sintan (problem 3.4.5)

Average Accuracy: 51.8% → 99.9%

Time: 23.3s

Precision: binary64

Cost: 45636

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\tan x}\\ t_1 := \sin x - x\\ \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left({t_0}^{2}, t_0, -x\right)}{t_1}}\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;\left({x}^{4} \cdot -0.009642857142857142 + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{t_1}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (tan x))) (t_1 (- (sin x) x)))
   (if (<= x -0.03)
     (/ 1.0 (/ (fma (pow t_0 2.0) t_0 (- x)) t_1))
     (if (<= x 0.028)
       (+ (+ (* (pow x 4.0) -0.009642857142857142) (* x (* x 0.225))) -0.5)
       (/ 1.0 (/ (- (tan x) x) t_1))))))

C

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

↓

double code(double x) {
	double t_0 = cbrt(tan(x));
	double t_1 = sin(x) - x;
	double tmp;
	if (x <= -0.03) {
		tmp = 1.0 / (fma(pow(t_0, 2.0), t_0, -x) / t_1);
	} else if (x <= 0.028) {
		tmp = ((pow(x, 4.0) * -0.009642857142857142) + (x * (x * 0.225))) + -0.5;
	} else {
		tmp = 1.0 / ((tan(x) - x) / t_1);
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = cbrt(tan(x))
	t_1 = Float64(sin(x) - x)
	tmp = 0.0
	if (x <= -0.03)
		tmp = Float64(1.0 / Float64(fma((t_0 ^ 2.0), t_0, Float64(-x)) / t_1));
	elseif (x <= 0.028)
		tmp = Float64(Float64(Float64((x ^ 4.0) * -0.009642857142857142) + Float64(x * Float64(x * 0.225))) + -0.5);
	else
		tmp = Float64(1.0 / Float64(Float64(tan(x) - x) / t_1));
	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[Power[N[Tan[x], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -0.03], N[(1.0 / N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$0 + (-x)), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.028], N[(N[(N[(N[Power[x, 4.0], $MachinePrecision] * -0.009642857142857142), $MachinePrecision] + N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(1.0 / N[(N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.029999999999999999

   1. Initial program 99.9%

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

   2. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      Proof

      [Start] 99.9	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 99.9	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 99.9	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 99.9	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 99.9	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 99.9	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 99.9	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 99.9	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 99.9	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 99.9	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Applied egg-rr 99.7%

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

      [Start] 99.9	\[ \frac{\sin x - x}{\tan x - x} \]
      div-inv [=>] 99.7	\[ \color{blue}{\left(\sin x - x\right) \cdot \frac{1}{\tan x - x}} \]
      *-commutative [=>] 99.7	\[ \color{blue}{\frac{1}{\tan x - x} \cdot \left(\sin x - x\right)} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\tan x - x}{\sin x - x}}} \]
      Proof

      [Start] 99.7	\[ \frac{1}{\tan x - x} \cdot \left(\sin x - x\right) \]
      associate-/r/ [<=] 99.9	\[ \color{blue}{\frac{1}{\frac{\tan x - x}{\sin x - x}}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\tan x}\right)}^{2}, \sqrt[3]{\tan x}, -x\right)}}{\sin x - x}} \]
      Proof

      [Start] 99.9	\[ \frac{1}{\frac{\tan x - x}{\sin x - x}} \]
      add-cube-cbrt [=>] 99.9	\[ \frac{1}{\frac{\color{blue}{\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}} - x}{\sin x - x}} \]
      fma-neg [=>] 99.9	\[ \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}, \sqrt[3]{\tan x}, -x\right)}}{\sin x - x}} \]
      pow2 [=>] 99.9	\[ \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\tan x}\right)}^{2}}, \sqrt[3]{\tan x}, -x\right)}{\sin x - x}} \]

   if -0.029999999999999999 < x < 0.0280000000000000006

   1. Initial program 1.1%

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

   2. Simplified 1.1%

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      Proof

      [Start] 1.1	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 1.1	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 1.1	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 1.1	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 1.1	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 1.1	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 1.1	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 1.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 1.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 1.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 1.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 1.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 1.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 1.1	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 1.1	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 1.1	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Applied egg-rr 1.1%

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

      [Start] 1.1	\[ \frac{\sin x - x}{\tan x - x} \]
      div-inv [=>] 1.1	\[ \color{blue}{\left(\sin x - x\right) \cdot \frac{1}{\tan x - x}} \]
      *-commutative [=>] 1.1	\[ \color{blue}{\frac{1}{\tan x - x} \cdot \left(\sin x - x\right)} \]

   4. Applied egg-rr 1.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\tan x - x}{\sin x - x}}} \]
      Proof

      [Start] 1.1	\[ \frac{1}{\tan x - x} \cdot \left(\sin x - x\right) \]
      associate-/r/ [<=] 1.1	\[ \color{blue}{\frac{1}{\frac{\tan x - x}{\sin x - x}}} \]

