2. Taylor expanded in z around 0 31.9

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

   3. Applied add-cube-cbrt_binary64 31.9

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

   4. Taylor expanded in y around 0 32.4

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{2} \cdot {x}^{0.16666666666666666}\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} - \frac{a}{b \cdot 3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -614615024502844300:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \leq 1.7974798995587906 \cdot 10^{+301}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \cdot \left(\left(\sqrt[3]{2} \cdot {x}^{0.16666666666666666}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y}\right) - \frac{a}{b \cdot 3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(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))))