sintan (problem 3.4.5)

Average Accuracy: 49.7% → 99.9%

Time: 22.0s

Precision: binary64

Cost: 20296

Math

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

↓

\[\begin{array}{l} t_0 := \sin x - x\\ \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;\frac{x + \left(t_0 - x\right)}{\tan x - x}\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x}{t_0} - \frac{x}{t_0}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sin x) x)))
   (if (<= x -0.033)
     (/ (+ x (- t_0 x)) (- (tan x) x))
     (if (<= x 0.032)
       (+
        (+ (* 0.225 (* x x)) (* -0.009642857142857142 (* (* x x) (* x x))))
        -0.5)
       (/ 1.0 (- (/ (tan x) t_0) (/ x t_0)))))))

C

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

↓

double code(double x) {
	double t_0 = sin(x) - x;
	double tmp;
	if (x <= -0.033) {
		tmp = (x + (t_0 - x)) / (tan(x) - x);
	} else if (x <= 0.032) {
		tmp = ((0.225 * (x * x)) + (-0.009642857142857142 * ((x * x) * (x * x)))) + -0.5;
	} else {
		tmp = 1.0 / ((tan(x) / t_0) - (x / t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x - sin(x)) / (x - tan(x))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x) - x
    if (x <= (-0.033d0)) then
        tmp = (x + (t_0 - x)) / (tan(x) - x)
    else if (x <= 0.032d0) then
        tmp = ((0.225d0 * (x * x)) + ((-0.009642857142857142d0) * ((x * x) * (x * x)))) + (-0.5d0)
    else
        tmp = 1.0d0 / ((tan(x) / t_0) - (x / t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (x - Math.sin(x)) / (x - Math.tan(x));
}

↓

public static double code(double x) {
	double t_0 = Math.sin(x) - x;
	double tmp;
	if (x <= -0.033) {
		tmp = (x + (t_0 - x)) / (Math.tan(x) - x);
	} else if (x <= 0.032) {
		tmp = ((0.225 * (x * x)) + (-0.009642857142857142 * ((x * x) * (x * x)))) + -0.5;
	} else {
		tmp = 1.0 / ((Math.tan(x) / t_0) - (x / t_0));
	}
	return tmp;
}

Python

def code(x):
	return (x - math.sin(x)) / (x - math.tan(x))

↓

def code(x):
	t_0 = math.sin(x) - x
	tmp = 0
	if x <= -0.033:
		tmp = (x + (t_0 - x)) / (math.tan(x) - x)
	elif x <= 0.032:
		tmp = ((0.225 * (x * x)) + (-0.009642857142857142 * ((x * x) * (x * x)))) + -0.5
	else:
		tmp = 1.0 / ((math.tan(x) / t_0) - (x / 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(sin(x) - x)
	tmp = 0.0
	if (x <= -0.033)
		tmp = Float64(Float64(x + Float64(t_0 - x)) / Float64(tan(x) - x));
	elseif (x <= 0.032)
		tmp = Float64(Float64(Float64(0.225 * Float64(x * x)) + Float64(-0.009642857142857142 * Float64(Float64(x * x) * Float64(x * x)))) + -0.5);
	else
		tmp = Float64(1.0 / Float64(Float64(tan(x) / t_0) - Float64(x / t_0)));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (x - sin(x)) / (x - tan(x));
end

↓

function tmp_2 = code(x)
	t_0 = sin(x) - x;
	tmp = 0.0;
	if (x <= -0.033)
		tmp = (x + (t_0 - x)) / (tan(x) - x);
	elseif (x <= 0.032)
		tmp = ((0.225 * (x * x)) + (-0.009642857142857142 * ((x * x) * (x * x)))) + -0.5;
	else
		tmp = 1.0 / ((tan(x) / t_0) - (x / t_0));
	end
	tmp_2 = 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[Sin[x], $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -0.033], N[(N[(x + N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.032], N[(N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.009642857142857142 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(1.0 / N[(N[(N[Tan[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.033000000000000002

