2tan (problem 3.3.2)

Average Error: 36.7 → 0.7

Time: 9.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.3331437575950957 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(-1, \tan x \cdot \tan \varepsilon, 1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.9254302127384425 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)} + \frac{\varepsilon}{{\cos x}^{2}} \cdot {\sin x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan \varepsilon, t_0, 1\right)}, t_1, t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -1.3331437575950957e-6)
     (- (/ t_1 (fma -1.0 (* (tan x) (tan eps)) 1.0)) (tan x))
     (if (<= eps 1.9254302127384425e-29)
       (+
        (/
         (sin eps)
         (* (cos eps) (- 1.0 (* (/ (sin eps) (cos eps)) (/ (sin x) (cos x))))))
        (* (/ eps (pow (cos x) 2.0)) (pow (sin x) 2.0)))
       (fma (/ 1.0 (fma (tan eps) t_0 1.0)) t_1 t_0)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -1.3331437575950957e-6) {
		tmp = (t_1 / fma(-1.0, (tan(x) * tan(eps)), 1.0)) - tan(x);
	} else if (eps <= 1.9254302127384425e-29) {
		tmp = (sin(eps) / (cos(eps) * (1.0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x)))))) + ((eps / pow(cos(x), 2.0)) * pow(sin(x), 2.0));
	} else {
		tmp = fma((1.0 / fma(tan(eps), t_0, 1.0)), t_1, t_0);
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = Float64(-tan(x))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -1.3331437575950957e-6)
		tmp = Float64(Float64(t_1 / fma(-1.0, Float64(tan(x) * tan(eps)), 1.0)) - tan(x));
	elseif (eps <= 1.9254302127384425e-29)
		tmp = Float64(Float64(sin(eps) / Float64(cos(eps) * Float64(1.0 - Float64(Float64(sin(eps) / cos(eps)) * Float64(sin(x) / cos(x)))))) + Float64(Float64(eps / (cos(x) ^ 2.0)) * (sin(x) ^ 2.0)));
	else
		tmp = fma(Float64(1.0 / fma(tan(eps), t_0, 1.0)), t_1, t_0);
	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[Tan[x], $MachinePrecision])}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.3331437575950957e-6], N[(N[(t$95$1 / N[(-1.0 * N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.9254302127384425e-29], N[(N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Tan[eps], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	14.8
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.33314375759509571e-6

   1. Initial program 29.3

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.4

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

   if -1.33314375759509571e-6 < eps < 1.92543021273844254e-29

   1. Initial program 44.7

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

   2. Applied egg-rr 44.4

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

   3. Taylor expanded in x around inf 44.4

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

   4. Simplified 25.7

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

   5. Taylor expanded in eps around 0 0.3

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

   6. Simplified 0.3

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

   if 1.92543021273844254e-29 < eps

   1. Initial program 29.5

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

   2. Applied egg-rr 1.8

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

   3. Applied egg-rr 1.8

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3331437575950957 \cdot 10^{-6}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(-1, \tan x \cdot \tan \varepsilon, 1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.9254302127384425 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)} + \frac{\varepsilon}{{\cos x}^{2}} \cdot {\sin x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)}, \tan x + \tan \varepsilon, -\tan x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022172 
(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)))