2tan (problem 3.3.2)

Average Error: 36.9 → 0.5

Time: 9.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -96117.97416021654:\\ \;\;\;\;\frac{t_1}{t_2} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.522072481790777 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)} + \frac{\varepsilon \cdot {\sin x}^{2}}{\cos x \cdot \left(t_2 \cdot \cos x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{\mathsf{fma}\left(\tan \varepsilon, t_0, 1\right)}, t_0\right)\\ \end{array} \]

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)))
        (t_2 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -96117.97416021654)
     (- (/ t_1 t_2) (tan x))
     (if (<= eps 1.522072481790777e-9)
       (+
        (/
         (sin eps)
         (* (cos eps) (- 1.0 (* (/ (sin eps) (cos eps)) (/ (sin x) (cos x))))))
        (/ (* eps (pow (sin x) 2.0)) (* (cos x) (* t_2 (cos x)))))
       (fma t_1 (/ 1.0 (fma (tan eps) t_0 1.0)) 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 t_2 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -96117.97416021654) {
		tmp = (t_1 / t_2) - tan(x);
	} else if (eps <= 1.522072481790777e-9) {
		tmp = (sin(eps) / (cos(eps) * (1.0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x)))))) + ((eps * pow(sin(x), 2.0)) / (cos(x) * (t_2 * cos(x))));
	} else {
		tmp = fma(t_1, (1.0 / fma(tan(eps), t_0, 1.0)), 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))
	t_2 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -96117.97416021654)
		tmp = Float64(Float64(t_1 / t_2) - tan(x));
	elseif (eps <= 1.522072481790777e-9)
		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 * (sin(x) ^ 2.0)) / Float64(cos(x) * Float64(t_2 * cos(x)))));
	else
		tmp = fma(t_1, Float64(1.0 / fma(tan(eps), t_0, 1.0)), 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]}, Block[{t$95$2 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -96117.97416021654], N[(N[(t$95$1 / t$95$2), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.522072481790777e-9], 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[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 / N[(N[Tan[eps], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]

Target

Original	36.9
Target	15.2
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if eps < -96117.974160216545

   1. Initial program 29.5

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

   2. Applied egg-rr 0.3

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

   if -96117.974160216545 < eps < 1.52207248179077694e-9

   1. Initial program 44.8

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

   2. Applied egg-rr 43.6

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

   3. Taylor expanded in x around inf 43.7

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

   4. Simplified 25.5

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

   5. Applied egg-rr 25.5

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

   6. Taylor expanded in eps around 0 0.7

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

   if 1.52207248179077694e-9 < eps

   1. Initial program 29.2

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

   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

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -96117.97416021654:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.522072481790777 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)} + \frac{\varepsilon \cdot {\sin x}^{2}}{\cos x \cdot \left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)}, -\tan x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022197 
(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)))