cos2 (problem 3.4.1)

Average Error: 31.2 → 0.1

Time: 11.8s

Precision: binary64

Cost: 19976

Math

\[\frac{1 - \cos x}{x \cdot x} \]

↓

\[\begin{array}{l} t_0 := \sin x \cdot \left(\tan \left(x \cdot 0.5\right) \cdot {x}^{-2}\right)\\ \mathbf{if}\;x \leq -0.030539271847426327:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.479060632992062 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (sin x) (* (tan (* x 0.5)) (pow x -2.0)))))
   (if (<= x -0.030539271847426327)
     t_0
     (if (<= x 2.479060632992062e-23)
       (fma
        (* x x)
        (fma (* x x) 0.001388888888888889 -0.041666666666666664)
        0.5)
       t_0))))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

↓

double code(double x) {
	double t_0 = sin(x) * (tan((x * 0.5)) * pow(x, -2.0));
	double tmp;
	if (x <= -0.030539271847426327) {
		tmp = t_0;
	} else if (x <= 2.479060632992062e-23) {
		tmp = fma((x * x), fma((x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

↓

function code(x)
	t_0 = Float64(sin(x) * Float64(tan(Float64(x * 0.5)) * (x ^ -2.0)))
	tmp = 0.0
	if (x <= -0.030539271847426327)
		tmp = t_0;
	elseif (x <= 2.479060632992062e-23)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.030539271847426327], t$95$0, If[LessEqual[x, 2.479060632992062e-23], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0305392718474263268 or 2.479060632992062e-23 < x

   1. Initial program 2.4

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Applied egg-rr 1.0

      \[\leadsto \frac{\color{blue}{\frac{\sin x \cdot \sin x}{1 + \cos x}}}{x \cdot x} \]

   3. Applied egg-rr 0.7

      \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x \cdot x} \]

   4. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\sin x \cdot \left(\tan \left(x \cdot 0.5\right) \cdot {x}^{-2}\right)} \]

   if -0.0305392718474263268 < x < 2.479060632992062e-23

   1. Initial program 62.7

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)} \]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)} \]
      Proof
      (fma.f64 (*.f64 x x) (fma.f64 (*.f64 x x) 1/720 -1/24) 1/2): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (fma.f64 (*.f64 x x) 1/720 -1/24) 1/2): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 x 2) (fma.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) 1/720 -1/24) 1/2): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 x 2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 x 2) 1/720) -1/24)) 1/2): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 x 2) (Rewrite=> +-commutative_binary64 (+.f64 -1/24 (*.f64 (pow.f64 x 2) 1/720))) 1/2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 x 2) (+.f64 -1/24 (*.f64 (pow.f64 x 2) 1/720))) 1/2)): 0 points increase in error, 1 points decrease in error
      (+.f64 (*.f64 (pow.f64 x 2) (+.f64 -1/24 (Rewrite=> *-commutative_binary64 (*.f64 1/720 (pow.f64 x 2))))) 1/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1/24 (pow.f64 x 2)) (*.f64 (*.f64 1/720 (pow.f64 x 2)) (pow.f64 x 2)))) 1/2): 1 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1/24 (pow.f64 x 2)) (Rewrite<= associate-*r*_binary64 (*.f64 1/720 (*.f64 (pow.f64 x 2) (pow.f64 x 2))))) 1/2): 9 points increase in error, 3 points decrease in error
      (+.f64 (+.f64 (*.f64 -1/24 (pow.f64 x 2)) (*.f64 1/720 (Rewrite=> pow-sqr_binary64 (pow.f64 x (*.f64 2 2))))) 1/2): 5 points increase in error, 6 points decrease in error
      (+.f64 (+.f64 (*.f64 -1/24 (pow.f64 x 2)) (*.f64 1/720 (pow.f64 x (Rewrite=> metadata-eval 4)))) 1/2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1/2 (+.f64 (*.f64 -1/24 (pow.f64 x 2)) (*.f64 1/720 (pow.f64 x 4))))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.030539271847426327:\\ \;\;\;\;\sin x \cdot \left(\tan \left(x \cdot 0.5\right) \cdot {x}^{-2}\right)\\ \mathbf{elif}\;x \leq 2.479060632992062 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\tan \left(x \cdot 0.5\right) \cdot {x}^{-2}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	7240

\[\begin{array}{l} t_0 := \frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{if}\;x \leq -0.030539271847426327:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.259081342341926 \cdot 10^{-23}:\\ \;\;\;\;\left(1 + \mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	14.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -97919.52281259478:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{elif}\;x \leq 4.259081342341926 \cdot 10^{-23}:\\ \;\;\;\;0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \end{array} \]

Alternative 3
Error	14.0
Cost	576

\[\frac{1}{\left(x \cdot x\right) \cdot 0.16666666666666666 + 2} \]

Alternative 4
Error	15.7
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -7.14495020679171 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.0905127883141194 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	31.2
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022289 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))