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

Average Error: 21.0 → 17.3

Time: 21.3s

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	21.0
Target	18.9
Herbie	17.3

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

Derivation

1. Initial program 21.0

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

2. Simplified 20.9

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), \frac{\frac{a}{-3}}{b}\right)} \]
   Proof
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (fma.f64 z (*.f64 t -1/3) y))) (/.f64 (/.f64 a -3) b)): 0 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (fma.f64 z (*.f64 t (Rewrite<= metadata-eval (/.f64 -1 3))) y))) (/.f64 (/.f64 a -3) b)): 0 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (fma.f64 z (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 -1 3) t)) y))) (/.f64 (/.f64 a -3) b)): 0 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (fma.f64 z (Rewrite<= associate-/r/_binary64 (/.f64 -1 (/.f64 3 t))) y))) (/.f64 (/.f64 a -3) b)): 20 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (fma.f64 z (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 t) 3)) y))) (/.f64 (/.f64 a -3) b)): 0 points increase in error, 20 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (fma.f64 z (/.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 t)) 3) y))) (/.f64 (/.f64 a -3) b)): 20 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (/.f64 (neg.f64 t) 3)) y)))) (/.f64 (/.f64 a -3) b)): 20 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 z (neg.f64 t)) 3)) y))) (/.f64 (/.f64 a -3) b)): 0 points increase in error, 20 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (+.f64 (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 z t))) 3) y))) (/.f64 (/.f64 a -3) b)): 20 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (+.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (*.f64 z t) 3))) y))) (/.f64 (/.f64 a -3) b)): 0 points increase in error, 20 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (Rewrite<= +-commutative_binary64 (+.f64 y (neg.f64 (/.f64 (*.f64 z t) 3)))))) (/.f64 (/.f64 a -3) b)): 20 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (Rewrite<= sub-neg_binary64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (/.f64 (/.f64 a -3) b)): 0 points increase in error, 20 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 (/.f64 a (Rewrite<= metadata-eval (/.f64 3 -1))) b)): 20 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a -1) 3)) b)): 0 points increase in error, 20 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 a)) 3) b)): 20 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 a)) 3) b)): 0 points increase in error, 20 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (Rewrite=> associate-/l/_binary64 (/.f64 (neg.f64 a) (*.f64 b 3)))): 0 points increase in error, 20 points decrease in error
   (fma.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 a (*.f64 b 3))))): 20 points increase in error, 0 points decrease in error
   (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (*.f64 (sqrt.f64 x) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (/.f64 a (*.f64 b 3)))): 0 points increase in error, 20 points decrease in error
   (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))))) (/.f64 a (*.f64 b 3))): 20 points increase in error, 0 points decrease in error

3. Taylor expanded in z around 0 17.3

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

4. Final simplification 17.3

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

Alternatives

Alternative 1
Error	21.0
Cost	14025

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-52} \lor \neg \left(t_1 \leq 10^{-34}\right):\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, \frac{\frac{a}{-3}}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \end{array} \]

Alternative 2
Error	21.0
Cost	13897

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-52} \lor \neg \left(t_1 \leq 10^{-34}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \end{array} \]

Alternative 3
Error	17.4
Cost	13504

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

Alternative 4
Error	17.4
Cost	13504

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

Alternative 5
Error	17.3
Cost	13504

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

Alternative 6
Error	25.3
Cost	6976

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

Alternative 7
Error	25.3
Cost	6976

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

Alternative 8
Error	25.2
Cost	6976

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

Alternative 9
Error	36.2
Cost	320

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

Alternative 10
Error	36.2
Cost	320

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

Alternative 11
Error	36.1
Cost	320

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

Alternative 12
Error	36.1
Cost	320

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

Reproduce

herbie shell --seed 2022340 
(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))))