sintan (problem 3.4.5)

Average Accuracy: 51.2% → 100.0%

Time: 24.5s

Precision: binary64

Cost: 26632

Math

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

↓

\[\begin{array}{l} t_0 := \tan x - x\\ \mathbf{if}\;x \leq -0.095:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\left(x \cdot \left(x \cdot 0.225\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \frac{1}{t_0}, \frac{-x}{t_0}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (tan x) x)))
   (if (<= x -0.095)
     (/ (- x (sin x)) (- x (tan x)))
     (if (<= x 0.095)
       (-
        (+
         (* x (* x 0.225))
         (+
          (* -0.009642857142857142 (pow x 4.0))
          (* 0.00024107142857142857 (pow x 6.0))))
        0.5)
       (fma (sin x) (/ 1.0 t_0) (/ (- x) t_0))))))

C

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

↓

double code(double x) {
	double t_0 = tan(x) - x;
	double tmp;
	if (x <= -0.095) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else if (x <= 0.095) {
		tmp = ((x * (x * 0.225)) + ((-0.009642857142857142 * pow(x, 4.0)) + (0.00024107142857142857 * pow(x, 6.0)))) - 0.5;
	} else {
		tmp = fma(sin(x), (1.0 / 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(tan(x) - x)
	tmp = 0.0
	if (x <= -0.095)
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	elseif (x <= 0.095)
		tmp = Float64(Float64(Float64(x * Float64(x * 0.225)) + Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.00024107142857142857 * (x ^ 6.0)))) - 0.5);
	else
		tmp = fma(sin(x), Float64(1.0 / t_0), Float64(Float64(-x) / 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[Tan[x], $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -0.095], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.095], N[(N[(N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision] + N[((-x) / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.095000000000000001

   1. Initial program 99.9%

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

   if -0.095000000000000001 < x < 0.095000000000000001

   1. Initial program 1.4%

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

   2. Simplified 1.4%

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

      [Start] 1.4	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 1.4	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 1.4	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 1.4	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 1.4	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 1.4	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 1.4	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 1.4	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 1.4	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 1.4	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(\color{blue}{\left(0 + x \cdot \left(x \cdot 0.225\right)\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      Proof

      [Start] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      add-log-exp [=>] 100.0	\[ \left(\color{blue}{\log \left(e^{0.225 \cdot {x}^{2}}\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      *-un-lft-identity [=>] 100.0	\[ \left(\log \color{blue}{\left(1 \cdot e^{0.225 \cdot {x}^{2}}\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      log-prod [=>] 100.0	\[ \left(\color{blue}{\left(\log 1 + \log \left(e^{0.225 \cdot {x}^{2}}\right)\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      metadata-eval [=>] 100.0	\[ \left(\left(\color{blue}{0} + \log \left(e^{0.225 \cdot {x}^{2}}\right)\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      add-log-exp [<=] 100.0	\[ \left(\left(0 + \color{blue}{0.225 \cdot {x}^{2}}\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      *-commutative [=>] 100.0	\[ \left(\left(0 + \color{blue}{{x}^{2} \cdot 0.225}\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      unpow2 [=>] 100.0	\[ \left(\left(0 + \color{blue}{\left(x \cdot x\right)} \cdot 0.225\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      associate-*l* [=>] 100.0	\[ \left(\left(0 + \color{blue}{x \cdot \left(x \cdot 0.225\right)}\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

   5. Simplified 100.0%

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

      [Start] 100.0	\[ \left(\left(0 + x \cdot \left(x \cdot 0.225\right)\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      +-lft-identity [=>] 100.0	\[ \left(\color{blue}{x \cdot \left(x \cdot 0.225\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

   if 0.095000000000000001 < x

   1. Initial program 100.0%

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

   2. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 100.0	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 100.0	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 100.0	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 100.0	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 100.0	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 100.0	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 100.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 100.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 100.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 100.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 100.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 100.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 100.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 100.0	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 100.0	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \frac{\sin x - x}{\tan x - x} \]
      div-sub [=>] 100.0	\[ \color{blue}{\frac{\sin x}{\tan x - x} - \frac{x}{\tan x - x}} \]
      div-inv [=>] 100.0	\[ \color{blue}{\sin x \cdot \frac{1}{\tan x - x}} - \frac{x}{\tan x - x} \]
      fma-neg [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(\sin x, \frac{1}{\tan x - x}, -\frac{x}{\tan x - x}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.095:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\left(x \cdot \left(x \cdot 0.225\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \frac{1}{\tan x - x}, \frac{-x}{\tan x - x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;x \leq -0.095:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\left(x \cdot \left(x \cdot 0.225\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -0.028:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.225 + x \cdot \left(x \cdot -0.009642857142857142\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \lor \neg \left(x \leq 2.6\right):\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.225 + x \cdot \left(x \cdot -0.009642857142857142\right)\right) + -0.5\\ \end{array} \]

Alternative 4
Accuracy	100.0%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -0.028 \lor \neg \left(x \leq 0.023\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.225 + x \cdot \left(x \cdot -0.009642857142857142\right)\right) + -0.5\\ \end{array} \]

Alternative 5
Accuracy	98.9%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.225 + x \cdot \left(x \cdot -0.009642857142857142\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	98.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.5:\\ \;\;\;\;-0.5 + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	98.5%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	49.9%
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023142 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))