2tan (problem 3.3.2)

Average Error: 36.8 → 0.4

Time: 14.0s

Precision: binary64

Cost: 59144

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0005537219591504346:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 4.141214802207624 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(t_0 + {t_0}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x)))
        (t_1 (- (tan x)))
        (t_2 (+ (tan x) (tan eps))))
   (if (<= eps -0.0005537219591504346)
     (fma t_2 (/ 1.0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) t_1)
     (if (<= eps 4.141214802207624e-10)
       (*
        eps
        (+
         (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
         (* eps (+ t_0 (pow t_0 3.0)))))
       (fma t_2 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_1)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	double t_1 = -tan(x);
	double t_2 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -0.0005537219591504346) {
		tmp = fma(t_2, (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))), t_1);
	} else if (eps <= 4.141214802207624e-10) {
		tmp = eps * ((1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0))) + (eps * (t_0 + pow(t_0, 3.0))));
	} else {
		tmp = fma(t_2, (1.0 / (1.0 - (tan(x) * tan(eps)))), t_1);
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	t_1 = Float64(-tan(x))
	t_2 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -0.0005537219591504346)
		tmp = fma(t_2, Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))), t_1);
	elseif (eps <= 4.141214802207624e-10)
		tmp = Float64(eps * Float64(Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))) + Float64(eps * Float64(t_0 + (t_0 ^ 3.0)))));
	else
		tmp = fma(t_2, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_1);
	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[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0005537219591504346], N[(t$95$2 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[eps, 4.141214802207624e-10], N[(eps * N[(N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(t$95$0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Target

Original	36.8
Target	15.5
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if eps < -5.53721959150434606e-4

   1. Initial program 29.9

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 0.3

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

   if -5.53721959150434606e-4 < eps < 4.141214802207624e-10

   1. Initial program 44.1

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

   2. Applied egg-rr 43.6

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

   3. Applied egg-rr 43.6

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

   4. Applied egg-rr 43.6

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}}, -\tan x\right) \]

   5. Taylor expanded in eps around 0 0.4

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

   6. Simplified 0.4

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

   if 4.141214802207624e-10 < eps

   1. Initial program 30.3

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

   2. Applied egg-rr 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0005537219591504346:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.141214802207624 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	45828

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0005537219591504346:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 4.141214802207624 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_0\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	39432

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0005537219591504346:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.141214802207624 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	32968

\[\begin{array}{l} t_0 := \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{if}\;\varepsilon \leq -0.0005537219591504346:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.141214802207624 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	14.9
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -869.9913763468928:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.141214802207624 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 5
Error	14.9
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -869.9913763468928:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.141214802207624 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 6
Error	27.0
Cost	6464

\[\tan \varepsilon \]

Alternative 7
Error	44.1
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022228 
(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)))