ENA, Section 1.4, Mentioned, A

Average Error: 30.5 → 0.0

Time: 5.0s

Precision: binary64

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

Math

\[1 - \cos x \]

↓

\[\mathsf{fma}\left(0.5, x \cdot x, \mathsf{fma}\left({x}^{4}, -0.041666666666666664, 0.001388888888888889 \cdot {x}^{6}\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (fma
  0.5
  (* x x)
  (fma
   (pow x 4.0)
   -0.041666666666666664
   (* 0.001388888888888889 (pow x 6.0)))))

C

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

↓

double code(double x) {
	return fma(0.5, (x * x), fma(pow(x, 4.0), -0.041666666666666664, (0.001388888888888889 * pow(x, 6.0))));
}

Julia

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

↓

function code(x)
	return fma(0.5, Float64(x * x), fma((x ^ 4.0), -0.041666666666666664, Float64(0.001388888888888889 * (x ^ 6.0))))
end

Wolfram

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

↓

code[x_] := N[(0.5 * N[(x * x), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * -0.041666666666666664 + N[(0.001388888888888889 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Target

Original	30.5
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 30.5

   \[1 - \cos x \]

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

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

4. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(0.5, x \cdot x, \mathsf{fma}\left({x}^{4}, -0.041666666666666664, 0.001388888888888889 \cdot {x}^{6}\right)\right) \]

Reproduce

herbie shell --seed 2022134 
(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)))