sintan (problem 3.4.5)

Average Error: 48.93% → 0.07%

Time: 18.6s

Precision: binary64

Cost: 26180

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0295:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x - x}{\tan x - x}\right)\right)\\ \mathbf{elif}\;x \leq 0.0295:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142 + 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.0295)
   (expm1 (log1p (/ (- (sin x) x) (- (tan x) x))))
   (if (<= x 0.0295)
     (+ (* (* x x) (+ (* (* x x) -0.009642857142857142) 0.225)) -0.5)
     (/ (- x (sin x)) (- x (tan x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.0295) {
		tmp = expm1(log1p(((sin(x) - x) / (tan(x) - x))));
	} else if (x <= 0.0295) {
		tmp = ((x * x) * (((x * x) * -0.009642857142857142) + 0.225)) + -0.5;
	} else {
		tmp = (x - sin(x)) / (x - tan(x));
	}
	return tmp;
}

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -0.0295) {
		tmp = Math.expm1(Math.log1p(((Math.sin(x) - x) / (Math.tan(x) - x))));
	} else if (x <= 0.0295) {
		tmp = ((x * x) * (((x * x) * -0.009642857142857142) + 0.225)) + -0.5;
	} else {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.0295:
		tmp = math.expm1(math.log1p(((math.sin(x) - x) / (math.tan(x) - x))))
	elif x <= 0.0295:
		tmp = ((x * x) * (((x * x) * -0.009642857142857142) + 0.225)) + -0.5
	else:
		tmp = (x - math.sin(x)) / (x - math.tan(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 <= -0.0295)
		tmp = expm1(log1p(Float64(Float64(sin(x) - x) / Float64(tan(x) - x))));
	elseif (x <= 0.0295)
		tmp = Float64(Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * -0.009642857142857142) + 0.225)) + -0.5);
	else
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(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, -0.0295], N[(Exp[N[Log[1 + N[(N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[x, 0.0295], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.009642857142857142), $MachinePrecision] + 0.225), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.029499999999999998

   1. Initial program 0.08

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

   2. Simplified 0.08

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

      [Start] 0.08	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 0.08	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 0.08	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 0.08	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 0.08	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 0.08	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 0.08	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 0.08	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 0.08	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 0.08	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 0.08	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 0.08	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 0.08	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 0.08	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 0.08	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 0.08	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Applied egg-rr 0.14

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x - x}{\tan x - x}\right)\right)} \]

   if -0.029499999999999998 < x < 0.029499999999999998

   1. Initial program 98.92

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

   2. Simplified 98.92

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

      [Start] 98.92	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 98.92	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 98.92	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 98.92	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 98.92	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 98.92	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 98.92	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 98.92	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 98.92	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 98.92	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 98.92	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 98.92	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 98.92	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 98.92	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 98.92	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 98.92	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 0.02

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

   4. Simplified 0.02

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

      [Start] 0.02	\[ \left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5 \]
      sub-neg [=>] 0.02	\[ \color{blue}{\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) + \left(-0.5\right)} \]
      unpow2 [=>] 0.02	\[ \left(0.225 \cdot \color{blue}{\left(x \cdot x\right)} + -0.009642857142857142 \cdot {x}^{4}\right) + \left(-0.5\right) \]
      fma-def [=>] 0.02	\[ \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.009642857142857142 \cdot {x}^{4}\right)} + \left(-0.5\right) \]
      metadata-eval [=>] 0.02	\[ \mathsf{fma}\left(0.225, x \cdot x, -0.009642857142857142 \cdot {x}^{4}\right) + \color{blue}{-0.5} \]

   5. Applied egg-rr 0.02

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

   6. Applied egg-rr 0.02

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

   if 0.029499999999999998 < x

   1. Initial program 0.08

      \[\frac{x - \sin x}{x - \tan x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.07

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0295:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x - x}{\tan x - x}\right)\right)\\ \mathbf{elif}\;x \leq 0.0295:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142 + 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.52%
Cost	13513

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

Alternative 2
Error	0.05%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0295 \lor \neg \left(x \leq 0.0295\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142 + 0.225\right) + -0.5\\ \end{array} \]

Alternative 3
Error	1.01%
Cost	6984

\[\begin{array}{l} t_0 := \frac{\sin x}{x}\\ \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;1 + t_0\\ \mathbf{elif}\;x \leq 2.75:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142 + 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \]

Alternative 4
Error	1.01%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;\frac{x + \sin x}{x}\\ \mathbf{elif}\;x \leq 2.75:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142 + 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\sin x}{x}\\ \end{array} \]

Alternative 5
Error	1%
Cost	6852

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

Alternative 6
Error	0.98%
Cost	1096

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

Alternative 7
Error	1.08%
Cost	712

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

Alternative 8
Error	1.36%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.56:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.58:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	50%
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))