sintan (problem 3.4.5)

Average Error: 31.0 → 0.2

Time: 17.6s

Precision: binary64

Cost: 13512

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1 - \frac{\sin x - \tan x}{x}\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142 + 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2.6)
   (- 1.0 (/ (- (sin x) (tan x)) x))
   (if (<= x 0.032)
     (+ (* (* 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 <= -2.6) {
		tmp = 1.0 - ((sin(x) - tan(x)) / x);
	} else if (x <= 0.032) {
		tmp = ((x * x) * (((x * x) * -0.009642857142857142) + 0.225)) + -0.5;
	} else {
		tmp = (x - sin(x)) / (x - tan(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 <= (-2.6d0)) then
        tmp = 1.0d0 - ((sin(x) - tan(x)) / x)
    else if (x <= 0.032d0) then
        tmp = ((x * x) * (((x * x) * (-0.009642857142857142d0)) + 0.225d0)) + (-0.5d0)
    else
        tmp = (x - sin(x)) / (x - tan(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 <= -2.6) {
		tmp = 1.0 - ((Math.sin(x) - Math.tan(x)) / x);
	} else if (x <= 0.032) {
		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 <= -2.6:
		tmp = 1.0 - ((math.sin(x) - math.tan(x)) / x)
	elif x <= 0.032:
		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 <= -2.6)
		tmp = Float64(1.0 - Float64(Float64(sin(x) - tan(x)) / x));
	elseif (x <= 0.032)
		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

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.6)
		tmp = 1.0 - ((sin(x) - tan(x)) / x);
	elseif (x <= 0.032)
		tmp = ((x * x) * (((x * x) * -0.009642857142857142) + 0.225)) + -0.5;
	else
		tmp = (x - sin(x)) / (x - tan(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, -2.6], N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.032], 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 < -2.60000000000000009

   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. Applied egg-rr 0.0

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

   4. Applied egg-rr 0.0

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

   5. Taylor expanded in x around inf 0.6

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

   6. Simplified 0.6

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

      [Start] 0.6	\[ \left(1 + -1 \cdot \frac{\sin x}{x}\right) - -1 \cdot \frac{\sin x}{\cos x \cdot x} \]
      associate--l+ [=>] 0.6	\[ \color{blue}{1 + \left(-1 \cdot \frac{\sin x}{x} - -1 \cdot \frac{\sin x}{\cos x \cdot x}\right)} \]
      distribute-lft-out-- [=>] 0.6	\[ 1 + \color{blue}{-1 \cdot \left(\frac{\sin x}{x} - \frac{\sin x}{\cos x \cdot x}\right)} \]
      associate-/r* [=>] 0.6	\[ 1 + -1 \cdot \left(\frac{\sin x}{x} - \color{blue}{\frac{\frac{\sin x}{\cos x}}{x}}\right) \]
      div-sub [<=] 0.6	\[ 1 + -1 \cdot \color{blue}{\frac{\sin x - \frac{\sin x}{\cos x}}{x}} \]
      mul-1-neg [=>] 0.6	\[ 1 + \color{blue}{\left(-\frac{\sin x - \frac{\sin x}{\cos x}}{x}\right)} \]
      unsub-neg [=>] 0.6	\[ \color{blue}{1 - \frac{\sin x - \frac{\sin x}{\cos x}}{x}} \]

   7. Applied egg-rr 0.6

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

   8. Simplified 0.6

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

      [Start] 0.6	\[ 1 - \frac{\sin x + \left(-\tan x\right)}{x} \]
      sub-neg [<=] 0.6	\[ 1 - \frac{\color{blue}{\sin x - \tan x}}{x} \]

   if -2.60000000000000009 < x < 0.032000000000000001

   1. Initial program 63.0

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

   2. Simplified 63.0

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

      [Start] 63.0	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 63.0	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 63.0	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 63.0	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 63.0	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 63.0	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 63.0	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 63.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 63.0	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 63.0	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 0.2

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

   4. Simplified 0.2

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

      [Start] 0.2	\[ \left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5 \]
      sub-neg [=>] 0.2	\[ \color{blue}{\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) + \left(-0.5\right)} \]
      unpow2 [=>] 0.2	\[ \left(0.225 \cdot \color{blue}{\left(x \cdot x\right)} + -0.009642857142857142 \cdot {x}^{4}\right) + \left(-0.5\right) \]
      fma-def [=>] 0.2	\[ \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.009642857142857142 \cdot {x}^{4}\right)} + \left(-0.5\right) \]
      metadata-eval [=>] 0.2	\[ \mathsf{fma}\left(0.225, x \cdot x, -0.009642857142857142 \cdot {x}^{4}\right) + \color{blue}{-0.5} \]

   5. Applied egg-rr 0.2

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

   if 0.032000000000000001 < x

   1. Initial program 0.0

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1 - \frac{\sin x - \tan x}{x}\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.009642857142857142 + 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13513

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

Alternative 2
Error	0.7
Cost	7496

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

Alternative 3
Error	0.7
Cost	6984

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

Alternative 4
Error	0.7
Cost	1096

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

Alternative 5
Error	0.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.55:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;-0.5 + \left(x \cdot x\right) \cdot 0.225\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	1.0
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 7
Error	32.3
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023066 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))