sintan (problem 3.4.5)

Average Error: 31.0 → 0.2

Time: 12.2s

Precision: binary64

Math

\[\frac{x - \sin x}{x - \tan x} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.09740715512336516:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 4.99533306627316:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, 0.00024107142857142857 \cdot {x}^{6}\right) - \mathsf{fma}\left(0.009642857142857142, {x}^{4}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\sin x}{x} \cdot \left(\frac{1}{\cos x} + -1\right)\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.09740715512336516)
   (/ (- x (sin x)) (- x (tan x)))
   (if (<= x 4.99533306627316)
     (-
      (fma 0.225 (* x x) (* 0.00024107142857142857 (pow x 6.0)))
      (fma 0.009642857142857142 (pow x 4.0) 0.5))
     (+ 1.0 (* (/ (sin x) x) (+ (/ 1.0 (cos x)) -1.0))))))

C

double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}

↓

double code(double x) {
	double tmp;
	if (x <= -0.09740715512336516) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else if (x <= 4.99533306627316) {
		tmp = fma(0.225, (x * x), (0.00024107142857142857 * pow(x, 6.0))) - fma(0.009642857142857142, pow(x, 4.0), 0.5);
	} else {
		tmp = 1.0 + ((sin(x) / x) * ((1.0 / cos(x)) + -1.0));
	}
	return tmp;
}

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
end

↓

function code(x)
	tmp = 0.0
	if (x <= -0.09740715512336516)
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	elseif (x <= 4.99533306627316)
		tmp = Float64(fma(0.225, Float64(x * x), Float64(0.00024107142857142857 * (x ^ 6.0))) - fma(0.009642857142857142, (x ^ 4.0), 0.5));
	else
		tmp = Float64(1.0 + Float64(Float64(sin(x) / x) * Float64(Float64(1.0 / cos(x)) + -1.0)));
	end
	return tmp
end

Wolfram

code[x_] := N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -0.09740715512336516], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.99533306627316], N[(N[(0.225 * N[(x * x), $MachinePrecision] + N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.0974071551233651606

   1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x} \]

   if -0.0974071551233651606 < x < 4.9953330662731599

   1. Initial program 62.9

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{\left(0.00024107142857142857 \cdot {x}^{6} + 0.225 \cdot {x}^{2}\right) - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)} \]

   3. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, 0.00024107142857142857 \cdot {x}^{6}\right) - \mathsf{fma}\left(0.009642857142857142, {x}^{4}, 0.5\right)} \]

   if 4.9953330662731599 < x

   1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Taylor expanded in x around inf 0.5

      \[\leadsto \color{blue}{\left(\frac{\sin x}{\cos x \cdot x} + 1\right) - \frac{\sin x}{x}} \]

   3. Simplified 0.5

      \[\leadsto \color{blue}{1 + \frac{\sin x}{x} \cdot \left(\frac{1}{\cos x} + -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.09740715512336516:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 4.99533306627316:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, 0.00024107142857142857 \cdot {x}^{6}\right) - \mathsf{fma}\left(0.009642857142857142, {x}^{4}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\sin x}{x} \cdot \left(\frac{1}{\cos x} + -1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))