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

Average Error: 20.6 → 15.1

Time: 23.7s

Precision: binary64

Cost: 48840

\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := y + t \cdot \left(z \cdot -0.3333333333333333\right)\\ t_3 := y - \frac{z \cdot t}{3}\\ t_4 := \mathsf{fma}\left(-z, t \cdot 0.3333333333333333, t \cdot \left(z \cdot 0.3333333333333333\right)\right)\\ \mathbf{if}\;t_3 \leq -4 \cdot 10^{+256}:\\ \;\;\;\;t_1 \cdot \cos y - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t_1 \cdot \left(\cos t_2 \cdot \cos t_4 - \sin t_2 \cdot \sin t_4\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1 - {\left(b \cdot \frac{3}{a}\right)}^{-1}\\ \end{array} \]

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 (* 2.0 (sqrt x)))
        (t_2 (+ y (* t (* z -0.3333333333333333))))
        (t_3 (- y (/ (* z t) 3.0)))
        (t_4
         (fma (- z) (* t 0.3333333333333333) (* t (* z 0.3333333333333333)))))
   (if (<= t_3 -4e+256)
     (- (* t_1 (cos y)) (/ (/ a 3.0) b))
     (if (<= t_3 2e+300)
       (-
        (* t_1 (- (* (cos t_2) (cos t_4)) (* (sin t_2) (sin t_4))))
        (/ a (* 3.0 b)))
       (- t_1 (pow (* b (/ 3.0 a)) -1.0))))))

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 = y + (t * (z * -0.3333333333333333));
	double t_3 = y - ((z * t) / 3.0);
	double t_4 = fma(-z, (t * 0.3333333333333333), (t * (z * 0.3333333333333333)));
	double tmp;
	if (t_3 <= -4e+256) {
		tmp = (t_1 * cos(y)) - ((a / 3.0) / b);
	} else if (t_3 <= 2e+300) {
		tmp = (t_1 * ((cos(t_2) * cos(t_4)) - (sin(t_2) * sin(t_4)))) - (a / (3.0 * b));
	} else {
		tmp = t_1 - pow((b * (3.0 / a)), -1.0);
	}
	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(y + Float64(t * Float64(z * -0.3333333333333333)))
	t_3 = Float64(y - Float64(Float64(z * t) / 3.0))
	t_4 = fma(Float64(-z), Float64(t * 0.3333333333333333), Float64(t * Float64(z * 0.3333333333333333)))
	tmp = 0.0
	if (t_3 <= -4e+256)
		tmp = Float64(Float64(t_1 * cos(y)) - Float64(Float64(a / 3.0) / b));
	elseif (t_3 <= 2e+300)
		tmp = Float64(Float64(t_1 * Float64(Float64(cos(t_2) * cos(t_4)) - Float64(sin(t_2) * sin(t_4)))) - Float64(a / Float64(3.0 * b)));
	else
		tmp = Float64(t_1 - (Float64(b * Float64(3.0 / a)) ^ -1.0));
	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[(y + N[(t * N[(z * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-z) * N[(t * 0.3333333333333333), $MachinePrecision] + N[(t * N[(z * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -4e+256], N[(N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+300], N[(N[(t$95$1 * N[(N[(N[Cos[t$95$2], $MachinePrecision] * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[Power[N[(b * N[(3.0 / a), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	20.6
Target	18.6
Herbie	15.1

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 y (/.f64 (*.f64 z t) 3)) < -4.0000000000000001e256

   1. Initial program 45.4

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

   2. Taylor expanded in z around 0 29.4

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

   3. Applied egg-rr 29.4

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

   if -4.0000000000000001e256 < (-.f64 y (/.f64 (*.f64 z t) 3)) < 2.0000000000000001e300

   1. Initial program 13.4

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

   2. Applied egg-rr 13.5

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

   3. Applied egg-rr 20.2

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

   4. Applied egg-rr 11.5

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

   if 2.0000000000000001e300 < (-.f64 y (/.f64 (*.f64 z t) 3))

   1. Initial program 60.3

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

   2. Taylor expanded in z around 0 32.3

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

   3. Applied egg-rr 32.3

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

   4. Applied egg-rr 32.3

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

   5. Taylor expanded in y around 0 33.0

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - {\left(\frac{3}{a} \cdot b\right)}^{-1} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 15.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -4 \cdot 10^{+256}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(y + t \cdot \left(z \cdot -0.3333333333333333\right)\right) \cdot \cos \left(\mathsf{fma}\left(-z, t \cdot 0.3333333333333333, t \cdot \left(z \cdot 0.3333333333333333\right)\right)\right) - \sin \left(y + t \cdot \left(z \cdot -0.3333333333333333\right)\right) \cdot \sin \left(\mathsf{fma}\left(-z, t \cdot 0.3333333333333333, t \cdot \left(z \cdot 0.3333333333333333\right)\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - {\left(b \cdot \frac{3}{a}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.1
Cost	13896

\[\begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-74}:\\ \;\;\;\;t_2 - t_1\\ \mathbf{elif}\;t_1 \leq 10^{-118}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 - \frac{\frac{a}{3}}{b}\\ \end{array} \]

Alternative 2
Error	17.1
Cost	13504

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

Alternative 3
Error	17.1
Cost	13504

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

Alternative 4
Error	17.1
Cost	13504

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

Alternative 5
Error	25.2
Cost	6976

\[2 \cdot \sqrt{x} + \frac{a}{b} \cdot -0.3333333333333333 \]

Alternative 6
Error	25.1
Cost	6976

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

Alternative 7
Error	25.1
Cost	6976

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

Alternative 8
Error	36.1
Cost	320

\[a \cdot \frac{-0.3333333333333333}{b} \]

Alternative 9
Error	36.1
Cost	320

\[\frac{a \cdot -0.3333333333333333}{b} \]

Reproduce

herbie shell --seed 2022300 
(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))))