   5. Taylor expanded in x around 0 100.0%

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

   6. Simplified 100.0%

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

      [Start] 100.0	\[ \left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5 \]
      sub-neg [=>] 100.0	\[ \color{blue}{\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) + \left(-0.5\right)} \]
      unpow2 [=>] 100.0	\[ \left(0.225 \cdot \color{blue}{\left(x \cdot x\right)} + -0.009642857142857142 \cdot {x}^{4}\right) + \left(-0.5\right) \]
      fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.009642857142857142 \cdot {x}^{4}\right)} + \left(-0.5\right) \]
      *-commutative [=>] 100.0	\[ \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{{x}^{4} \cdot -0.009642857142857142}\right) + \left(-0.5\right) \]
      metadata-eval [=>] 100.0	\[ \mathsf{fma}\left(0.225, x \cdot x, {x}^{4} \cdot -0.009642857142857142\right) + \color{blue}{-0.5} \]

   7. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \mathsf{fma}\left(0.225, x \cdot x, {x}^{4} \cdot -0.009642857142857142\right) + -0.5 \]
      fma-udef [=>] 100.0	\[ \color{blue}{\left(0.225 \cdot \left(x \cdot x\right) + {x}^{4} \cdot -0.009642857142857142\right)} + -0.5 \]
      +-commutative [=>] 100.0	\[ \color{blue}{\left({x}^{4} \cdot -0.009642857142857142 + 0.225 \cdot \left(x \cdot x\right)\right)} + -0.5 \]
      associate-*r* [=>] 100.0	\[ \left({x}^{4} \cdot -0.009642857142857142 + \color{blue}{\left(0.225 \cdot x\right) \cdot x}\right) + -0.5 \]

   if 0.0280000000000000006 < x

   1. Initial program 99.9%

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

   2. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      Proof

      [Start] 99.9	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 99.9	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 99.9	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 99.9	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 99.9	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 99.9	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 99.9	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 99.9	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 99.9	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 99.9	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 99.9	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Applied egg-rr 99.7%

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

      [Start] 99.9	\[ \frac{\sin x - x}{\tan x - x} \]
      div-inv [=>] 99.7	\[ \color{blue}{\left(\sin x - x\right) \cdot \frac{1}{\tan x - x}} \]
      *-commutative [=>] 99.7	\[ \color{blue}{\frac{1}{\tan x - x} \cdot \left(\sin x - x\right)} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\tan x - x}{\sin x - x}}} \]
      Proof

      [Start] 99.7	\[ \frac{1}{\tan x - x} \cdot \left(\sin x - x\right) \]
      associate-/r/ [<=] 99.9	\[ \color{blue}{\frac{1}{\frac{\tan x - x}{\sin x - x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left({\left(\sqrt[3]{\tan x}\right)}^{2}, \sqrt[3]{\tan x}, -x\right)}{\sin x - x}}\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;\left({x}^{4} \cdot -0.009642857142857142 + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	45508

\[\begin{array}{l} t_0 := \sin x - x\\ t_1 := \sqrt[3]{\tan x}\\ \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left({t_1}^{2}, t_1, -x\right)}\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;\left({x}^{4} \cdot -0.009642857142857142 + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{t_0}}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -0.026:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;\left({x}^{4} \cdot -0.009642857142857142 + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	13640

\[\begin{array}{l} t_0 := \sin x - x\\ \mathbf{if}\;x \leq -0.026:\\ \;\;\;\;\frac{t_0}{\frac{1}{\frac{1}{\tan x}} - x}\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;\left({x}^{4} \cdot -0.009642857142857142 + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{t_0}}\\ \end{array} \]

Alternative 4
Accuracy	100.0%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -0.026 \lor \neg \left(x \leq 0.028\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot -0.009642857142857142 + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \end{array} \]

Alternative 5
Accuracy	98.8%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;x \leq -3.05:\\ \;\;\;\;\frac{\frac{3}{x}}{x} - \frac{x}{\tan x - x}\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;\left({x}^{4} \cdot -0.009642857142857142 + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x}}\\ \end{array} \]

Alternative 6
Accuracy	98.7%
Cost	7236

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot 0.225\right)\\ \mathbf{if}\;x \leq -2.8:\\ \;\;\;\;\frac{\frac{3}{x}}{x} - \frac{x}{\tan x - x}\\ \mathbf{elif}\;x \leq 2.2:\\ \;\;\;\;\frac{t_0 \cdot t_0 + -0.25}{t_0 + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x}}\\ \end{array} \]

Alternative 7
Accuracy	98.7%
Cost	7112

\[\begin{array}{l} t_0 := x - \tan x\\ t_1 := x \cdot \left(x \cdot 0.225\right)\\ \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;x \leq 2.2:\\ \;\;\;\;\frac{t_1 \cdot t_1 + -0.25}{t_1 + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{x}}\\ \end{array} \]

Alternative 8
Accuracy	98.7%
Cost	6985

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot 0.225\right)\\ \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 2.2\right):\\ \;\;\;\;\frac{x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot t_0 + -0.25}{t_0 + 0.5}\\ \end{array} \]

Alternative 9
Accuracy	98.7%
Cost	1608

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot 0.225\right)\\ \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;\frac{t_0 \cdot t_0 + -0.25}{t_0 + 0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{3}{x}}{x}\\ \end{array} \]

Alternative 10
Accuracy	98.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;-0.5 + \left(-1 + \left(1 + 0.225 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{3}{x}}{x}\\ \end{array} \]

Alternative 11
Accuracy	98.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Accuracy	98.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{3}{x}}{x}\\ \end{array} \]

Alternative 13
Accuracy	98.4%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.56:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	49.3%
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))