ENA, Section 1.4, Mentioned, A

Average Error: 30.1 → 0.0

Time: 4.8s

Precision: binary64

\[-0.01 \leq x \land x \leq 0.01\]

Math

\[1 - \cos x \]

↓

\[\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right), 0.001388888888888889 \cdot {x}^{6}\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (fma
  (* x x)
  (fma (* x x) -0.041666666666666664 0.5)
  (* 0.001388888888888889 (pow x 6.0))))

C

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

↓

double code(double x) {
	return fma((x * x), fma((x * x), -0.041666666666666664, 0.5), (0.001388888888888889 * pow(x, 6.0)));
}

Julia

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

↓

function code(x)
	return fma(Float64(x * x), fma(Float64(x * x), -0.041666666666666664, 0.5), Float64(0.001388888888888889 * (x ^ 6.0)))
end

Wolfram

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

↓

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Target

Original	30.1
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 30.1

   \[1 - \cos x \]

2. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot {x}^{6} + 0.5 \cdot {x}^{2}\right) - 0.041666666666666664 \cdot {x}^{4}} \]

3. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right), 0.001388888888888889 \cdot {x}^{6}\right)} \]

4. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right), 0.001388888888888889 \cdot {x}^{6}\right) \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, A"
  :precision binary64
  :pre (and (<= -0.01 x) (<= x 0.01))

  :herbie-target
  (/ (* (sin x) (sin x)) (+ 1.0 (cos x)))

  (- 1.0 (cos x)))