sintan (problem 3.4.5)

Average Accuracy: 51.7% → 99.9%

Time: 19.7s

Precision: binary64

Cost: 13513

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 0.0052\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0054) (not (<= x 0.0052)))
   (/ (- x (sin x)) (- x (tan x)))
   (+ (* x (* x 0.225)) -0.5)))

C

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

↓

double code(double x) {
	double tmp;
	if ((x <= -0.0054) || !(x <= 0.0052)) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else {
		tmp = (x * (x * 0.225)) + -0.5;
	}
	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) :: tmp
    if ((x <= (-0.0054d0)) .or. (.not. (x <= 0.0052d0))) then
        tmp = (x - sin(x)) / (x - tan(x))
    else
        tmp = (x * (x * 0.225d0)) + (-0.5d0)
    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 tmp;
	if ((x <= -0.0054) || !(x <= 0.0052)) {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	} else {
		tmp = (x * (x * 0.225)) + -0.5;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if (x <= -0.0054) or not (x <= 0.0052):
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	else:
		tmp = (x * (x * 0.225)) + -0.5
	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 <= -0.0054) || !(x <= 0.0052))
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	else
		tmp = Float64(Float64(x * Float64(x * 0.225)) + -0.5);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.0054) || ~((x <= 0.0052)))
		tmp = (x - sin(x)) / (x - tan(x));
	else
		tmp = (x * (x * 0.225)) + -0.5;
	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_] := If[Or[LessEqual[x, -0.0054], N[Not[LessEqual[x, 0.0052]], $MachinePrecision]], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0054000000000000003 or 0.0051999999999999998 < x

   1. Initial program 99.9%

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

   if -0.0054000000000000003 < x < 0.0051999999999999998

   1. Initial program 1.3%

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

   2. Simplified 1.3%

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

      [Start] 1.3	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 1.3	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 1.3	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 1.3	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 1.3	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 1.3	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 1.3	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 1.3	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 1.3	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 1.3	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 1.3	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 1.3	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 1.3	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 1.3	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 1.3	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 1.3	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]

   4. Simplified 99.9%

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

      [Start] 99.9	\[ 0.225 \cdot {x}^{2} - 0.5 \]
      unpow2 [=>] 99.9	\[ 0.225 \cdot \color{blue}{\left(x \cdot x\right)} - 0.5 \]
      fma-neg [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
      metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{-0.5}\right) \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(0.225 \cdot x\right) \cdot x + -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 0.0052\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -2.45 \lor \neg \left(x \leq 2.45\right):\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \end{array} \]

Alternative 2
Accuracy	98.7%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -1.46:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x}}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Accuracy	98.7%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -1.46:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
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 5
Accuracy	98.4%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -17.5:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	49.4%
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023129 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))