sintan (problem 3.4.5)

Average Accuracy: 50.5% → 99.9%

Time: 20.6s

Precision: binary64

Cost: 13640

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.026)
   (/ (- x (sin x)) (- x (tan x)))
   (if (<= x 0.026)
     (+ (* (* x x) (fma x (* x -0.009642857142857142) 0.225)) -0.5)
     (/ 1.0 (/ (- (tan x) x) (- (sin x) x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.026) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else if (x <= 0.026) {
		tmp = ((x * x) * fma(x, (x * -0.009642857142857142), 0.225)) + -0.5;
	} else {
		tmp = 1.0 / ((tan(x) - x) / (sin(x) - x));
	}
	return tmp;
}

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -0.026)
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	elseif (x <= 0.026)
		tmp = Float64(Float64(Float64(x * x) * fma(x, Float64(x * -0.009642857142857142), 0.225)) + -0.5);
	else
		tmp = Float64(1.0 / Float64(Float64(tan(x) - x) / Float64(sin(x) - x)));
	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_] := If[LessEqual[x, -0.026], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.026], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.009642857142857142), $MachinePrecision] + 0.225), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(1.0 / N[(N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0259999999999999988

   1. Initial program 99.9%

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

   if -0.0259999999999999988 < x < 0.0259999999999999988

   1. Initial program 2.8%

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

   2. Simplified 2.8%

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

      [Start] 2.8	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 2.8	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 2.8	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 2.8	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 2.8	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 2.8	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 2.8	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 2.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 2.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 2.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 2.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 2.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 2.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 2.8	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 2.8	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 2.8	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   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. Taylor expanded in x around 0 100.0%

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

   7. Simplified 100.0%

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

      [Start] 100.0	\[ \left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot {x}^{2}\right) + -0.5 \]
      *-commutative [=>] 100.0	\[ \left(\color{blue}{{x}^{4} \cdot -0.009642857142857142} + 0.225 \cdot {x}^{2}\right) + -0.5 \]
      metadata-eval [<=] 100.0	\[ \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot -0.009642857142857142 + 0.225 \cdot {x}^{2}\right) + -0.5 \]
      pow-plus [<=] 100.0	\[ \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot -0.009642857142857142 + 0.225 \cdot {x}^{2}\right) + -0.5 \]
      unpow3 [=>] 100.0	\[ \left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot x\right) \cdot -0.009642857142857142 + 0.225 \cdot {x}^{2}\right) + -0.5 \]
      associate-*r* [<=] 100.0	\[ \left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot -0.009642857142857142 + 0.225 \cdot {x}^{2}\right) + -0.5 \]
      associate-*r* [<=] 100.0	\[ \left(\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142\right)} + 0.225 \cdot {x}^{2}\right) + -0.5 \]
      unpow2 [=>] 100.0	\[ \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142\right) + 0.225 \cdot \color{blue}{\left(x \cdot x\right)}\right) + -0.5 \]
      *-commutative [=>] 100.0	\[ \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142\right) + \color{blue}{\left(x \cdot x\right) \cdot 0.225}\right) + -0.5 \]
      distribute-lft-out [=>] 100.0	\[ \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142 + 0.225\right)} + -0.5 \]
      associate-*l* [=>] 100.0	\[ \left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot -0.009642857142857142\right)} + 0.225\right) + -0.5 \]
      fma-def [=>] 100.0	\[ \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.009642857142857142, 0.225\right)} + -0.5 \]

   if 0.0259999999999999988 < 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%

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

      [Start] 99.9	\[ \frac{\sin x - x}{\tan x - x} \]
      clear-num [=>] 99.9	\[ \color{blue}{\frac{1}{\frac{\tan x - x}{\sin x - x}}} \]
      inv-pow [=>] 99.9	\[ \color{blue}{{\left(\frac{\tan x - x}{\sin x - x}\right)}^{-1}} \]

   4. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ {\left(\frac{\tan x - x}{\sin x - x}\right)}^{-1} \]
      unpow-1 [=>] 99.9	\[ \color{blue}{\frac{1}{\frac{\tan x - x}{\sin x - x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13513

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

Alternative 2
Accuracy	98.9%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \lor \neg \left(x \leq 2.75\right):\\ \;\;\;\;\frac{x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot -0.009642857142857142, 0.225\right) + -0.5\\ \end{array} \]

Alternative 3
Accuracy	98.8%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \lor \neg \left(x \leq 2.3\right):\\ \;\;\;\;\frac{x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;-0.5 + \left(x \cdot x\right) \cdot 0.225\\ \end{array} \]

Alternative 4
Accuracy	98.8%
Cost	712

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

Alternative 5
Accuracy	98.5%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	50.5%
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023160 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))