2tan (problem 3.3.2)

Average Error: 36.6 → 0.7

Time: 11.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := {\cos x}^{2}\\ t_2 := -\tan x\\ t_3 := \tan x + \tan \varepsilon\\ t_4 := \frac{{\sin x}^{2}}{t_1}\\ \mathbf{if}\;\varepsilon \leq -0.00022303564752948558:\\ \;\;\;\;\mathsf{fma}\left(t_3, \frac{1}{1 - \sqrt[3]{{t_0}^{3}}}, t_2\right)\\ \mathbf{elif}\;\varepsilon \leq 1.8570140171349828 \cdot 10^{-29}:\\ \;\;\;\;\left(\varepsilon + \mathsf{fma}\left(\frac{{\varepsilon}^{3}}{{\cos x}^{4}}, {\sin x}^{4}, \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{\left(\frac{\cos x}{\sin x}\right)}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, {\varepsilon}^{3} \cdot t_4, \mathsf{fma}\left(\varepsilon, t_4, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_3, \frac{1}{1 - {\tan x}^{3} \cdot \left(\tan \varepsilon \cdot {\tan \varepsilon}^{2}\right)} \cdot \mathsf{fma}\left(t_0, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), t_2\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps)))
        (t_1 (pow (cos x) 2.0))
        (t_2 (- (tan x)))
        (t_3 (+ (tan x) (tan eps)))
        (t_4 (/ (pow (sin x) 2.0) t_1)))
   (if (<= eps -0.00022303564752948558)
     (fma t_3 (/ 1.0 (- 1.0 (cbrt (pow t_0 3.0)))) t_2)
     (if (<= eps 1.8570140171349828e-29)
       (+
        (+
         eps
         (fma
          (/ (pow eps 3.0) (pow (cos x) 4.0))
          (pow (sin x) 4.0)
          (fma
           1.6666666666666667
           (/ (pow eps 4.0) (pow (/ (cos x) (sin x)) 3.0))
           (fma
            (/ (pow eps 4.0) (pow (cos x) 5.0))
            (pow (sin x) 5.0)
            (fma
             1.3333333333333333
             (* (pow eps 3.0) t_4)
             (fma
              eps
              t_4
              (fma
               0.3333333333333333
               (pow eps 3.0)
               (*
                0.6666666666666666
                (* (pow eps 4.0) (/ (sin x) (cos x)))))))))))
        (* (/ (* eps eps) (cos x)) (+ (sin x) (/ (pow (sin x) 3.0) t_1))))
       (fma
        t_3
        (*
         (/
          1.0
          (- 1.0 (* (pow (tan x) 3.0) (* (tan eps) (pow (tan eps) 2.0)))))
         (fma t_0 (fma (tan x) (tan eps) 1.0) 1.0))
        t_2)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = -tan(x);
	double t_3 = tan(x) + tan(eps);
	double t_4 = pow(sin(x), 2.0) / t_1;
	double tmp;
	if (eps <= -0.00022303564752948558) {
		tmp = fma(t_3, (1.0 / (1.0 - cbrt(pow(t_0, 3.0)))), t_2);
	} else if (eps <= 1.8570140171349828e-29) {
		tmp = (eps + fma((pow(eps, 3.0) / pow(cos(x), 4.0)), pow(sin(x), 4.0), fma(1.6666666666666667, (pow(eps, 4.0) / pow((cos(x) / sin(x)), 3.0)), fma((pow(eps, 4.0) / pow(cos(x), 5.0)), pow(sin(x), 5.0), fma(1.3333333333333333, (pow(eps, 3.0) * t_4), fma(eps, t_4, fma(0.3333333333333333, pow(eps, 3.0), (0.6666666666666666 * (pow(eps, 4.0) * (sin(x) / cos(x))))))))))) + (((eps * eps) / cos(x)) * (sin(x) + (pow(sin(x), 3.0) / t_1)));
	} else {
		tmp = fma(t_3, ((1.0 / (1.0 - (pow(tan(x), 3.0) * (tan(eps) * pow(tan(eps), 2.0))))) * fma(t_0, fma(tan(x), tan(eps), 1.0), 1.0)), t_2);
	}
	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) * tan(eps))
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(-tan(x))
	t_3 = Float64(tan(x) + tan(eps))
	t_4 = Float64((sin(x) ^ 2.0) / t_1)
	tmp = 0.0
	if (eps <= -0.00022303564752948558)
		tmp = fma(t_3, Float64(1.0 / Float64(1.0 - cbrt((t_0 ^ 3.0)))), t_2);
	elseif (eps <= 1.8570140171349828e-29)
		tmp = Float64(Float64(eps + fma(Float64((eps ^ 3.0) / (cos(x) ^ 4.0)), (sin(x) ^ 4.0), fma(1.6666666666666667, Float64((eps ^ 4.0) / (Float64(cos(x) / sin(x)) ^ 3.0)), fma(Float64((eps ^ 4.0) / (cos(x) ^ 5.0)), (sin(x) ^ 5.0), fma(1.3333333333333333, Float64((eps ^ 3.0) * t_4), fma(eps, t_4, fma(0.3333333333333333, (eps ^ 3.0), Float64(0.6666666666666666 * Float64((eps ^ 4.0) * Float64(sin(x) / cos(x))))))))))) + Float64(Float64(Float64(eps * eps) / cos(x)) * Float64(sin(x) + Float64((sin(x) ^ 3.0) / t_1))));
	else
		tmp = fma(t_3, Float64(Float64(1.0 / Float64(1.0 - Float64((tan(x) ^ 3.0) * Float64(tan(eps) * (tan(eps) ^ 2.0))))) * fma(t_0, fma(tan(x), tan(eps), 1.0), 1.0)), t_2);
	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[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$3 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[eps, -0.00022303564752948558], N[(t$95$3 * N[(1.0 / N[(1.0 - N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[eps, 1.8570140171349828e-29], N[(N[(eps + 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] + N[(1.6666666666666667 * N[(N[Power[eps, 4.0], $MachinePrecision] / N[Power[N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 5.0], $MachinePrecision] + N[(1.3333333333333333 * N[(N[Power[eps, 3.0], $MachinePrecision] * t$95$4), $MachinePrecision] + N[(eps * t$95$4 + N[(0.3333333333333333 * N[Power[eps, 3.0], $MachinePrecision] + N[(0.6666666666666666 * N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(N[(1.0 / N[(1.0 - N[(N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[eps], $MachinePrecision] * N[Power[N[Tan[eps], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]]

