2tan (problem 3.3.2)

Average Error: 37.1 → 0.3

Time: 11.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := \frac{\cos x}{\sin x}\\ t_2 := {\sin x}^{2}\\ t_3 := {\cos x}^{2}\\ t_4 := \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -0.00018896478073308958:\\ \;\;\;\;\frac{t_0}{1 - t_4} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.923034667284573 \cdot 10^{-6}:\\ \;\;\;\;\left(\varepsilon + \left(\frac{{\varepsilon}^{3}}{{t_1}^{4}} + \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{t_1}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_2}{t_3}, \mathsf{fma}\left(\varepsilon, \frac{t_2}{t_3}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t_0}{1 - {\tan x}^{3} \cdot {\tan \varepsilon}^{3}}, \mathsf{fma}\left(t_4, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x)))
        (t_1 (/ (cos x) (sin x)))
        (t_2 (pow (sin x) 2.0))
        (t_3 (pow (cos x) 2.0))
        (t_4 (* (tan eps) (tan x))))
   (if (<= eps -0.00018896478073308958)
     (- (/ t_0 (- 1.0 t_4)) (tan x))
     (if (<= eps 1.923034667284573e-6)
       (+
        (+
         eps
         (+
          (/ (pow eps 3.0) (pow t_1 4.0))
          (fma
           1.6666666666666667
           (/ (pow eps 4.0) (pow t_1 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_2) t_3)
             (fma
              eps
              (/ t_2 t_3)
              (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_3))))
       (fma
        (/ t_0 (- 1.0 (* (pow (tan x) 3.0) (pow (tan eps) 3.0))))
        (fma t_4 (fma (tan x) (tan eps) 1.0) 1.0)
        (- (tan x)))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double t_1 = cos(x) / sin(x);
	double t_2 = pow(sin(x), 2.0);
	double t_3 = pow(cos(x), 2.0);
	double t_4 = tan(eps) * tan(x);
	double tmp;
	if (eps <= -0.00018896478073308958) {
		tmp = (t_0 / (1.0 - t_4)) - tan(x);
	} else if (eps <= 1.923034667284573e-6) {
		tmp = (eps + ((pow(eps, 3.0) / pow(t_1, 4.0)) + fma(1.6666666666666667, (pow(eps, 4.0) / pow(t_1, 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_2) / t_3), fma(eps, (t_2 / t_3), 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_3)));
	} else {
		tmp = fma((t_0 / (1.0 - (pow(tan(x), 3.0) * pow(tan(eps), 3.0)))), fma(t_4, fma(tan(x), tan(eps), 1.0), 1.0), -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(tan(eps) + tan(x))
	t_1 = Float64(cos(x) / sin(x))
	t_2 = sin(x) ^ 2.0
	t_3 = cos(x) ^ 2.0
	t_4 = Float64(tan(eps) * tan(x))
	tmp = 0.0
	if (eps <= -0.00018896478073308958)
		tmp = Float64(Float64(t_0 / Float64(1.0 - t_4)) - tan(x));
	elseif (eps <= 1.923034667284573e-6)
		tmp = Float64(Float64(eps + Float64(Float64((eps ^ 3.0) / (t_1 ^ 4.0)) + fma(1.6666666666666667, Float64((eps ^ 4.0) / (t_1 ^ 3.0)), fma(Float64((eps ^ 4.0) / (cos(x) ^ 5.0)), (sin(x) ^ 5.0), fma(1.3333333333333333, Float64(Float64((eps ^ 3.0) * t_2) / t_3), fma(eps, Float64(t_2 / t_3), 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_3))));
	else
		tmp = fma(Float64(t_0 / Float64(1.0 - Float64((tan(x) ^ 3.0) * (tan(eps) ^ 3.0)))), fma(t_4, fma(tan(x), tan(eps), 1.0), 1.0), 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[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00018896478073308958], N[(N[(t$95$0 / N[(1.0 - t$95$4), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.923034667284573e-6], N[(N[(eps + N[(N[(N[Power[eps, 3.0], $MachinePrecision] / N[Power[t$95$1, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.6666666666666667 * N[(N[Power[eps, 4.0], $MachinePrecision] / N[Power[t$95$1, 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[(N[Power[eps, 3.0], $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(eps * N[(t$95$2 / t$95$3), $MachinePrecision] + 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$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision] * N[Power[N[Tan[eps], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]]]]

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.8896478073308958e-4

   1. Initial program 30.1

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

   2. Applied tan-sum_binary64 0.4

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

   3. Applied +-commutative_binary64 0.4

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

   if -1.8896478073308958e-4 < eps < 1.9230346672845731e-6

   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

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

   if 1.9230346672845731e-6 < eps

   1. Initial program 30.1

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

   2. Applied tan-sum_binary64 0.4

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

   3. Applied add-cube-cbrt_binary64 0.7

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

   4. Applied flip3--_binary64 0.8

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

   5. Applied associate-/r/_binary64 0.7

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

   6. Applied prod-diff_binary64 0.7

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

   7. Simplified 0.4

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

   8. Simplified 0.4

      \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right) + \color{blue}{0} \]

   9. Applied unpow-prod-down_binary64 0.4

      \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \color{blue}{{\tan x}^{3} \cdot {\tan \varepsilon}^{3}}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right) + 0 \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018896478073308958:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.923034667284573 \cdot 10^{-6}:\\ \;\;\;\;\left(\varepsilon + \left(\frac{{\varepsilon}^{3}}{{\left(\frac{\cos x}{\sin x}\right)}^{4}} + \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{\left(\frac{\cos x}{\sin x}\right)}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan \varepsilon + \tan x}{1 - {\tan x}^{3} \cdot {\tan \varepsilon}^{3}}, \mathsf{fma}\left(\tan \varepsilon \cdot \tan x, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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)))