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

Average Error: 20.5 → 16.2

Time: 12.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{b \cdot 3}\\ t_3 := \left(z \cdot t\right) \cdot -0.3333333333333333\\ \mathbf{if}\;z \cdot t \leq -3.5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, t_1\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+168}:\\ \;\;\;\;t_1 \cdot \left(\cos y \cdot \cos t_3 - \sin y \cdot \sin t_3\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x \cdot 4\right) \cdot {\cos y}^{2}} - t_2\\ \end{array} \]

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 (* 2.0 (sqrt x)))
        (t_2 (/ a (* b 3.0)))
        (t_3 (* (* z t) -0.3333333333333333)))
   (if (<= (* z t) -3.5e+217)
     (fma (/ a b) -0.3333333333333333 t_1)
     (if (<= (* z t) 5e+168)
       (- (* t_1 (- (* (cos y) (cos t_3)) (* (sin y) (sin t_3)))) t_2)
       (- (sqrt (* (* x 4.0) (pow (cos y) 2.0))) t_2)))))

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 = 2.0 * sqrt(x);
	double t_2 = a / (b * 3.0);
	double t_3 = (z * t) * -0.3333333333333333;
	double tmp;
	if ((z * t) <= -3.5e+217) {
		tmp = fma((a / b), -0.3333333333333333, t_1);
	} else if ((z * t) <= 5e+168) {
		tmp = (t_1 * ((cos(y) * cos(t_3)) - (sin(y) * sin(t_3)))) - t_2;
	} else {
		tmp = sqrt(((x * 4.0) * pow(cos(y), 2.0))) - t_2;
	}
	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(2.0 * sqrt(x))
	t_2 = Float64(a / Float64(b * 3.0))
	t_3 = Float64(Float64(z * t) * -0.3333333333333333)
	tmp = 0.0
	if (Float64(z * t) <= -3.5e+217)
		tmp = fma(Float64(a / b), -0.3333333333333333, t_1);
	elseif (Float64(z * t) <= 5e+168)
		tmp = Float64(Float64(t_1 * Float64(Float64(cos(y) * cos(t_3)) - Float64(sin(y) * sin(t_3)))) - t_2);
	else
		tmp = Float64(sqrt(Float64(Float64(x * 4.0) * (cos(y) ^ 2.0))) - t_2);
	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[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -3.5e+217], N[(N[(a / b), $MachinePrecision] * -0.3333333333333333 + t$95$1), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+168], N[(N[(t$95$1 * N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Sqrt[N[(N[(x * 4.0), $MachinePrecision] * N[Power[N[Cos[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t$95$2), $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	20.5
Target	18.6
Herbie	16.2

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -3.4999999999999998e217

   1. Initial program 50.9

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

   2. Taylor expanded in z around 0 32.5

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

   3. Applied egg-rr 32.6

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

   4. Taylor expanded in y around 0 32.3

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

   5. Simplified 32.3

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

   if -3.4999999999999998e217 < (*.f64 z t) < 4.99999999999999967e168

   1. Initial program 11.4

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

   2. Applied egg-rr 10.8

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

   if 4.99999999999999967e168 < (*.f64 z t)

   1. Initial program 46.4

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

   2. Taylor expanded in z around 0 33.2

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

   3. Applied egg-rr 33.2

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot 4\right) \cdot {\cos y}^{2}}} - \frac{a}{b \cdot 3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -3.5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+168}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(z \cdot t\right) \cdot -0.3333333333333333\right) - \sin y \cdot \sin \left(\left(z \cdot t\right) \cdot -0.3333333333333333\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x \cdot 4\right) \cdot {\cos y}^{2}} - \frac{a}{b \cdot 3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))