sintan (problem 3.4.5)

Average Error: 31.1 → 0.0

Time: 12.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \leq -0.03244483507352827:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\\ \mathbf{elif}\;x \leq 0.02803816352959124:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left({x}^{4}, -0.009642857142857142, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- x (sin x)) (- x (tan x)))))
   (if (<= x -0.03244483507352827)
     (expm1 (log1p t_0))
     (if (<= x 0.02803816352959124)
       (fma 0.225 (* x x) (fma (pow x 4.0) -0.009642857142857142 -0.5))
       t_0))))

C

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

↓

double code(double x) {
	double t_0 = (x - sin(x)) / (x - tan(x));
	double tmp;
	if (x <= -0.03244483507352827) {
		tmp = expm1(log1p(t_0));
	} else if (x <= 0.02803816352959124) {
		tmp = fma(0.225, (x * x), fma(pow(x, 4.0), -0.009642857142857142, -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
	tmp = 0.0
	if (x <= -0.03244483507352827)
		tmp = expm1(log1p(t_0));
	elseif (x <= 0.02803816352959124)
		tmp = fma(0.225, Float64(x * x), fma((x ^ 4.0), -0.009642857142857142, -0.5));
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.03244483507352827], N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[x, 0.02803816352959124], N[(0.225 * N[(x * x), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * -0.009642857142857142 + -0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.032444835073528272

   1. Initial program 0.1

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

   2. Applied expm1-log1p-u_binary64 0.1

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - \sin x}{x - \tan x}\right)\right)} \]

   if -0.032444835073528272 < x < 0.0280381635295912403

   1. Initial program 63.1

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

   2. Taylor expanded in x around 0 0.0

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

   3. Simplified 0.0

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

   4. Taylor expanded in x around 0 0.0

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

   5. Simplified 0.0

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

   if 0.0280381635295912403 < x

   1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03244483507352827:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - \sin x}{x - \tan x}\right)\right)\\ \mathbf{elif}\;x \leq 0.02803816352959124:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left({x}^{4}, -0.009642857142857142, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))