2tan (problem 3.3.2)

Average Error: 37.1 → 0.3

Time: 22.2s

Precision: binary64

Cost: 72264

Math

\[\tan \left(x + \varepsilon\right) - \tan x \]

↓

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \frac{\tan \varepsilon}{t_0}\\ \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;t_1 + \left(\frac{\tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;t_1 + \left(\frac{\varepsilon}{{\cos x}^{2}} \cdot {\sin x}^{2} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_0}, \tan \varepsilon + \tan x, -\tan x\right)\\ \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))) (t_1 (/ (tan eps) t_0)))
   (if (<= eps -3.2e-6)
     (+
      t_1
      (- (/ (tan x) (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x)))
     (if (<= eps 3.6e-6)
       (+
        t_1
        (+
         (* (/ eps (pow (cos x) 2.0)) (pow (sin x) 2.0))
         (/ (* eps eps) (/ (pow (cos x) 3.0) (pow (sin x) 3.0)))))
       (fma (/ 1.0 t_0) (+ (tan eps) (tan x)) (- (tan x)))))))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

↓

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double t_1 = tan(eps) / t_0;
	double tmp;
	if (eps <= -3.2e-6) {
		tmp = t_1 + ((tan(x) / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	} else if (eps <= 3.6e-6) {
		tmp = t_1 + (((eps / pow(cos(x), 2.0)) * pow(sin(x), 2.0)) + ((eps * eps) / (pow(cos(x), 3.0) / pow(sin(x), 3.0))));
	} else {
		tmp = fma((1.0 / t_0), (tan(eps) + tan(x)), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

↓

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_1 = Float64(tan(eps) / t_0)
	tmp = 0.0
	if (eps <= -3.2e-6)
		tmp = Float64(t_1 + Float64(Float64(tan(x) / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x)));
	elseif (eps <= 3.6e-6)
		tmp = Float64(t_1 + Float64(Float64(Float64(eps / (cos(x) ^ 2.0)) * (sin(x) ^ 2.0)) + Float64(Float64(eps * eps) / Float64((cos(x) ^ 3.0) / (sin(x) ^ 3.0)))));
	else
		tmp = fma(Float64(1.0 / t_0), Float64(tan(eps) + tan(x)), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[eps, -3.2e-6], N[(t$95$1 + N[(N[(N[Tan[x], $MachinePrecision] / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.6e-6], N[(t$95$1 + N[(N[(N[(eps / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] / N[(N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Target

Original	37.1
Target	15.4
Herbie	0.3

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

1. Split input into 3 regimes

2. if eps < -3.1999999999999999e-6

   1. Initial program 30.0

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.4

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

   4. Simplified 0.4

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

      [Start] 0.4	\[ \frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right) \]
      +-commutative [=>] 0.4	\[ \frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\left(-\tan x\right) + \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)} \]
      associate-+r+ [=>] 0.4	\[ \color{blue}{\left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right) + \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} \]
      sub-neg [<=] 0.4	\[ \color{blue}{\left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)} + \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \]
      +-commutative [<=] 0.4	\[ \color{blue}{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)} \]

   5. Applied egg-rr 0.4

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

   if -3.1999999999999999e-6 < eps < 3.59999999999999984e-6

   1. Initial program 44.8

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Applied egg-rr 44.3

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

   3. Applied egg-rr 44.3

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

   4. Simplified 25.5

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

      [Start] 44.3	\[ \frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right) \]
      +-commutative [=>] 44.3	\[ \frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\left(-\tan x\right) + \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)} \]
      associate-+r+ [=>] 25.5	\[ \color{blue}{\left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right) + \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} \]
      sub-neg [<=] 25.5	\[ \color{blue}{\left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)} + \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \]
      +-commutative [<=] 25.5	\[ \color{blue}{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)} \]

   5. Taylor expanded in eps around 0 0.2

      \[\leadsto \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right)} \]

   6. Simplified 0.2

      \[\leadsto \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\frac{\varepsilon}{{\cos x}^{2}} \cdot {\sin x}^{2} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)} \]
      Proof

      [Start] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right) \]
      associate-/l* [=>] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\color{blue}{\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}} + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right) \]
      associate-/r/ [=>] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\color{blue}{\frac{\varepsilon}{{\cos x}^{2}} \cdot {\sin x}^{2}} + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right) \]
      associate-/l* [=>] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{{\cos x}^{2}} \cdot {\sin x}^{2} + \color{blue}{\frac{{\varepsilon}^{2}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}}\right) \]
      unpow2 [=>] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{{\cos x}^{2}} \cdot {\sin x}^{2} + \frac{\color{blue}{\varepsilon \cdot \varepsilon}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right) \]

   if 3.59999999999999984e-6 < eps

   1. Initial program 29.8

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}, \tan x + \tan \varepsilon, -\tan x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\varepsilon}{{\cos x}^{2}} \cdot {\sin x}^{2} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}, \tan \varepsilon + \tan x, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	65736

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \left(\frac{\tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_0}, \tan \varepsilon + \tan x, -\tan x\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	52548

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \left(\frac{\tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.25 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_0}, \tan \varepsilon + \tan x, -\tan x\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	46020

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.25 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_0}, \tan \varepsilon + \tan x, -\tan x\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	39432

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.22 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}, t_0, -\tan x\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	39172

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.25 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 6
Error	0.4
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.25 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 7
Error	14.5
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0009:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 8
Error	14.5
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0009:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 9
Error	26.8
Cost	6464

\[\tan \varepsilon \]

Alternative 10
Error	44.2
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022354 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))