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

Average Error: 20.5 → 16.3

Time: 10.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(z \cdot t\right) \cdot -0.3333333333333333\\ t_2 := y - \frac{z \cdot t}{3}\\ t_3 := a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b \cdot -3}\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x} \cdot \left(\cos y \cdot \cos t_1 - \sin t_1 \cdot \sin y\right), t_3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x \cdot {\cos y}^{2}}, t_3\right)\\ \end{array} \]

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 (* (* z t) -0.3333333333333333))
        (t_2 (- y (/ (* z t) 3.0)))
        (t_3 (* a (/ -0.3333333333333333 b))))
   (if (<= t_2 -1e+210)
     (fma 2.0 (* (sqrt x) (cos y)) (/ a (* b -3.0)))
     (if (<= t_2 2e+305)
       (fma
        2.0
        (* (sqrt x) (- (* (cos y) (cos t_1)) (* (sin t_1) (sin y))))
        t_3)
       (fma 2.0 (sqrt (* x (pow (cos y) 2.0))) t_3)))))

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 = (z * t) * -0.3333333333333333;
	double t_2 = y - ((z * t) / 3.0);
	double t_3 = a * (-0.3333333333333333 / b);
	double tmp;
	if (t_2 <= -1e+210) {
		tmp = fma(2.0, (sqrt(x) * cos(y)), (a / (b * -3.0)));
	} else if (t_2 <= 2e+305) {
		tmp = fma(2.0, (sqrt(x) * ((cos(y) * cos(t_1)) - (sin(t_1) * sin(y)))), t_3);
	} else {
		tmp = fma(2.0, sqrt((x * pow(cos(y), 2.0))), t_3);
	}
	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(Float64(z * t) * -0.3333333333333333)
	t_2 = Float64(y - Float64(Float64(z * t) / 3.0))
	t_3 = Float64(a * Float64(-0.3333333333333333 / b))
	tmp = 0.0
	if (t_2 <= -1e+210)
		tmp = fma(2.0, Float64(sqrt(x) * cos(y)), Float64(a / Float64(b * -3.0)));
	elseif (t_2 <= 2e+305)
		tmp = fma(2.0, Float64(sqrt(x) * Float64(Float64(cos(y) * cos(t_1)) - Float64(sin(t_1) * sin(y)))), t_3);
	else
		tmp = fma(2.0, sqrt(Float64(x * (cos(y) ^ 2.0))), t_3);
	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[(N[(z * t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]}, Block[{t$95$2 = N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a * N[(-0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+210], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+305], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$1], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(2.0 * N[Sqrt[N[(x * N[Power[N[Cos[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$3), $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	18.4
Herbie	16.3

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 y (/.f64 (*.f64 z t) 3)) < -9.99999999999999927e209

   1. Initial program 37.9

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

   2. Simplified 37.8

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

   3. Taylor expanded in z around 0 26.1

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

   4. Applied egg-rr 26.1

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

   if -9.99999999999999927e209 < (-.f64 y (/.f64 (*.f64 z t) 3)) < 1.9999999999999999e305

   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. Simplified 13.2

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

   3. Applied egg-rr 12.7

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

   if 1.9999999999999999e305 < (-.f64 y (/.f64 (*.f64 z t) 3))

   1. Initial program 62.0

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

   2. Simplified 62.1

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

   3. Taylor expanded in z around 0 32.7

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

   4. Applied egg-rr 32.8

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{\sqrt{x \cdot {\cos y}^{2}}}, a \cdot \frac{-0.3333333333333333}{b}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -1 \cdot 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b \cdot -3}\right)\\ \mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x} \cdot \left(\cos y \cdot \cos \left(\left(z \cdot t\right) \cdot -0.3333333333333333\right) - \sin \left(\left(z \cdot t\right) \cdot -0.3333333333333333\right) \cdot \sin y\right), a \cdot \frac{-0.3333333333333333}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x \cdot {\cos y}^{2}}, a \cdot \frac{-0.3333333333333333}{b}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))