sintan (problem 3.4.5)

Average Accuracy: 50.2% → 99.6%

Time: 21.7s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.09:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.09)
   (/ (- x (sin x)) (- x (tan x)))
   (if (<= x 5.0)
     (-
      (+
       (* 0.225 (pow x 2.0))
       (+
        (* -0.009642857142857142 (pow x 4.0))
        (* 0.00024107142857142857 (pow x 6.0))))
      0.5)
     (+ 1.0 (/ (- (tan x) (sin x)) x)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.09) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else if (x <= 5.0) {
		tmp = ((0.225 * pow(x, 2.0)) + ((-0.009642857142857142 * pow(x, 4.0)) + (0.00024107142857142857 * pow(x, 6.0)))) - 0.5;
	} else {
		tmp = 1.0 + ((tan(x) - sin(x)) / x);
	}
	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 <= (-0.09d0)) then
        tmp = (x - sin(x)) / (x - tan(x))
    else if (x <= 5.0d0) then
        tmp = ((0.225d0 * (x ** 2.0d0)) + (((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.00024107142857142857d0 * (x ** 6.0d0)))) - 0.5d0
    else
        tmp = 1.0d0 + ((tan(x) - sin(x)) / x)
    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 <= -0.09) {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	} else if (x <= 5.0) {
		tmp = ((0.225 * Math.pow(x, 2.0)) + ((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.00024107142857142857 * Math.pow(x, 6.0)))) - 0.5;
	} else {
		tmp = 1.0 + ((Math.tan(x) - Math.sin(x)) / x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.09:
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	elif x <= 5.0:
		tmp = ((0.225 * math.pow(x, 2.0)) + ((-0.009642857142857142 * math.pow(x, 4.0)) + (0.00024107142857142857 * math.pow(x, 6.0)))) - 0.5
	else:
		tmp = 1.0 + ((math.tan(x) - math.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 <= -0.09)
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	elseif (x <= 5.0)
		tmp = Float64(Float64(Float64(0.225 * (x ^ 2.0)) + Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.00024107142857142857 * (x ^ 6.0)))) - 0.5);
	else
		tmp = Float64(1.0 + Float64(Float64(tan(x) - sin(x)) / x));
	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 <= -0.09)
		tmp = (x - sin(x)) / (x - tan(x));
	elseif (x <= 5.0)
		tmp = ((0.225 * (x ^ 2.0)) + ((-0.009642857142857142 * (x ^ 4.0)) + (0.00024107142857142857 * (x ^ 6.0)))) - 0.5;
	else
		tmp = 1.0 + ((tan(x) - sin(x)) / x);
	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[LessEqual[x, -0.09], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.0], N[(N[(N[(0.225 * N[Power[x, 2.0], $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[(1.0 + N[(N[(N[Tan[x], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.089999999999999997

   1. Initial program 99.9%

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

   if -0.089999999999999997 < x < 5

   1. Initial program 1.6%

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

   2. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      Step-by-step derivation

      [Start] 1.6	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 1.6	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 1.6	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 1.6	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 1.6	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 1.6	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 1.6	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 1.6	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 1.6	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 1.6	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 1.6	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 1.6	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 1.6	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 1.6	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 1.6	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 1.6	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 100.0%

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

   if 5 < x

   1. Initial program 100.0%

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

   2. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      Step-by-step derivation

      [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. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{\sin x}{x}\right) - -1 \cdot \frac{\sin x}{\cos x \cdot x}} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\sin x - \frac{\sin x}{\cos x}}{x}} \]
      Step-by-step derivation

      [Start] 100.0	\[ \left(1 + -1 \cdot \frac{\sin x}{x}\right) - -1 \cdot \frac{\sin x}{\cos x \cdot x} \]
      associate--l+ [=>] 100.0	\[ \color{blue}{1 + \left(-1 \cdot \frac{\sin x}{x} - -1 \cdot \frac{\sin x}{\cos x \cdot x}\right)} \]
      associate-*r/ [=>] 100.0	\[ 1 + \left(\color{blue}{\frac{-1 \cdot \sin x}{x}} - -1 \cdot \frac{\sin x}{\cos x \cdot x}\right) \]
      associate-/r* [=>] 100.0	\[ 1 + \left(\frac{-1 \cdot \sin x}{x} - -1 \cdot \color{blue}{\frac{\frac{\sin x}{\cos x}}{x}}\right) \]
      associate-*r/ [=>] 100.0	\[ 1 + \left(\frac{-1 \cdot \sin x}{x} - \color{blue}{\frac{-1 \cdot \frac{\sin x}{\cos x}}{x}}\right) \]
      div-sub [<=] 100.0	\[ 1 + \color{blue}{\frac{-1 \cdot \sin x - -1 \cdot \frac{\sin x}{\cos x}}{x}} \]
      distribute-lft-out-- [=>] 100.0	\[ 1 + \frac{\color{blue}{-1 \cdot \left(\sin x - \frac{\sin x}{\cos x}\right)}}{x} \]
      associate-*r/ [<=] 100.0	\[ 1 + \color{blue}{-1 \cdot \frac{\sin x - \frac{\sin x}{\cos x}}{x}} \]
      mul-1-neg [=>] 100.0	\[ 1 + \color{blue}{\left(-\frac{\sin x - \frac{\sin x}{\cos x}}{x}\right)} \]
      unsub-neg [=>] 100.0	\[ \color{blue}{1 - \frac{\sin x - \frac{\sin x}{\cos x}}{x}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{\color{blue}{\sin x + \left(-\tan x\right)}}{x} \]
      Step-by-step derivation

      [Start] 100.0	\[ 1 - \frac{\sin x - \frac{\sin x}{\cos x}}{x} \]
      sub-neg [=>] 100.0	\[ 1 - \frac{\color{blue}{\sin x + \left(-\frac{\sin x}{\cos x}\right)}}{x} \]
      quot-tan [=>] 100.0	\[ 1 - \frac{\sin x + \left(-\color{blue}{\tan x}\right)}{x} \]

   6. Simplified 100.0%

      \[\leadsto 1 - \frac{\color{blue}{\sin x - \tan x}}{x} \]
      Step-by-step derivation

      [Start] 100.0	\[ 1 - \frac{\sin x + \left(-\tan x\right)}{x} \]
      sub-neg [<=] 100.0	\[ 1 - \frac{\color{blue}{\sin x - \tan x}}{x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.09:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \lor \neg \left(x \leq 4.8\right):\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 4.8:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \end{array} \]

Alternative 3
Accuracy	98.8%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.9:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	98.7%
Cost	712

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

Alternative 5
Accuracy	98.4%
Cost	328

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

Alternative 6
Accuracy	50.9%
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023157 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))