2tan (problem 3.3.2)

Average Error: 37.5 → 0.3

Time: 8.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := {\cos x}^{2}\\ \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \log \left({\left(e^{\frac{\frac{\sin x}{\cos \varepsilon}}{\cos x}}\right)}^{\sin \varepsilon}\right)}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\varepsilon}{t_1}, {\sin x}^{2}, \varepsilon\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

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)))
   (if (<= eps -1.9e-7)
     (fma
      t_0
      (/
       1.0
       (- 1.0 (log (pow (exp (/ (/ (sin x) (cos eps)) (cos x))) (sin eps)))))
      (- (tan x)))
     (if (<= eps 5e-7)
       (+
        (fma (/ eps t_1) (pow (sin x) 2.0) eps)
        (* (/ (* eps eps) (cos x)) (+ (sin x) (/ (pow (sin x) 3.0) t_1))))
       (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (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(x) + tan(eps);
	double t_1 = pow(cos(x), 2.0);
	double tmp;
	if (eps <= -1.9e-7) {
		tmp = fma(t_0, (1.0 / (1.0 - log(pow(exp(((sin(x) / cos(eps)) / cos(x))), sin(eps))))), -tan(x));
	} else if (eps <= 5e-7) {
		tmp = fma((eps / t_1), pow(sin(x), 2.0), eps) + (((eps * eps) / cos(x)) * (sin(x) + (pow(sin(x), 3.0) / t_1)));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - 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(x) + tan(eps))
	t_1 = cos(x) ^ 2.0
	tmp = 0.0
	if (eps <= -1.9e-7)
		tmp = fma(t_0, Float64(1.0 / Float64(1.0 - log((exp(Float64(Float64(sin(x) / cos(eps)) / cos(x))) ^ sin(eps))))), Float64(-tan(x)));
	elseif (eps <= 5e-7)
		tmp = Float64(fma(Float64(eps / t_1), (sin(x) ^ 2.0), eps) + Float64(Float64(Float64(eps * eps) / cos(x)) * Float64(sin(x) + Float64((sin(x) ^ 3.0) / t_1))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - 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[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[eps, -1.9e-7], N[(t$95$0 * N[(1.0 / N[(1.0 - N[Log[N[Power[N[Exp[N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[eps], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 5e-7], N[(N[(N[(eps / t$95$1), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $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[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus eps

Target

Original	37.5
Target	15.0
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.90000000000000007e-7

   1. Initial program 30.1

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

   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. Taylor expanded in x around inf 0.4

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

   4. Simplified 0.4

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

   5. Applied egg-rr 0.5

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

   if -1.90000000000000007e-7 < eps < 4.99999999999999977e-7

   1. Initial program 45.7

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

   2. Taylor expanded in eps around 0 0.2

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

   3. Simplified 0.2

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

   if 4.99999999999999977e-7 < eps

   1. Initial program 29.0

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

   2. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \log \left({\left(e^{\frac{\frac{\sin x}{\cos \varepsilon}}{\cos x}}\right)}^{\sin \varepsilon}\right)}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\varepsilon}{{\cos x}^{2}}, {\sin x}^{2}, \varepsilon\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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)))