sintan (problem 3.4.5)

Average Accuracy: 51.0% → 100.0%

Time: 19.4s

Precision: binary64

Cost: 20425

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.08 \lor \neg \left(x \leq 0.095\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + {x}^{4} \cdot \mathsf{fma}\left(x, x \cdot 0.00024107142857142857, -0.009642857142857142\right)\right) - 0.5\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.08) (not (<= x 0.095)))
   (/ (- x (sin x)) (- x (tan x)))
   (-
    (+
     (* 0.225 (pow x 2.0))
     (*
      (pow x 4.0)
      (fma x (* x 0.00024107142857142857) -0.009642857142857142)))
    0.5)))

C

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

↓

double code(double x) {
	double tmp;
	if ((x <= -0.08) || !(x <= 0.095)) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else {
		tmp = ((0.225 * pow(x, 2.0)) + (pow(x, 4.0) * fma(x, (x * 0.00024107142857142857), -0.009642857142857142))) - 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 <= -0.08) || !(x <= 0.095))
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	else
		tmp = Float64(Float64(Float64(0.225 * (x ^ 2.0)) + Float64((x ^ 4.0) * fma(x, Float64(x * 0.00024107142857142857), -0.009642857142857142))) - 0.5);
	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_] := If[Or[LessEqual[x, -0.08], N[Not[LessEqual[x, 0.095]], $MachinePrecision]], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.225 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * N[(x * N[(x * 0.00024107142857142857), $MachinePrecision] + -0.009642857142857142), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0800000000000000017 or 0.095000000000000001 < x

   1. Initial program 100.0%

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

   if -0.0800000000000000017 < x < 0.095000000000000001

   1. Initial program 1.4%

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

   2. Simplified 1.4%

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

      [Start] 1.4	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 1.4	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 1.4	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 1.4	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 1.4	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 1.4	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 1.4	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 1.4	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 1.4	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 1.4	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 1.4	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 100.0%

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

   4. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
      expm1-log1p-u [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right)}\right) - 0.5 \]
      expm1-udef [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)} - 1\right)}\right) - 0.5 \]
      +-commutative [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(e^{\mathsf{log1p}\left(\color{blue}{0.00024107142857142857 \cdot {x}^{6} + -0.009642857142857142 \cdot {x}^{4}}\right)} - 1\right)\right) - 0.5 \]
      fma-def [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.00024107142857142857, {x}^{6}, -0.009642857142857142 \cdot {x}^{4}\right)}\right)} - 1\right)\right) - 0.5 \]

   5. Simplified 100.0%

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

      [Start] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.00024107142857142857, {x}^{6}, -0.009642857142857142 \cdot {x}^{4}\right)\right)} - 1\right)\right) - 0.5 \]
      expm1-def [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.00024107142857142857, {x}^{6}, -0.009642857142857142 \cdot {x}^{4}\right)\right)\right)}\right) - 0.5 \]
      expm1-log1p [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \color{blue}{\mathsf{fma}\left(0.00024107142857142857, {x}^{6}, -0.009642857142857142 \cdot {x}^{4}\right)}\right) - 0.5 \]
      fma-udef [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \color{blue}{\left(0.00024107142857142857 \cdot {x}^{6} + -0.009642857142857142 \cdot {x}^{4}\right)}\right) - 0.5 \]
      *-commutative [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(\color{blue}{{x}^{6} \cdot 0.00024107142857142857} + -0.009642857142857142 \cdot {x}^{4}\right)\right) - 0.5 \]
      metadata-eval [<=] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left({x}^{\color{blue}{\left(2 \cdot 3\right)}} \cdot 0.00024107142857142857 + -0.009642857142857142 \cdot {x}^{4}\right)\right) - 0.5 \]
      pow-sqr [<=] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(\color{blue}{\left({x}^{3} \cdot {x}^{3}\right)} \cdot 0.00024107142857142857 + -0.009642857142857142 \cdot {x}^{4}\right)\right) - 0.5 \]
      cube-mult [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(\left({x}^{3} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot 0.00024107142857142857 + -0.009642857142857142 \cdot {x}^{4}\right)\right) - 0.5 \]
      associate-*r* [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(\color{blue}{\left(\left({x}^{3} \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot 0.00024107142857142857 + -0.009642857142857142 \cdot {x}^{4}\right)\right) - 0.5 \]
      pow-plus [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(\left(\color{blue}{{x}^{\left(3 + 1\right)}} \cdot \left(x \cdot x\right)\right) \cdot 0.00024107142857142857 + -0.009642857142857142 \cdot {x}^{4}\right)\right) - 0.5 \]
      metadata-eval [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(\left({x}^{\color{blue}{4}} \cdot \left(x \cdot x\right)\right) \cdot 0.00024107142857142857 + -0.009642857142857142 \cdot {x}^{4}\right)\right) - 0.5 \]
      associate-*r* [<=] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left(\color{blue}{{x}^{4} \cdot \left(\left(x \cdot x\right) \cdot 0.00024107142857142857\right)} + -0.009642857142857142 \cdot {x}^{4}\right)\right) - 0.5 \]
      *-commutative [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \left({x}^{4} \cdot \left(\left(x \cdot x\right) \cdot 0.00024107142857142857\right) + \color{blue}{{x}^{4} \cdot -0.009642857142857142}\right)\right) - 0.5 \]
      distribute-lft-out [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + \color{blue}{{x}^{4} \cdot \left(\left(x \cdot x\right) \cdot 0.00024107142857142857 + -0.009642857142857142\right)}\right) - 0.5 \]
      associate-*l* [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + {x}^{4} \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.00024107142857142857\right)} + -0.009642857142857142\right)\right) - 0.5 \]
      fma-def [=>] 100.0	\[ \left(0.225 \cdot {x}^{2} + {x}^{4} \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot 0.00024107142857142857, -0.009642857142857142\right)}\right) - 0.5 \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.08 \lor \neg \left(x \leq 0.095\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + {x}^{4} \cdot \mathsf{fma}\left(x, x \cdot 0.00024107142857142857, -0.009642857142857142\right)\right) - 0.5\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	14409

\[\begin{array}{l} \mathbf{if}\;x \leq -0.08 \lor \neg \left(x \leq 0.095\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 + 0.225 \cdot \left(x \cdot x\right)\right) + -1\right) + \left({x}^{4} \cdot -0.009642857142857142 + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	13705

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0255 \lor \neg \left(x \leq 0.029\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, {x}^{4} \cdot -0.009642857142857142\right) + -0.5\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 0.0046\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \end{array} \]

Alternative 4
Accuracy	98.9%
Cost	8456

\[\begin{array}{l} t_0 := 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;\frac{{t_0}^{3} + -0.125}{0.050625 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(0.25 - t_0 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{3}{x}}{x} - \frac{x}{\tan x - x}\\ \end{array} \]

Alternative 5
Accuracy	98.9%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{3}{x}}{x} - \frac{x}{\tan x - x}\\ \end{array} \]

Alternative 6
Accuracy	98.9%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.2:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \]

Alternative 7
Accuracy	98.9%
Cost	712

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

Alternative 8
Accuracy	98.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.85:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{3}{x}}{x} - -1\\ \end{array} \]

Alternative 9
Accuracy	98.6%
Cost	328

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

Alternative 10
Accuracy	50.2%
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))