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

Average Error: 20.3 → 16.8

Time: 22.3s

Precision: binary64

Cost: 19776

Math

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

↓

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

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
 (fma 2.0 (* (sqrt x) (cos y)) (/ (/ a -3.0) b)))

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) {
	return fma(2.0, (sqrt(x) * cos(y)), ((a / -3.0) / b));
}

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)
	return fma(2.0, Float64(sqrt(x) * cos(y)), Float64(Float64(a / -3.0) / b))
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_] := N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(a / -3.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Target

Original	20.3
Target	18.3
Herbie	16.8

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

Derivation

1. Initial program 20.3

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

2. Simplified 20.3

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

   [Start] 20.3	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   associate-*l* [=>] 20.3	\[ \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]
   fma-neg [=>] 20.3	\[ \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right), -\frac{a}{b \cdot 3}\right)} \]
   remove-double-neg [<=] 20.3	\[ \mathsf{fma}\left(2, \color{blue}{-\left(-\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)}, -\frac{a}{b \cdot 3}\right) \]
   remove-double-neg [=>] 20.3	\[ \mathsf{fma}\left(2, \color{blue}{\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)}, -\frac{a}{b \cdot 3}\right) \]
   sub-neg [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \color{blue}{\left(y + \left(-\frac{z \cdot t}{3}\right)\right)}, -\frac{a}{b \cdot 3}\right) \]
   +-commutative [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \color{blue}{\left(\left(-\frac{z \cdot t}{3}\right) + y\right)}, -\frac{a}{b \cdot 3}\right) \]
   distribute-neg-frac [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\color{blue}{\frac{-z \cdot t}{3}} + y\right), -\frac{a}{b \cdot 3}\right) \]
   distribute-rgt-neg-in [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\frac{\color{blue}{z \cdot \left(-t\right)}}{3} + y\right), -\frac{a}{b \cdot 3}\right) \]
   associate-*r/ [<=] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\color{blue}{z \cdot \frac{-t}{3}} + y\right), -\frac{a}{b \cdot 3}\right) \]
   fma-def [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(z, \frac{-t}{3}, y\right)\right)}, -\frac{a}{b \cdot 3}\right) \]
   mul-1-neg [<=] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, \frac{\color{blue}{-1 \cdot t}}{3}, y\right)\right), -\frac{a}{b \cdot 3}\right) \]
   associate-/l* [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, \color{blue}{\frac{-1}{\frac{3}{t}}}, y\right)\right), -\frac{a}{b \cdot 3}\right) \]
   associate-/r/ [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, \color{blue}{\frac{-1}{3} \cdot t}, y\right)\right), -\frac{a}{b \cdot 3}\right) \]
   *-commutative [<=] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, \color{blue}{t \cdot \frac{-1}{3}}, y\right)\right), -\frac{a}{b \cdot 3}\right) \]
   metadata-eval [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot \color{blue}{-0.3333333333333333}, y\right)\right), -\frac{a}{b \cdot 3}\right) \]
   distribute-neg-frac [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), \color{blue}{\frac{-a}{b \cdot 3}}\right) \]
   associate-/l/ [<=] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), \color{blue}{\frac{\frac{-a}{3}}{b}}\right) \]
   neg-mul-1 [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), \frac{\frac{\color{blue}{-1 \cdot a}}{3}}{b}\right) \]
   *-commutative [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), \frac{\frac{\color{blue}{a \cdot -1}}{3}}{b}\right) \]
   associate-/l* [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), \frac{\color{blue}{\frac{a}{\frac{3}{-1}}}}{b}\right) \]
   metadata-eval [=>] 20.3	\[ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), \frac{\frac{a}{\color{blue}{-3}}}{b}\right) \]

3. Taylor expanded in z around 0 16.8

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

4. Final simplification 16.8

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

Alternatives

Alternative 1
Error	20.4
Cost	14025

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-42} \lor \neg \left(t_1 \leq 5 \cdot 10^{-153}\right):\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, \frac{\frac{a}{-3}}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \end{array} \]

Alternative 2
Error	20.4
Cost	13897

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-42} \lor \neg \left(t_1 \leq 5 \cdot 10^{-153}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \end{array} \]

Alternative 3
Error	16.9
Cost	13504

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

Alternative 4
Error	16.8
Cost	13504

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

Alternative 5
Error	25.1
Cost	6976

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

Alternative 6
Error	25.0
Cost	6976

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

Alternative 7
Error	36.1
Cost	320

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

Alternative 8
Error	36.1
Cost	320

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

Alternative 9
Error	36.1
Cost	320

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

Alternative 10
Error	36.1
Cost	320

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

Alternative 11
Error	36.1
Cost	320

\[\frac{\frac{a}{-3}}{b} \]

Reproduce

herbie shell --seed 2022364 
(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))))