   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 99.9%

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

      [Start] 99.9	\[ \frac{\sin x - x}{\tan x - x} \]
      add-cube-cbrt [=>] 99.9	\[ \frac{\color{blue}{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}} - x}{\tan x - x} \]
      *-un-lft-identity [=>] 99.9	\[ \frac{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x} - \color{blue}{1 \cdot x}}{\tan x - x} \]
      prod-diff [=>] 99.9	\[ \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x}, -x \cdot 1\right) + \mathsf{fma}\left(-x, 1, x \cdot 1\right)}}{\tan x - x} \]
      *-commutative [<=] 99.9	\[ \frac{\mathsf{fma}\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x}, -\color{blue}{1 \cdot x}\right) + \mathsf{fma}\left(-x, 1, x \cdot 1\right)}{\tan x - x} \]
      *-un-lft-identity [<=] 99.9	\[ \frac{\mathsf{fma}\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x}, -\color{blue}{x}\right) + \mathsf{fma}\left(-x, 1, x \cdot 1\right)}{\tan x - x} \]
      fma-neg [<=] 99.9	\[ \frac{\color{blue}{\left(\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x} - x\right)} + \mathsf{fma}\left(-x, 1, x \cdot 1\right)}{\tan x - x} \]
      add-cube-cbrt [<=] 99.9	\[ \frac{\left(\color{blue}{\sin x} - x\right) + \mathsf{fma}\left(-x, 1, x \cdot 1\right)}{\tan x - x} \]
      *-commutative [<=] 99.9	\[ \frac{\left(\sin x - x\right) + \mathsf{fma}\left(-x, 1, \color{blue}{1 \cdot x}\right)}{\tan x - x} \]
      *-un-lft-identity [<=] 99.9	\[ \frac{\left(\sin x - x\right) + \mathsf{fma}\left(-x, 1, \color{blue}{x}\right)}{\tan x - x} \]

   4. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{\left(\sin x - x\right) + \mathsf{fma}\left(-x, 1, x\right)}{\tan x - x} \]
      fma-udef [=>] 99.9	\[ \frac{\left(\sin x - x\right) + \color{blue}{\left(\left(-x\right) \cdot 1 + x\right)}}{\tan x - x} \]
      *-rgt-identity [=>] 99.9	\[ \frac{\left(\sin x - x\right) + \left(\color{blue}{\left(-x\right)} + x\right)}{\tan x - x} \]
      associate-+r+ [=>] 99.9	\[ \frac{\color{blue}{\left(\left(\sin x - x\right) + \left(-x\right)\right) + x}}{\tan x - x} \]
      +-commutative [=>] 99.9	\[ \frac{\color{blue}{x + \left(\left(\sin x - x\right) + \left(-x\right)\right)}}{\tan x - x} \]
      unsub-neg [=>] 99.9	\[ \frac{x + \color{blue}{\left(\left(\sin x - x\right) - x\right)}}{\tan x - x} \]

   if -0.033000000000000002 < x < 0.032000000000000001

   1. Initial program 1.2%

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

   2. Simplified 1.2%

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

      [Start] 1.2	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 1.2	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 1.2	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 1.2	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 1.2	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 1.2	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 1.2	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 1.2	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 1.2	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 1.2	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 1.2	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 1.2	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 1.2	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 1.2	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 1.2	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 1.2	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 100.0%

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

   4. Simplified 100.0%

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

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

   5. Applied egg-rr 100.0%

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

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

   6. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right) + -0.5 \]
      sqr-pow [=>] 100.0	\[ \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right)}\right) + -0.5 \]
      metadata-eval [=>] 100.0	\[ \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left({x}^{\color{blue}{2}} \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right) + -0.5 \]
      pow2 [<=] 100.0	\[ \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right) + -0.5 \]
      metadata-eval [=>] 100.0	\[ \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\left(x \cdot x\right) \cdot {x}^{\color{blue}{2}}\right)\right) + -0.5 \]
      pow2 [<=] 100.0	\[ \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) + -0.5 \]

   if 0.032000000000000001 < 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 99.9%

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

      [Start] 99.9	\[ \frac{\sin x - x}{\tan x - x} \]
      div-sub [=>] 99.9	\[ \color{blue}{\frac{\sin x}{\tan x - x} - \frac{x}{\tan x - x}} \]

   4. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ \frac{\sin x}{\tan x - x} - \frac{x}{\tan x - x} \]
      sub-div [=>] 99.9	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      clear-num [=>] 99.9	\[ \color{blue}{\frac{1}{\frac{\tan x - x}{\sin x - x}}} \]

   5. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ \frac{1}{\frac{\tan x - x}{\sin x - x}} \]
      div-sub [=>] 99.9	\[ \frac{1}{\color{blue}{\frac{\tan x}{\sin x - x} - \frac{x}{\sin x - x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;\frac{x + \left(\left(\sin x - x\right) - x\right)}{\tan x - x}\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x}{\sin x - x} - \frac{x}{\sin x - x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13768

\[\begin{array}{l} t_0 := \sin x - x\\ t_1 := \tan x - x\\ \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;\frac{x + \left(t_0 - x\right)}{t_1}\\ \mathbf{elif}\;x \leq 0.044:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(t_1 + 1\right) + -1}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	13640

\[\begin{array}{l} t_0 := \sin x - x\\ t_1 := \tan x - x\\ \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;\frac{x + \left(t_0 - x\right)}{t_1}\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{t_0}}\\ \end{array} \]

Alternative 4
Accuracy	100.0%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -0.033 \lor \neg \left(x \leq 0.026\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + -0.5\\ \end{array} \]

Alternative 5
Accuracy	98.9%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8:\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \mathbf{elif}\;x \leq 2.9:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	98.9%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \leq -2.95:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.9:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	98.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.55:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	98.5%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.56:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	51.4%
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))