2tan (problem 3.3.2)

Average Error: 42.2% → 99.5%

Time: 22.5s

Precision: binary64

Cost: 117512.00

Math

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

↓

\[\begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ t_2 := \tan x + \tan \varepsilon\\ t_3 := 1 + t_0\\ \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_2}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(0.5, {t_3}^{2}, t_3 \cdot \frac{\sin x}{\cos x}\right), \varepsilon + \varepsilon \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_1 (- 1.0 (* (tan x) (tan eps))))
        (t_2 (+ (tan x) (tan eps)))
        (t_3 (+ 1.0 t_0)))
   (if (<= eps -1.45e-7)
     (- (/ t_2 t_1) (tan x))
     (if (<= eps 1.85e-7)
       (log1p
        (fma
         (* eps eps)
         (fma 0.5 (pow t_3 2.0) (* t_3 (/ (sin x) (cos x))))
         (+ eps (* eps t_0))))
       (- (* t_2 (/ 1.0 t_1)) (tan x))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double t_2 = tan(x) + tan(eps);
	double t_3 = 1.0 + t_0;
	double tmp;
	if (eps <= -1.45e-7) {
		tmp = (t_2 / t_1) - tan(x);
	} else if (eps <= 1.85e-7) {
		tmp = log1p(fma((eps * eps), fma(0.5, pow(t_3, 2.0), (t_3 * (sin(x) / cos(x)))), (eps + (eps * t_0))));
	} else {
		tmp = (t_2 * (1.0 / t_1)) - 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((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	t_2 = Float64(tan(x) + tan(eps))
	t_3 = Float64(1.0 + t_0)
	tmp = 0.0
	if (eps <= -1.45e-7)
		tmp = Float64(Float64(t_2 / t_1) - tan(x));
	elseif (eps <= 1.85e-7)
		tmp = log1p(fma(Float64(eps * eps), fma(0.5, (t_3 ^ 2.0), Float64(t_3 * Float64(sin(x) / cos(x)))), Float64(eps + Float64(eps * t_0))));
	else
		tmp = Float64(Float64(t_2 * Float64(1.0 / t_1)) - 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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[eps, -1.45e-7], N[(N[(t$95$2 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.85e-7], N[Log[1 + N[(N[(eps * eps), $MachinePrecision] * N[(0.5 * N[Power[t$95$3, 2.0], $MachinePrecision] + N[(t$95$3 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps + N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$2 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	42.2%
Target	77.3%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.4499999999999999e-7

   1. Initial program 54.9

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

   2. Applied egg-rr 99.3

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

   3. Simplified 99.4

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

      [Start] 99.3	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 99.4	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 99.4	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   if -1.4499999999999999e-7 < eps < 1.85000000000000002e-7

   1. Initial program 29.8

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

   2. Applied egg-rr 29.8

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

   3. Taylor expanded in eps around 0 99.6

      \[\leadsto \mathsf{log1p}\left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + 0.5 \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}\right) + \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right) \]

   4. Simplified 99.7

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(0.5, {\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}, \frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right), \varepsilon + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon\right)}\right) \]
      Proof

      [Start] 99.6	\[ \mathsf{log1p}\left({\varepsilon}^{2} \cdot \left(\frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + 0.5 \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}\right) + \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
      fma-def [=>] 99.6	\[ \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + 0.5 \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}, \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right) \]

   if 1.85000000000000002e-7 < eps

   1. Initial program 54.2

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

   2. Applied egg-rr 99.3

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

4. Final simplification 99.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(0.5, {\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}, \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Error	99.5%
Cost	72136.00

\[\begin{array}{l} t_0 := 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{t_2} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t_0, \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{1}{t_2} - \tan x\\ \end{array} \]

Alternative 2
Error	99.5%
Cost	65736.00

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \]

Alternative 3
Error	99.3%
Cost	33096.00

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \]

Alternative 4
Error	99.3%
Cost	32969.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 5
Error	78.4%
Cost	26952.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.95 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333}} - \tan x\\ \end{array} \]

Alternative 6
Error	77.8%
Cost	20360.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 640:\\ \;\;\;\;\varepsilon + \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\varepsilon \cdot {\cos x}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 7
Error	77.8%
Cost	20360.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 640:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 8
Error	77.8%
Cost	19976.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 640:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 9
Error	77.9%
Cost	19976.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 640:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 10
Error	58.2%
Cost	6464.00

\[\tan \varepsilon \]

Alternative 11
Error	31.4%
Cost	64.00

\[\varepsilon \]

Reproduce

herbie shell --seed 2023097 
(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)))