Average Error: 31.9 → 0.0
Time: 11.8s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x} \]
↓
\[\begin{array}{l} t_0 := x - \sin x\\ t_1 := \frac{-1}{\cos x}\\ \mathbf{if}\;x \leq -0.09:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(x, 1, \sin x \cdot t_1\right) + \mathsf{fma}\left(t_1, \sin x, \sin x \cdot \frac{1}{\cos x}\right)}\\ \mathbf{elif}\;x \leq 0.088:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, -0.009642857142857142, \mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(0.00024107142857142857, {x}^{6}, -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x - \tan x}\\ \end{array} \]
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- x (sin x))) (t_1 (/ -1.0 (cos x))))
   (if (<= x -0.09)
     (/
      t_0
      (+
       (fma x 1.0 (* (sin x) t_1))
       (fma t_1 (sin x) (* (sin x) (/ 1.0 (cos x))))))
     (if (<= x 0.088)
       (fma
        (pow x 4.0)
        -0.009642857142857142
        (fma 0.225 (* x x) (fma 0.00024107142857142857 (pow x 6.0) -0.5)))
       (/ t_0 (- x (tan x)))))))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double t_0 = x - sin(x);
	double t_1 = -1.0 / cos(x);
	double tmp;
	if (x <= -0.09) {
		tmp = t_0 / (fma(x, 1.0, (sin(x) * t_1)) + fma(t_1, sin(x), (sin(x) * (1.0 / cos(x)))));
	} else if (x <= 0.088) {
		tmp = fma(pow(x, 4.0), -0.009642857142857142, fma(0.225, (x * x), fma(0.00024107142857142857, pow(x, 6.0), -0.5)));
	} else {
		tmp = t_0 / (x - tan(x));
	}
	return tmp;
}

function code(x)
	return Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
end
↓
function code(x)
	t_0 = Float64(x - sin(x))
	t_1 = Float64(-1.0 / cos(x))
	tmp = 0.0
	if (x <= -0.09)
		tmp = Float64(t_0 / Float64(fma(x, 1.0, Float64(sin(x) * t_1)) + fma(t_1, sin(x), Float64(sin(x) * Float64(1.0 / cos(x))))));
	elseif (x <= 0.088)
		tmp = fma((x ^ 4.0), -0.009642857142857142, fma(0.225, Float64(x * x), fma(0.00024107142857142857, (x ^ 6.0), -0.5)));
	else
		tmp = Float64(t_0 / Float64(x - tan(x)));
	end
	return tmp
end

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[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.09], N[(t$95$0 / N[(N[(x * 1.0 + N[(N[Sin[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Sin[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.088], N[(N[Power[x, 4.0], $MachinePrecision] * -0.009642857142857142 + N[(0.225 * N[(x * x), $MachinePrecision] + N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
t_0 := x - \sin x\\
t_1 := \frac{-1}{\cos x}\\
\mathbf{if}\;x \leq -0.09:\\
\;\;\;\;\frac{t_0}{\mathsf{fma}\left(x, 1, \sin x \cdot t_1\right) + \mathsf{fma}\left(t_1, \sin x, \sin x \cdot \frac{1}{\cos x}\right)}\\

\mathbf{elif}\;x \leq 0.088:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, -0.009642857142857142, \mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(0.00024107142857142857, {x}^{6}, -0.5\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{t_0}{x - \tan x}\\


\end{array}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.089999999999999997

    1. Initial program 0.0

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

    2. Applied egg-rr0.0

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

    if -0.089999999999999997 < x < 0.087999999999999995

    1. Initial program 63.0

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

    2. Taylor expanded in x around 0 0.0

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

    3. Simplified0.0

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

    4. Taylor expanded in x around 0 0.0

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

    5. Simplified0.0

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

    if 0.087999999999999995 < x

    1. Initial program 0.0

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

    2. Applied egg-rr0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.09:\\ \;\;\;\;\frac{x - \sin x}{\mathsf{fma}\left(x, 1, \sin x \cdot \frac{-1}{\cos x}\right) + \mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \sin x \cdot \frac{1}{\cos x}\right)}\\ \mathbf{elif}\;x \leq 0.088:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, -0.009642857142857142, \mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(0.00024107142857142857, {x}^{6}, -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))