sintan (problem 3.4.5)

Average Error: 31.3 → 0.4

Time: 17.0s

Precision: binary64

Cost: 20745

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + \left(\left(0.00024107142857142857 \cdot {x}^{6} + -1\right) + \left(1 + -0.009642857142857142 \cdot {x}^{4}\right)\right)\right) + -0.5\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.9) (not (<= x 5.0)))
   (+ 1.0 (/ (- (tan x) (sin x)) x))
   (+
    (+
     (* 0.225 (pow x 2.0))
     (+
      (+ (* 0.00024107142857142857 (pow x 6.0)) -1.0)
      (+ 1.0 (* -0.009642857142857142 (pow x 4.0)))))
    -0.5)))

C

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

↓

double code(double x) {
	double tmp;
	if ((x <= -2.9) || !(x <= 5.0)) {
		tmp = 1.0 + ((tan(x) - sin(x)) / x);
	} else {
		tmp = ((0.225 * pow(x, 2.0)) + (((0.00024107142857142857 * pow(x, 6.0)) + -1.0) + (1.0 + (-0.009642857142857142 * pow(x, 4.0))))) + -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 <= (-2.9d0)) .or. (.not. (x <= 5.0d0))) then
        tmp = 1.0d0 + ((tan(x) - sin(x)) / x)
    else
        tmp = ((0.225d0 * (x ** 2.0d0)) + (((0.00024107142857142857d0 * (x ** 6.0d0)) + (-1.0d0)) + (1.0d0 + ((-0.009642857142857142d0) * (x ** 4.0d0))))) + (-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 <= -2.9) || !(x <= 5.0)) {
		tmp = 1.0 + ((Math.tan(x) - Math.sin(x)) / x);
	} else {
		tmp = ((0.225 * Math.pow(x, 2.0)) + (((0.00024107142857142857 * Math.pow(x, 6.0)) + -1.0) + (1.0 + (-0.009642857142857142 * Math.pow(x, 4.0))))) + -0.5;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if (x <= -2.9) or not (x <= 5.0):
		tmp = 1.0 + ((math.tan(x) - math.sin(x)) / x)
	else:
		tmp = ((0.225 * math.pow(x, 2.0)) + (((0.00024107142857142857 * math.pow(x, 6.0)) + -1.0) + (1.0 + (-0.009642857142857142 * math.pow(x, 4.0))))) + -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 <= -2.9) || !(x <= 5.0))
		tmp = Float64(1.0 + Float64(Float64(tan(x) - sin(x)) / x));
	else
		tmp = Float64(Float64(Float64(0.225 * (x ^ 2.0)) + Float64(Float64(Float64(0.00024107142857142857 * (x ^ 6.0)) + -1.0) + Float64(1.0 + Float64(-0.009642857142857142 * (x ^ 4.0))))) + -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 <= -2.9) || ~((x <= 5.0)))
		tmp = 1.0 + ((tan(x) - sin(x)) / x);
	else
		tmp = ((0.225 * (x ^ 2.0)) + (((0.00024107142857142857 * (x ^ 6.0)) + -1.0) + (1.0 + (-0.009642857142857142 * (x ^ 4.0))))) + -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, -2.9], N[Not[LessEqual[x, 5.0]], $MachinePrecision]], N[(1.0 + N[(N[(N[Tan[x], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.225 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[(1.0 + N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.89999999999999991 or 5 < x

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 0.0	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 0.0	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 0.0	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 0.0	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 0.0	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 0.0	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 0.0	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 0.0	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around -inf 0.6

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

   4. Simplified 0.6

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

      [Start] 0.6	\[ 1 + -1 \cdot \frac{\sin x - \frac{\sin x}{\cos x}}{x} \]
      associate-*r/ [=>] 0.6	\[ 1 + \color{blue}{\frac{-1 \cdot \left(\sin x - \frac{\sin x}{\cos x}\right)}{x}} \]
      distribute-lft-out-- [<=] 0.6	\[ 1 + \frac{\color{blue}{-1 \cdot \sin x - -1 \cdot \frac{\sin x}{\cos x}}}{x} \]
      cancel-sign-sub-inv [=>] 0.6	\[ 1 + \frac{\color{blue}{-1 \cdot \sin x + \left(--1\right) \cdot \frac{\sin x}{\cos x}}}{x} \]
      +-commutative [=>] 0.6	\[ 1 + \frac{\color{blue}{\left(--1\right) \cdot \frac{\sin x}{\cos x} + -1 \cdot \sin x}}{x} \]
      metadata-eval [=>] 0.6	\[ 1 + \frac{\color{blue}{1} \cdot \frac{\sin x}{\cos x} + -1 \cdot \sin x}{x} \]
      associate-*r/ [=>] 0.6	\[ 1 + \frac{\color{blue}{\frac{1 \cdot \sin x}{\cos x}} + -1 \cdot \sin x}{x} \]
      *-lft-identity [=>] 0.6	\[ 1 + \frac{\frac{\color{blue}{\sin x}}{\cos x} + -1 \cdot \sin x}{x} \]
      mul-1-neg [=>] 0.6	\[ 1 + \frac{\frac{\sin x}{\cos x} + \color{blue}{\left(-\sin x\right)}}{x} \]

   5. Applied egg-rr 0.6

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

   if -2.89999999999999991 < x < 5

   1. Initial program 62.8

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

   2. Simplified 62.8

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

      [Start] 62.8	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 62.8	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 62.8	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 62.8	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 62.8	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 62.8	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 62.8	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 62.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 62.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 62.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 62.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 62.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 62.8	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 62.8	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 62.8	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 62.8	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 0.2

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

   4. Applied egg-rr 0.2

      \[\leadsto \left(0.225 \cdot {x}^{2} + \color{blue}{\left(\left(0.00024107142857142857 \cdot {x}^{6} + e^{\mathsf{log1p}\left(-0.009642857142857142 \cdot {x}^{4}\right)}\right) - 1\right)}\right) - 0.5 \]

   5. Applied egg-rr 0.2

      \[\leadsto \left(0.225 \cdot {x}^{2} + \color{blue}{\left(\left(0.00024107142857142857 \cdot {x}^{6} + -1\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 1\right)\right)}\right) - 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + \left(\left(0.00024107142857142857 \cdot {x}^{6} + -1\right) + \left(1 + -0.009642857142857142 \cdot {x}^{4}\right)\right)\right) + -0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + {x}^{4} \cdot \left(-0.009642857142857142 + 0.00024107142857142857 \cdot \left(x \cdot x\right)\right)\right) + -0.5\\ \end{array} \]

Alternative 2
Error	0.7
Cost	7816

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.2:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + {x}^{4} \cdot \left(-0.009642857142857142 + 0.00024107142857142857 \cdot \left(x \cdot x\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \end{array} \]

Alternative 3
Error	0.7
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.225 + -0.009642857142857142 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \end{array} \]

Alternative 4
Error	0.7
Cost	1096

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

Alternative 5
Error	0.8
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 6
Error	1.0
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.56:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	32.0
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))