2tan (problem 3.3.2)

Average Error: 57.23% → 0.67%

Time: 19.8s

Precision: binary64

Cost: 32968

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -3.5e-9)
     (- (/ t_0 (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))
     (if (<= eps 4.5e-9)
       (* eps (+ 1.0 (/ (- 0.5 (* 0.5 (cos (* x 2.0)))) (pow (cos x) 2.0))))
       (- (/ 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 tmp;
	if (eps <= -3.5e-9) {
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	} else if (eps <= 4.5e-9) {
		tmp = eps * (1.0 + ((0.5 - (0.5 * cos((x * 2.0)))) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    if (eps <= (-3.5d-9)) then
        tmp = (t_0 / (1.0d0 - (tan(x) / (1.0d0 / tan(eps))))) - tan(x)
    else if (eps <= 4.5d-9) then
        tmp = eps * (1.0d0 + ((0.5d0 - (0.5d0 * cos((x * 2.0d0)))) / (cos(x) ** 2.0d0)))
    else
        tmp = (t_0 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

↓

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -3.5e-9) {
		tmp = (t_0 / (1.0 - (Math.tan(x) / (1.0 / Math.tan(eps))))) - Math.tan(x);
	} else if (eps <= 4.5e-9) {
		tmp = eps * (1.0 + ((0.5 - (0.5 * Math.cos((x * 2.0)))) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_0 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

↓

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -3.5e-9:
		tmp = (t_0 / (1.0 - (math.tan(x) / (1.0 / math.tan(eps))))) - math.tan(x)
	elif eps <= 4.5e-9:
		tmp = eps * (1.0 + ((0.5 - (0.5 * math.cos((x * 2.0)))) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_0 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.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))
	tmp = 0.0
	if (eps <= -3.5e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - tan(x));
	elseif (eps <= 4.5e-9)
		tmp = Float64(eps * Float64(1.0 + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x * 2.0)))) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

↓

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -3.5e-9)
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	elseif (eps <= 4.5e-9)
		tmp = eps * (1.0 + ((0.5 - (0.5 * cos((x * 2.0)))) / (cos(x) ^ 2.0)));
	else
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	end
	tmp_2 = 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]}, If[LessEqual[eps, -3.5e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.5e-9], N[(eps * N[(1.0 + N[(N[(0.5 - N[(0.5 * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $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]]]]

Target

Original	57.23%
Target	22.92%
Herbie	0.67%

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

Derivation

1. Split input into 3 regimes

2. if eps < -3.4999999999999999e-9

   1. Initial program 43.89

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

   2. Applied egg-rr 0.72

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

   3. Simplified 0.68

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

      [Start] 0.72	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 0.68	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 0.68	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 0.69

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

   if -3.4999999999999999e-9 < eps < 4.49999999999999976e-9

   1. Initial program 69.08

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

   2. Applied egg-rr 68.53

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

   3. Simplified 68.53

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

      [Start] 68.53	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      swap-sqr [=>] 68.53	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\tan x \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      unpow2 [<=] 68.53	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{{\tan x}^{2}} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

   4. Taylor expanded in eps around 0 0.65

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

   5. Applied egg-rr 0.64

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

   if 4.49999999999999976e-9 < eps

   1. Initial program 47.38

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

   2. Applied egg-rr 0.74

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

   3. Simplified 0.71

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

      [Start] 0.74	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 0.71	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 0.71	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.67

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.67%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 2
Error	21.57%
Cost	26953

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 3
Error	21.81%
Cost	20360

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 4
Error	21.83%
Cost	19976

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 5
Error	40.93%
Cost	6464

\[\tan \varepsilon \]

Alternative 6
Error	67.84%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023090 
(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)))