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	15.0
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.23035647529485579e-4

   1. Initial program 29.5

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

   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 0.4

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

   if -2.23035647529485579e-4 < eps < 1.85701401713498277e-29

   1. Initial program 44.3

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

   2. Taylor expanded in eps around 0 0.2

      \[\leadsto \color{blue}{\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{{\varepsilon}^{4} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.6666666666666666 \cdot \frac{{\varepsilon}^{4} \cdot \sin x}{\cos x} + 0.3333333333333333 \cdot {\varepsilon}^{3}\right)\right)\right)\right)\right)\right)\right)\right)} \]

   3. Simplified 0.2

      \[\leadsto \color{blue}{\left(\varepsilon + \mathsf{fma}\left(\frac{{\varepsilon}^{3}}{{\cos x}^{4}}, {\sin x}^{4}, \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{\left(\frac{\cos x}{\sin x}\right)}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left(\frac{\sin x}{\cos x} \cdot {\varepsilon}^{4}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)} \]

   if 1.85701401713498277e-29 < eps

   1. Initial program 29.9

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

   2. Applied egg-rr 2.0

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

   3. Applied egg-rr 2.1

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

   4. Applied egg-rr 2.0

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

   5. Applied egg-rr 2.0

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\tan x}^{3} \cdot \color{blue}{\left({\tan \varepsilon}^{2} \cdot \tan \varepsilon\right)}} \cdot \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00022303564752948558:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.8570140171349828 \cdot 10^{-29}:\\ \;\;\;\;\left(\varepsilon + \mathsf{fma}\left(\frac{{\varepsilon}^{3}}{{\cos x}^{4}}, {\sin x}^{4}, \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{\left(\frac{\cos x}{\sin x}\right)}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, {\varepsilon}^{3} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\tan x}^{3} \cdot \left(\tan \varepsilon \cdot {\tan \varepsilon}^{2}\right)} \cdot \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))