2tan (problem 3.3.2)

Average Error: 36.9 → 0.3

Time: 10.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;\varepsilon \leq -5.0008209901859945 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 4.0929675985837755 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\varepsilon}{\frac{\cos x}{\varepsilon}} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_0}\right) + \mathsf{fma}\left(\frac{{\varepsilon}^{3}}{{\cos x}^{4}}, {\sin x}^{4}, \varepsilon\right)\right) + \mathsf{fma}\left(\varepsilon, \frac{t_3}{t_0}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_3}{t_0}, {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - \left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) + -1\right)}, t_1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (- (tan x)))
        (t_2 (+ (tan x) (tan eps)))
        (t_3 (pow (sin x) 2.0)))
   (if (<= eps -5.0008209901859945e-5)
     (fma t_2 (/ 1.0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) t_1)
     (if (<= eps 4.0929675985837755e-5)
       (+
        (+
         (* (/ eps (/ (cos x) eps)) (+ (sin x) (/ (pow (sin x) 3.0) t_0)))
         (fma (/ (pow eps 3.0) (pow (cos x) 4.0)) (pow (sin x) 4.0) eps))
        (fma
         eps
         (/ t_3 t_0)
         (fma
          1.3333333333333333
          (/ (* (pow eps 3.0) t_3) t_0)
          (* (pow eps 3.0) 0.3333333333333333))))
       (fma t_2 (/ 1.0 (- 1.0 (+ (fma (tan x) (tan eps) 1.0) -1.0))) t_1)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = -tan(x);
	double t_2 = tan(x) + tan(eps);
	double t_3 = pow(sin(x), 2.0);
	double tmp;
	if (eps <= -5.0008209901859945e-5) {
		tmp = fma(t_2, (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))), t_1);
	} else if (eps <= 4.0929675985837755e-5) {
		tmp = (((eps / (cos(x) / eps)) * (sin(x) + (pow(sin(x), 3.0) / t_0))) + fma((pow(eps, 3.0) / pow(cos(x), 4.0)), pow(sin(x), 4.0), eps)) + fma(eps, (t_3 / t_0), fma(1.3333333333333333, ((pow(eps, 3.0) * t_3) / t_0), (pow(eps, 3.0) * 0.3333333333333333)));
	} else {
		tmp = fma(t_2, (1.0 / (1.0 - (fma(tan(x), tan(eps), 1.0) + -1.0))), t_1);
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = Float64(-tan(x))
	t_2 = Float64(tan(x) + tan(eps))
	t_3 = sin(x) ^ 2.0
	tmp = 0.0
	if (eps <= -5.0008209901859945e-5)
		tmp = fma(t_2, Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))), t_1);
	elseif (eps <= 4.0929675985837755e-5)
		tmp = Float64(Float64(Float64(Float64(eps / Float64(cos(x) / eps)) * Float64(sin(x) + Float64((sin(x) ^ 3.0) / t_0))) + fma(Float64((eps ^ 3.0) / (cos(x) ^ 4.0)), (sin(x) ^ 4.0), eps)) + fma(eps, Float64(t_3 / t_0), fma(1.3333333333333333, Float64(Float64((eps ^ 3.0) * t_3) / t_0), Float64((eps ^ 3.0) * 0.3333333333333333))));
	else
		tmp = fma(t_2, Float64(1.0 / Float64(1.0 - Float64(fma(tan(x), tan(eps), 1.0) + -1.0))), 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[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[eps, -5.0008209901859945e-5], 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, 4.0929675985837755e-5], N[(N[(N[(N[(eps / N[(N[Cos[x], $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(t$95$3 / t$95$0), $MachinePrecision] + N[(1.3333333333333333 * N[(N[(N[Power[eps, 3.0], $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.2
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -5.0008209901859945e-5

   1. Initial program 29.9

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.4

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

   if -5.0008209901859945e-5 < eps < 4.0929675985837755e-5

   1. Initial program 44.3

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

   2. Applied egg-rr 43.7

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

   3. Applied egg-rr 43.7

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

   4. Taylor expanded in eps around 0 0.2

      \[\leadsto \color{blue}{\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(\varepsilon + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}} + 0.3333333333333333 \cdot {\varepsilon}^{3}\right)\right)\right)\right)\right)} \]

   5. Simplified 0.2

      \[\leadsto \color{blue}{\left(\frac{\varepsilon}{\frac{\cos x}{\varepsilon}} \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x\right) + \mathsf{fma}\left(\frac{{\varepsilon}^{3}}{{\cos x}^{4}}, {\sin x}^{4}, \varepsilon\right)\right) + \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}}, {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)} \]

   if 4.0929675985837755e-5 < eps

   1. Initial program 29.7

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.4

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

   4. Applied egg-rr 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.0008209901859945 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.0929675985837755 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\varepsilon}{\frac{\cos x}{\varepsilon}} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right) + \mathsf{fma}\left(\frac{{\varepsilon}^{3}}{{\cos x}^{4}}, {\sin x}^{4}, \varepsilon\right)\right) + \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}}, {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) + -1\right)}, -\tan x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))