2tan (problem 3.3.2)

Average Error: 58.01% → 0.78%

Time: 21.8s

Precision: binary64

Cost: 65736

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{-1 + \tan x \cdot \tan \varepsilon}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-17}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

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 (- (tan x))))
   (if (<= eps -2.1e-7)
     (fma t_0 (/ -1.0 (+ -1.0 (* (tan x) (tan eps)))) t_1)
     (if (<= eps 4.5e-17)
       (+
        (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         (* eps eps)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (- t_1 (/ t_0 (fma (tan x) (tan eps) -1.0)))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = -tan(x);
	double tmp;
	if (eps <= -2.1e-7) {
		tmp = fma(t_0, (-1.0 / (-1.0 + (tan(x) * tan(eps)))), t_1);
	} else if (eps <= 4.5e-17) {
		tmp = (eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = t_1 - (t_0 / fma(tan(x), tan(eps), -1.0));
	}
	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) + tan(eps))
	t_1 = Float64(-tan(x))
	tmp = 0.0
	if (eps <= -2.1e-7)
		tmp = fma(t_0, Float64(-1.0 / Float64(-1.0 + Float64(tan(x) * tan(eps)))), t_1);
	elseif (eps <= 4.5e-17)
		tmp = Float64(Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(t_1 - Float64(t_0 / fma(tan(x), tan(eps), -1.0)));
	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[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Tan[x], $MachinePrecision])}, If[LessEqual[eps, -2.1e-7], N[(t$95$0 * N[(-1.0 / N[(-1.0 + N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[eps, 4.5e-17], N[(N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	58.01%
Target	23.66%
Herbie	0.78%

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.1e-7

   1. Initial program 47.29

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

   2. Applied egg-rr 0.68

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

   3. Simplified 0.64

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

      [Start] 0.68	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 0.64	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 0.64	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 0.66

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

   if -2.1e-7 < eps < 4.49999999999999978e-17

   1. Initial program 71.19

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

   2. Applied egg-rr 70.87

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

   3. Taylor expanded in eps around 0 0.4

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

   4. Simplified 0.4

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

      [Start] 0.4	\[ \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + -1 \cdot \left({\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right)\right) \]
      mul-1-neg [=>] 0.4	\[ \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \color{blue}{\left(-{\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right)\right)} \]
      unsub-neg [=>] 0.4	\[ \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - {\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right)} \]
      sub-neg [=>] 0.4	\[ \varepsilon \cdot \color{blue}{\left(1 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} - {\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right) \]
      mul-1-neg [=>] 0.4	\[ \varepsilon \cdot \left(1 + \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) - {\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right) \]
      remove-double-neg [=>] 0.4	\[ \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) - {\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right) \]
      distribute-lft-out [=>] 0.4	\[ \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - {\varepsilon}^{2} \cdot \color{blue}{\left(-1 \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)} \]

   if 4.49999999999999978e-17 < eps

   1. Initial program 44.77

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

   2. Applied egg-rr 1.63

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

   3. Simplified 1.59

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

      [Start] 1.63	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 1.59	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 1.59	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 1.63

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

   5. Simplified 1.58

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

      [Start] 1.63	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{-1 + \tan x \cdot \tan \varepsilon} - \tan x \]
      *-commutative [<=] 1.63	\[ \color{blue}{\frac{-1}{-1 + \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]
      associate-*l/ [=>] 1.59	\[ \color{blue}{\frac{-1 \cdot \left(\tan x + \tan \varepsilon\right)}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      associate-*r/ [<=] 1.59	\[ \color{blue}{-1 \cdot \frac{\tan x + \tan \varepsilon}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      neg-mul-1 [<=] 1.59	\[ \color{blue}{\left(-\frac{\tan x + \tan \varepsilon}{-1 + \tan x \cdot \tan \varepsilon}\right)} - \tan x \]
      distribute-neg-frac [=>] 1.59	\[ \color{blue}{\frac{-\left(\tan x + \tan \varepsilon\right)}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      +-commutative [=>] 1.59	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\tan x \cdot \tan \varepsilon + -1}} - \tan x \]
      metadata-eval [<=] 1.59	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\tan x \cdot \tan \varepsilon + \color{blue}{\left(-1\right)}} - \tan x \]
      sub-neg [<=] 1.59	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\tan x \cdot \tan \varepsilon - 1}} - \tan x \]
      fma-neg [=>] 1.58	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}} - \tan x \]
      metadata-eval [=>] 1.58	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, \color{blue}{-1}\right)} - \tan x \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.78

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{-1 + \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-17}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8%
Cost	39305

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-17}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 2
Error	0.8%
Cost	39304

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{-1 + \tan x \cdot \tan \varepsilon}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-17}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 3
Error	0.81%
Cost	33096

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-17}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

Alternative 4
Error	0.8%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 5
Error	22.74%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-17}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 6
Error	22.72%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-17}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 7
Error	42.23%
Cost	6464

\[\tan \varepsilon \]

Alternative 8
Error	69.44%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023121 
(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)))