2tan (problem 3.3.2)

Average Error: 36.6 → 14.1

Time: 14.5s

Precision: binary64

Cost: 46408

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \left(x \cdot \left(-\sin \varepsilon\right) + \cos \varepsilon \cdot \left(-0.5 \cdot {x}^{2}\right)\right)} - \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -0.0006:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;-\varepsilon \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (-
          (/
           (sin (+ eps x))
           (+
            (cos eps)
            (+ (* x (- (sin eps))) (* (cos eps) (* -0.5 (pow x 2.0))))))
          (/ (sin x) (cos x)))))
   (if (<= eps -0.0006)
     t_0
     (if (<= eps 1.3e-7)
       (- (* eps (- -1.0 (/ (pow (sin x) 2.0) (pow (cos x) 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 = (sin((eps + x)) / (cos(eps) + ((x * -sin(eps)) + (cos(eps) * (-0.5 * pow(x, 2.0)))))) - (sin(x) / cos(x));
	double tmp;
	if (eps <= -0.0006) {
		tmp = t_0;
	} else if (eps <= 1.3e-7) {
		tmp = -(eps * (-1.0 - (pow(sin(x), 2.0) / pow(cos(x), 2.0))));
	} else {
		tmp = t_0;
	}
	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 = (sin((eps + x)) / (cos(eps) + ((x * -sin(eps)) + (cos(eps) * ((-0.5d0) * (x ** 2.0d0)))))) - (sin(x) / cos(x))
    if (eps <= (-0.0006d0)) then
        tmp = t_0
    else if (eps <= 1.3d-7) then
        tmp = -(eps * ((-1.0d0) - ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0))))
    else
        tmp = t_0
    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.sin((eps + x)) / (Math.cos(eps) + ((x * -Math.sin(eps)) + (Math.cos(eps) * (-0.5 * Math.pow(x, 2.0)))))) - (Math.sin(x) / Math.cos(x));
	double tmp;
	if (eps <= -0.0006) {
		tmp = t_0;
	} else if (eps <= 1.3e-7) {
		tmp = -(eps * (-1.0 - (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	t_0 = (math.sin((eps + x)) / (math.cos(eps) + ((x * -math.sin(eps)) + (math.cos(eps) * (-0.5 * math.pow(x, 2.0)))))) - (math.sin(x) / math.cos(x))
	tmp = 0
	if eps <= -0.0006:
		tmp = t_0
	elif eps <= 1.3e-7:
		tmp = -(eps * (-1.0 - (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0))))
	else:
		tmp = 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(Float64(sin(Float64(eps + x)) / Float64(cos(eps) + Float64(Float64(x * Float64(-sin(eps))) + Float64(cos(eps) * Float64(-0.5 * (x ^ 2.0)))))) - Float64(sin(x) / cos(x)))
	tmp = 0.0
	if (eps <= -0.0006)
		tmp = t_0;
	elseif (eps <= 1.3e-7)
		tmp = Float64(-Float64(eps * Float64(-1.0 - Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))));
	else
		tmp = t_0;
	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 = (sin((eps + x)) / (cos(eps) + ((x * -sin(eps)) + (cos(eps) * (-0.5 * (x ^ 2.0)))))) - (sin(x) / cos(x));
	tmp = 0.0;
	if (eps <= -0.0006)
		tmp = t_0;
	elseif (eps <= 1.3e-7)
		tmp = -(eps * (-1.0 - ((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = t_0;
	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[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] + N[(N[(x * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision] + N[(N[Cos[eps], $MachinePrecision] * N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0006], t$95$0, If[LessEqual[eps, 1.3e-7], (-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]), t$95$0]]]

Target

Original	36.6
Target	14.7
Herbie	14.1

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

Derivation

1. Split input into 2 regimes

2. if eps < -5.99999999999999947e-4 or 1.29999999999999999e-7 < eps

   1. Initial program 29.3

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

   2. Taylor expanded in x around inf 29.4

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

   3. Taylor expanded in x around 0 27.5

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

   4. Simplified 27.5

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

      [Start] 27.5	\[ \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \left(-0.5 \cdot \left(\cos \varepsilon \cdot {x}^{2}\right) + -1 \cdot \left(x \cdot \sin \varepsilon\right)\right)} - \frac{\sin x}{\cos x} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 27.5	\[ \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \color{blue}{\left(-1 \cdot \left(x \cdot \sin \varepsilon\right) + -0.5 \cdot \left(\cos \varepsilon \cdot {x}^{2}\right)\right)}} - \frac{\sin x}{\cos x} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.5	\[ \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \left(\color{blue}{x \cdot \left(-1 \cdot \sin \varepsilon\right)} + -0.5 \cdot \left(\cos \varepsilon \cdot {x}^{2}\right)\right)} - \frac{\sin x}{\cos x} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 27.5	\[ \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \left(x \cdot \color{blue}{\left(\sin \varepsilon \cdot -1\right)} + -0.5 \cdot \left(\cos \varepsilon \cdot {x}^{2}\right)\right)} - \frac{\sin x}{\cos x} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 27.5	\[ \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \left(x \cdot \color{blue}{\left(-\sin \varepsilon\right)} + -0.5 \cdot \left(\cos \varepsilon \cdot {x}^{2}\right)\right)} - \frac{\sin x}{\cos x} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.5	\[ \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \left(x \cdot \left(-\sin \varepsilon\right) + \color{blue}{\cos \varepsilon \cdot \left(-0.5 \cdot {x}^{2}\right)}\right)} - \frac{\sin x}{\cos x} \]

   if -5.99999999999999947e-4 < eps < 1.29999999999999999e-7

   1. Initial program 44.0

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

   2. Taylor expanded in eps around 0 0.6

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

   3. Applied egg-rr 0.6

      \[\leadsto \color{blue}{-\varepsilon \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0006:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \left(x \cdot \left(-\sin \varepsilon\right) + \cos \varepsilon \cdot \left(-0.5 \cdot {x}^{2}\right)\right)} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;-\varepsilon \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \left(x \cdot \left(-\sin \varepsilon\right) + \cos \varepsilon \cdot \left(-0.5 \cdot {x}^{2}\right)\right)} - \frac{\sin x}{\cos x}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.1
Cost	33160

\[\begin{array}{l} t_0 := \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + x \cdot \left(-\sin \varepsilon\right)} - \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;-\varepsilon \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	14.0
Cost	26504

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;-\varepsilon \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	14.0
Cost	26440

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	26.3
Cost	12992

\[\frac{\sin \varepsilon}{\cos \varepsilon} \]

Alternative 5
Error	43.4
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)))