Average Error: 37.4 → 0.3
Time: 16.9s
Precision: binary64
Cost: 170184
\[\tan \left(x + \varepsilon\right) - \tan x \]
↓
\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -0.11399096168961476:\\ \;\;\;\;\mathsf{fma}\left(\frac{t_1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.0687464007627642 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_2}{1 - t_2 \cdot t_0} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} - {\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(t_0 \cdot 0.16666666666666666 - t_0 \cdot 0.5\right)}{\cos x} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x\\ \end{array} \]
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x)))
        (t_1 (+ (tan x) (tan eps)))
        (t_2 (/ (sin eps) (cos eps))))
   (if (<= eps -0.11399096168961476)
     (fma
      (/ t_1 (- 1.0 (pow (* (tan x) (tan eps)) 2.0)))
      (fma (tan x) (tan eps) 1.0)
      (- (tan x)))
     (if (<= eps 2.0687464007627642e-7)
       (+
        (/ t_2 (- 1.0 (* t_2 t_0)))
        (+
         (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0))
         (-
          (/ (* (pow eps 2.0) (pow (sin x) 3.0)) (pow (cos x) 3.0))
          (*
           (pow eps 3.0)
           (-
            (/ (* (sin x) (- (* t_0 0.16666666666666666) (* t_0 0.5))) (cos x))
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))))))
       (- (/ t_1 (fma (tan x) (- (tan eps)) 1.0)) (tan x))))))
double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}
↓
double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	double t_1 = tan(x) + tan(eps);
	double t_2 = sin(eps) / cos(eps);
	double tmp;
	if (eps <= -0.11399096168961476) {
		tmp = fma((t_1 / (1.0 - pow((tan(x) * tan(eps)), 2.0))), fma(tan(x), tan(eps), 1.0), -tan(x));
	} else if (eps <= 2.0687464007627642e-7) {
		tmp = (t_2 / (1.0 - (t_2 * t_0))) + (((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0)) + (((pow(eps, 2.0) * pow(sin(x), 3.0)) / pow(cos(x), 3.0)) - (pow(eps, 3.0) * (((sin(x) * ((t_0 * 0.16666666666666666) - (t_0 * 0.5))) / cos(x)) - (pow(sin(x), 4.0) / pow(cos(x), 4.0))))));
	} else {
		tmp = (t_1 / fma(tan(x), -tan(eps), 1.0)) - tan(x);
	}
	return tmp;
}

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end
↓
function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	t_1 = Float64(tan(x) + tan(eps))
	t_2 = Float64(sin(eps) / cos(eps))
	tmp = 0.0
	if (eps <= -0.11399096168961476)
		tmp = fma(Float64(t_1 / Float64(1.0 - (Float64(tan(x) * tan(eps)) ^ 2.0))), fma(tan(x), tan(eps), 1.0), Float64(-tan(x)));
	elseif (eps <= 2.0687464007627642e-7)
		tmp = Float64(Float64(t_2 / Float64(1.0 - Float64(t_2 * t_0))) + Float64(Float64(Float64(eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)) + Float64(Float64(Float64((eps ^ 2.0) * (sin(x) ^ 3.0)) / (cos(x) ^ 3.0)) - Float64((eps ^ 3.0) * Float64(Float64(Float64(sin(x) * Float64(Float64(t_0 * 0.16666666666666666) - Float64(t_0 * 0.5))) / cos(x)) - Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)))))));
	else
		tmp = Float64(Float64(t_1 / fma(tan(x), Float64(-tan(eps)), 1.0)) - tan(x));
	end
	return tmp
end

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[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.11399096168961476], N[(N[(t$95$1 / N[(1.0 - N[Power[N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 2.0687464007627642e-7], N[(N[(t$95$2 / N[(1.0 - N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(t$95$0 * 0.16666666666666666), $MachinePrecision] - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[Tan[x], $MachinePrecision] * (-N[Tan[eps], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]

\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
t_0 := \frac{\sin x}{\cos x}\\
t_1 := \tan x + \tan \varepsilon\\
t_2 := \frac{\sin \varepsilon}{\cos \varepsilon}\\
\mathbf{if}\;\varepsilon \leq -0.11399096168961476:\\
\;\;\;\;\mathsf{fma}\left(\frac{t_1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right)\\

\mathbf{elif}\;\varepsilon \leq 2.0687464007627642 \cdot 10^{-7}:\\
\;\;\;\;\frac{t_2}{1 - t_2 \cdot t_0} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} - {\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(t_0 \cdot 0.16666666666666666 - t_0 \cdot 0.5\right)}{\cos x} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{t_1}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x\\


\end{array}

Error

Target

Original37.4
Target15.5
Herbie0.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.11399096168961476

    1. Initial program 30.2

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

    2. Applied egg-rr0.4

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

    3. Applied egg-rr18.7

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\tan x + \tan \varepsilon\right)} - 1\right)} \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

    4. Applied egg-rr0.4

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

    if -0.11399096168961476 < eps < 2.0687464007627642e-7

    1. Initial program 45.2

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

    2. Applied egg-rr44.4

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

    3. Taylor expanded in x around inf 44.4

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

    4. Simplified26.0

      \[\leadsto \color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right)} \]
      Proof
      (+.f64 (/.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (-.f64 1 (*.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (/.f64 (sin.f64 x) (cos.f64 x))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (*.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (/.f64 (sin.f64 x) (cos.f64 x))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (sin.f64 eps) (sin.f64 x)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (*.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (/.f64 (sin.f64 x) (cos.f64 x))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 3 points increase in error, 5 points decrease in error
      (+.f64 (/.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (-.f64 1 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (sin.f64 eps))) (*.f64 (cos.f64 eps) (cos.f64 x))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (*.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (/.f64 (sin.f64 x) (cos.f64 x))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x))))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (*.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (/.f64 (sin.f64 x) (cos.f64 x))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 6 points increase in error, 6 points decrease in error
      (+.f64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (sin.f64 eps) (sin.f64 x)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 8 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (sin.f64 eps))) (*.f64 (cos.f64 eps) (cos.f64 x))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (-.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (sin.f64 x) (*.f64 (cos.f64 x) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x))))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 11 points increase in error, 5 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (/.f64 (sin.f64 x) (*.f64 (cos.f64 x) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x))))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 109 points increase in error, 3 points decrease in error

    5. Taylor expanded in eps around 0 0.3

      \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \color{blue}{\left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\right)} \]

    if 2.0687464007627642e-7 < eps

    1. Initial program 29.9

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

    2. Applied egg-rr0.4

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

    3. Applied egg-rr0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.11399096168961476:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.0687464007627642 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} - {\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(\frac{\sin x}{\cos x} \cdot 0.16666666666666666 - \frac{\sin x}{\cos x} \cdot 0.5\right)}{\cos x} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Error0.5
Cost32968
\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.8111344076700385 \cdot 10^{-8}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.676746825118407 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]
Alternative 2
Error14.8
Cost13448
\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -7.077230593705174 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.0687464007627642 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error44.5
Cost64
\[\varepsilon \]

Error

Reproduce

herbie shell --seed 2022289 
(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)))