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

Average Error: 21.5 → 17.0

Time: 14.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{z \cdot t}{3}\\ t_2 := \frac{a}{b \cdot 3}\\ t_3 := 2 \cdot \sqrt{x}\\ t_4 := \sqrt{t_3}\\ t_5 := \mathsf{fma}\left(t_4, t_4 \cdot \cos y, -t_2\right)\\ \mathbf{if}\;z \cdot t \leq -3.4987150309268354 \cdot 10^{+47}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;z \cdot t \leq 7.053942751863652 \cdot 10^{+25}:\\ \;\;\;\;\left(t_3 \cdot \left(\cos y \cdot \cos t_1\right) + t_3 \cdot \left(\sin y \cdot \sin t_1\right)\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

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) 3.0))
        (t_2 (/ a (* b 3.0)))
        (t_3 (* 2.0 (sqrt x)))
        (t_4 (sqrt t_3))
        (t_5 (fma t_4 (* t_4 (cos y)) (- t_2))))
   (if (<= (* z t) -3.4987150309268354e+47)
     t_5
     (if (<= (* z t) 7.053942751863652e+25)
       (- (+ (* t_3 (* (cos y) (cos t_1))) (* t_3 (* (sin y) (sin t_1)))) t_2)
       t_5))))

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) / 3.0;
	double t_2 = a / (b * 3.0);
	double t_3 = 2.0 * sqrt(x);
	double t_4 = sqrt(t_3);
	double t_5 = fma(t_4, (t_4 * cos(y)), -t_2);
	double tmp;
	if ((z * t) <= -3.4987150309268354e+47) {
		tmp = t_5;
	} else if ((z * t) <= 7.053942751863652e+25) {
		tmp = ((t_3 * (cos(y) * cos(t_1))) + (t_3 * (sin(y) * sin(t_1)))) - t_2;
	} else {
		tmp = t_5;
	}
	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) / 3.0)
	t_2 = Float64(a / Float64(b * 3.0))
	t_3 = Float64(2.0 * sqrt(x))
	t_4 = sqrt(t_3)
	t_5 = fma(t_4, Float64(t_4 * cos(y)), Float64(-t_2))
	tmp = 0.0
	if (Float64(z * t) <= -3.4987150309268354e+47)
		tmp = t_5;
	elseif (Float64(z * t) <= 7.053942751863652e+25)
		tmp = Float64(Float64(Float64(t_3 * Float64(cos(y) * cos(t_1))) + Float64(t_3 * Float64(sin(y) * sin(t_1)))) - t_2);
	else
		tmp = t_5;
	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] / 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + (-t$95$2)), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -3.4987150309268354e+47], t$95$5, If[LessEqual[N[(z * t), $MachinePrecision], 7.053942751863652e+25], N[(N[(N[(t$95$3 * N[(N[Cos[y], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(N[Sin[y], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], t$95$5]]]]]]]

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	21.5
Target	19.3
Herbie	17.0

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -3.4987150309268354e47 or 7.0539427518636517e25 < (*.f64 z t)

   1. Initial program 42.8

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

   2. Taylor expanded in z around 0 33.9

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

   3. Applied egg-rr 33.9

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

   if -3.4987150309268354e47 < (*.f64 z t) < 7.0539427518636517e25

   1. Initial program 3.5

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

   2. Applied egg-rr 2.8

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

4. Final simplification 17.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -3.4987150309268354 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2 \cdot \sqrt{x}}, \sqrt{2 \cdot \sqrt{x}} \cdot \cos y, -\frac{a}{b \cdot 3}\right)\\ \mathbf{elif}\;z \cdot t \leq 7.053942751863652 \cdot 10^{+25}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2 \cdot \sqrt{x}}, \sqrt{2 \cdot \sqrt{x}} \cdot \cos y, -\frac{a}{b \cdot 3}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))))