2tan (problem 3.3.2)

Average Error: 37.7 → 0.3

Time: 20.8s

Precision: binary64

Cost: 72008

Math

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

↓

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \varepsilon \cdot \varepsilon, \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{t_1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, t_0\right)\\ \end{array} \]

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))))
   (if (<= eps -2.15e-7)
     (fma t_1 (/ 1.0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) t_0)
     (if (<= eps 3.5e-7)
       (fma
        (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))
        (* eps eps)
        (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))
       (fma
        (fma (tan x) (tan eps) 1.0)
        (/ t_1 (- 1.0 (pow (* (tan x) (tan eps)) 2.0)))
        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 tmp;
	if (eps <= -2.15e-7) {
		tmp = fma(t_1, (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))), t_0);
	} else if (eps <= 3.5e-7) {
		tmp = fma(((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))), (eps * eps), (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))));
	} else {
		tmp = fma(fma(tan(x), tan(eps), 1.0), (t_1 / (1.0 - pow((tan(x) * tan(eps)), 2.0))), 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))
	tmp = 0.0
	if (eps <= -2.15e-7)
		tmp = fma(t_1, Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))), t_0);
	elseif (eps <= 3.5e-7)
		tmp = fma(Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))), Float64(eps * eps), Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))));
	else
		tmp = fma(fma(tan(x), tan(eps), 1.0), Float64(t_1 / Float64(1.0 - (Float64(tan(x) * tan(eps)) ^ 2.0))), 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]}, If[LessEqual[eps, -2.15e-7], N[(t$95$1 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[eps, 3.5e-7], N[(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] * N[(eps * eps), $MachinePrecision] + N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$1 / N[(1.0 - N[Power[N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Target

Original	37.7
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 < -2.1500000000000001e-7

   1. Initial program 30.7

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

   2. Applied egg-rr 8.6

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

   3. Applied egg-rr 1.7

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)\right)} - \tan x \]

   4. Applied egg-rr 0.4

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

   5. Applied egg-rr 0.5

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

   if -2.1500000000000001e-7 < eps < 3.49999999999999984e-7

   1. Initial program 45.7

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

   2. Applied egg-rr 45.2

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

   3. Simplified 45.2

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

      [Start] 45.2	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      swap-sqr [=>] 45.2	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\tan x \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      unpow2 [<=] 45.2	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{{\tan x}^{2}} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

   4. Taylor expanded in eps around 0 0.3

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

   5. Simplified 0.2

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

      [Start] 0.3	\[ \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
      fma-def [=>] 0.3	\[ \color{blue}{\mathsf{fma}\left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}, {\varepsilon}^{2}, \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]
      sub-neg [=>] 0.3	\[ \mathsf{fma}\left(\color{blue}{\frac{\sin x}{\cos x} + \left(--1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)}, {\varepsilon}^{2}, \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
      mul-1-neg [=>] 0.3	\[ \mathsf{fma}\left(\frac{\sin x}{\cos x} + \left(-\color{blue}{\left(-\frac{{\sin x}^{3}}{{\cos x}^{3}}\right)}\right), {\varepsilon}^{2}, \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
      remove-double-neg [=>] 0.3	\[ \mathsf{fma}\left(\frac{\sin x}{\cos x} + \color{blue}{\frac{{\sin x}^{3}}{{\cos x}^{3}}}, {\varepsilon}^{2}, \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
      unpow2 [=>] 0.3	\[ \mathsf{fma}\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \color{blue}{\varepsilon \cdot \varepsilon}, \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
      distribute-lft-in [=>] 0.2	\[ \mathsf{fma}\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \varepsilon \cdot \varepsilon, \color{blue}{\varepsilon \cdot 1 + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]
      *-rgt-identity [=>] 0.2	\[ \mathsf{fma}\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \varepsilon \cdot \varepsilon, \color{blue}{\varepsilon} + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

   if 3.49999999999999984e-7 < eps

   1. Initial program 29.1

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

   2. Applied egg-rr 0.4

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

   3. Simplified 0.5

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

      [Start] 0.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      swap-sqr [=>] 0.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\tan x \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      unpow2 [<=] 0.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{{\tan x}^{2}} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

   4. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \varepsilon \cdot \varepsilon, \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	72008

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{t_1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, t_0\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	65224

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 1.22 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{t_1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, t_0\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	45828

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_0\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	45704

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_1\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	39433

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.9 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \end{array} \]

Alternative 6
Error	0.4
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \end{array} \]

Alternative 7
Error	14.3
Cost	19720

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.26:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 8
Error	14.3
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.26:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot \tan x}{\frac{1}{\tan x}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 9
Error	14.3
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.26:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 10
Error	14.3
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.26:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 11
Error	27.2
Cost	6464

\[\tan \varepsilon \]

Alternative 12
Error	44.2
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023018 
(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)))