sintan (problem 3.4.5)

Average Error: 31.4 → 0.3

Time: 22.0s

Precision: binary64

Cost: 33608

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin x - x}{\tan x - x}\\ \mathbf{if}\;x \leq -2953.836196677259:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00010546888065621142:\\ \;\;\;\;\left(x \cdot \left(x \cdot 0.225\right) + \log \left(\left(1 + \mathsf{expm1}\left(0.00024107142857142857 \cdot {x}^{6}\right)\right) \cdot \left(1 + \mathsf{expm1}\left(-0.009642857142857142 \cdot {x}^{4}\right)\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- (sin x) x) (- (tan x) x))))
   (if (<= x -2953.836196677259)
     t_0
     (if (<= x 0.00010546888065621142)
       (+
        (+
         (* x (* x 0.225))
         (log
          (*
           (+ 1.0 (expm1 (* 0.00024107142857142857 (pow x 6.0))))
           (+ 1.0 (expm1 (* -0.009642857142857142 (pow x 4.0)))))))
        -0.5)
       t_0))))

C

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

↓

double code(double x) {
	double t_0 = (sin(x) - x) / (tan(x) - x);
	double tmp;
	if (x <= -2953.836196677259) {
		tmp = t_0;
	} else if (x <= 0.00010546888065621142) {
		tmp = ((x * (x * 0.225)) + log(((1.0 + expm1((0.00024107142857142857 * pow(x, 6.0)))) * (1.0 + expm1((-0.009642857142857142 * pow(x, 4.0))))))) + -0.5;
	} else {
		tmp = t_0;
	}
	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 t_0 = (Math.sin(x) - x) / (Math.tan(x) - x);
	double tmp;
	if (x <= -2953.836196677259) {
		tmp = t_0;
	} else if (x <= 0.00010546888065621142) {
		tmp = ((x * (x * 0.225)) + Math.log(((1.0 + Math.expm1((0.00024107142857142857 * Math.pow(x, 6.0)))) * (1.0 + Math.expm1((-0.009642857142857142 * Math.pow(x, 4.0))))))) + -0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = (math.sin(x) - x) / (math.tan(x) - x)
	tmp = 0
	if x <= -2953.836196677259:
		tmp = t_0
	elif x <= 0.00010546888065621142:
		tmp = ((x * (x * 0.225)) + math.log(((1.0 + math.expm1((0.00024107142857142857 * math.pow(x, 6.0)))) * (1.0 + math.expm1((-0.009642857142857142 * math.pow(x, 4.0))))))) + -0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
end

↓

function code(x)
	t_0 = Float64(Float64(sin(x) - x) / Float64(tan(x) - x))
	tmp = 0.0
	if (x <= -2953.836196677259)
		tmp = t_0;
	elseif (x <= 0.00010546888065621142)
		tmp = Float64(Float64(Float64(x * Float64(x * 0.225)) + log(Float64(Float64(1.0 + expm1(Float64(0.00024107142857142857 * (x ^ 6.0)))) * Float64(1.0 + expm1(Float64(-0.009642857142857142 * (x ^ 4.0))))))) + -0.5);
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2953.836196677259], t$95$0, If[LessEqual[x, 0.00010546888065621142], N[(N[(N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(1.0 + N[(Exp[N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(Exp[N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2953.8361966772591 or 1.0546888065621142e-4 < x

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      Proof
      (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (sin.f64 x) x)) (*.f64 -1 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (sin.f64 x) x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sin.f64 x) x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (sin.f64 x)) x)) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sin.f64 x))) x) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (sin.f64 x)))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (sin.f64 x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (tan.f64 x)) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (tan.f64 x))) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (tan.f64 x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= sub-neg_binary64 (-.f64 x (tan.f64 x)))): 0 points increase in error, 0 points decrease in error

   if -2953.8361966772591 < x < 1.0546888065621142e-4

   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
      (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (sin.f64 x) x)) (*.f64 -1 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (sin.f64 x) x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sin.f64 x) x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (sin.f64 x)) x)) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sin.f64 x))) x) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (sin.f64 x)))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (sin.f64 x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (tan.f64 x)) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (tan.f64 x))) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (tan.f64 x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= sub-neg_binary64 (-.f64 x (tan.f64 x)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 0.5

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

   4. Applied egg-rr 0.5

      \[\leadsto \left(\color{blue}{\left(0 + x \cdot \left(x \cdot 0.225\right)\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

   5. Applied egg-rr 0.5

      \[\leadsto \left(\left(0 + x \cdot \left(x \cdot 0.225\right)\right) + \color{blue}{\log \left(\left(1 + \mathsf{expm1}\left(0.00024107142857142857 \cdot {x}^{6}\right)\right) \cdot \left(1 + \mathsf{expm1}\left(-0.009642857142857142 \cdot {x}^{4}\right)\right)\right)}\right) - 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2953.836196677259:\\ \;\;\;\;\frac{\sin x - x}{\tan x - x}\\ \mathbf{elif}\;x \leq 0.00010546888065621142:\\ \;\;\;\;\left(x \cdot \left(x \cdot 0.225\right) + \log \left(\left(1 + \mathsf{expm1}\left(0.00024107142857142857 \cdot {x}^{6}\right)\right) \cdot \left(1 + \mathsf{expm1}\left(-0.009642857142857142 \cdot {x}^{4}\right)\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x - x}{\tan x - x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	14408

\[\begin{array}{l} t_0 := \frac{\sin x - x}{\tan x - x}\\ \mathbf{if}\;x \leq -2953.836196677259:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00010546888065621142:\\ \;\;\;\;\left(x \cdot \left(x \cdot 0.225\right) + \left(-0.009642857142857142 \cdot {x}^{4} + \left(\left(1 + 0.00024107142857142857 \cdot {x}^{6}\right) + -1\right)\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.3
Cost	14152

\[\begin{array}{l} t_0 := \frac{\sin x - x}{\tan x - x}\\ \mathbf{if}\;x \leq -2953.836196677259:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00010546888065621142:\\ \;\;\;\;\left(x \cdot \left(x \cdot 0.225\right) + \left(0.00024107142857142857 \cdot {x}^{6} + -0.009642857142857142 \cdot {x}^{4}\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.1
Cost	13512

\[\begin{array}{l} t_0 := \frac{\sin x - x}{\tan x - x}\\ \mathbf{if}\;x \leq -0.058172189929390154:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00010546888065621142:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.0
Cost	7432

\[\begin{array}{l} \mathbf{if}\;x \leq -2953.836196677259:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00010546888065621142:\\ \;\;\;\;-0.5 + \left(x \cdot \left(x \cdot 0.225\right) + -0.009642857142857142 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \end{array} \]

Alternative 5
Error	1.0
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -2953.836196677259:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00010546888065621142:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \end{array} \]

Alternative 6
Error	1.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2953.836196677259:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00010546888065621142:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	1.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -2953.836196677259:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00010546888065621142:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	32.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022297 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))