2tan (problem 3.3.2)

Average Error: 36.6 → 13.4

Time: 38.9s

Precision: binary64

Cost: 268744

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := \frac{\sin x}{\cos x}\\ t_2 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_3 := t_2 - -1\\ t_4 := 0.16666666666666666 + \left(0.16666666666666666 \cdot t_2 + t_3 \cdot \left(-0.5 + \left(-t_2\right)\right)\right)\\ \mathbf{if}\;\varepsilon \leq -0.0075:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.018:\\ \;\;\;\;t_3 \cdot \left(\varepsilon + t_1 \cdot {\varepsilon}^{2}\right) + \left(-\left({\varepsilon}^{3} \cdot t_4 + \left(\sin x \cdot \frac{t_4}{\cos x} + \left(t_1 \cdot t_3\right) \cdot -0.3333333333333333\right) \cdot {\varepsilon}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps)))
        (t_1 (/ (sin x) (cos x)))
        (t_2 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_3 (- t_2 -1.0))
        (t_4
         (+
          0.16666666666666666
          (+ (* 0.16666666666666666 t_2) (* t_3 (+ -0.5 (- t_2)))))))
   (if (<= eps -0.0075)
     t_0
     (if (<= eps 0.018)
       (+
        (* t_3 (+ eps (* t_1 (pow eps 2.0))))
        (-
         (+
          (* (pow eps 3.0) t_4)
          (*
           (+ (* (sin x) (/ t_4 (cos x))) (* (* t_1 t_3) -0.3333333333333333))
           (pow eps 4.0)))))
       (- t_0 (tan x))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double t_1 = sin(x) / cos(x);
	double t_2 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_3 = t_2 - -1.0;
	double t_4 = 0.16666666666666666 + ((0.16666666666666666 * t_2) + (t_3 * (-0.5 + -t_2)));
	double tmp;
	if (eps <= -0.0075) {
		tmp = t_0;
	} else if (eps <= 0.018) {
		tmp = (t_3 * (eps + (t_1 * pow(eps, 2.0)))) + -((pow(eps, 3.0) * t_4) + (((sin(x) * (t_4 / cos(x))) + ((t_1 * t_3) * -0.3333333333333333)) * pow(eps, 4.0)));
	} else {
		tmp = t_0 - 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sin(eps) / cos(eps)
    t_1 = sin(x) / cos(x)
    t_2 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    t_3 = t_2 - (-1.0d0)
    t_4 = 0.16666666666666666d0 + ((0.16666666666666666d0 * t_2) + (t_3 * ((-0.5d0) + -t_2)))
    if (eps <= (-0.0075d0)) then
        tmp = t_0
    else if (eps <= 0.018d0) then
        tmp = (t_3 * (eps + (t_1 * (eps ** 2.0d0)))) + -(((eps ** 3.0d0) * t_4) + (((sin(x) * (t_4 / cos(x))) + ((t_1 * t_3) * (-0.3333333333333333d0))) * (eps ** 4.0d0)))
    else
        tmp = t_0 - 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.sin(eps) / Math.cos(eps);
	double t_1 = Math.sin(x) / Math.cos(x);
	double t_2 = Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	double t_3 = t_2 - -1.0;
	double t_4 = 0.16666666666666666 + ((0.16666666666666666 * t_2) + (t_3 * (-0.5 + -t_2)));
	double tmp;
	if (eps <= -0.0075) {
		tmp = t_0;
	} else if (eps <= 0.018) {
		tmp = (t_3 * (eps + (t_1 * Math.pow(eps, 2.0)))) + -((Math.pow(eps, 3.0) * t_4) + (((Math.sin(x) * (t_4 / Math.cos(x))) + ((t_1 * t_3) * -0.3333333333333333)) * Math.pow(eps, 4.0)));
	} else {
		tmp = t_0 - 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.sin(eps) / math.cos(eps)
	t_1 = math.sin(x) / math.cos(x)
	t_2 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	t_3 = t_2 - -1.0
	t_4 = 0.16666666666666666 + ((0.16666666666666666 * t_2) + (t_3 * (-0.5 + -t_2)))
	tmp = 0
	if eps <= -0.0075:
		tmp = t_0
	elif eps <= 0.018:
		tmp = (t_3 * (eps + (t_1 * math.pow(eps, 2.0)))) + -((math.pow(eps, 3.0) * t_4) + (((math.sin(x) * (t_4 / math.cos(x))) + ((t_1 * t_3) * -0.3333333333333333)) * math.pow(eps, 4.0)))
	else:
		tmp = t_0 - 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(sin(eps) / cos(eps))
	t_1 = Float64(sin(x) / cos(x))
	t_2 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_3 = Float64(t_2 - -1.0)
	t_4 = Float64(0.16666666666666666 + Float64(Float64(0.16666666666666666 * t_2) + Float64(t_3 * Float64(-0.5 + Float64(-t_2)))))
	tmp = 0.0
	if (eps <= -0.0075)
		tmp = t_0;
	elseif (eps <= 0.018)
		tmp = Float64(Float64(t_3 * Float64(eps + Float64(t_1 * (eps ^ 2.0)))) + Float64(-Float64(Float64((eps ^ 3.0) * t_4) + Float64(Float64(Float64(sin(x) * Float64(t_4 / cos(x))) + Float64(Float64(t_1 * t_3) * -0.3333333333333333)) * (eps ^ 4.0)))));
	else
		tmp = Float64(t_0 - 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 = sin(eps) / cos(eps);
	t_1 = sin(x) / cos(x);
	t_2 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	t_3 = t_2 - -1.0;
	t_4 = 0.16666666666666666 + ((0.16666666666666666 * t_2) + (t_3 * (-0.5 + -t_2)));
	tmp = 0.0;
	if (eps <= -0.0075)
		tmp = t_0;
	elseif (eps <= 0.018)
		tmp = (t_3 * (eps + (t_1 * (eps ^ 2.0)))) + -(((eps ^ 3.0) * t_4) + (((sin(x) * (t_4 / cos(x))) + ((t_1 * t_3) * -0.3333333333333333)) * (eps ^ 4.0)));
	else
		tmp = t_0 - 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[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(0.16666666666666666 + N[(N[(0.16666666666666666 * t$95$2), $MachinePrecision] + N[(t$95$3 * N[(-0.5 + (-t$95$2)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0075], t$95$0, If[LessEqual[eps, 0.018], N[(N[(t$95$3 * N[(eps + N[(t$95$1 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(N[(N[Power[eps, 3.0], $MachinePrecision] * t$95$4), $MachinePrecision] + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$4 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * t$95$3), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(t$95$0 - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	36.6
Target	14.1
Herbie	13.4

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

Derivation

1. Split input into 3 regimes

2. if eps < -0.0074999999999999997

   1. Initial program 28.9

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

   2. Applied egg-rr 29.1

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

   3. Taylor expanded in x around 0 27.1

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

   if -0.0074999999999999997 < eps < 0.0179999999999999986

   1. Initial program 44.3

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

   2. Taylor expanded in eps around 0 0.3

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + \left(-1 \cdot \left(\left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{\left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \cdot \sin x}{\cos x} + 0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) \cdot {\varepsilon}^{4}\right) + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right)\right)} \]

   3. Simplified 0.3

      \[\leadsto \color{blue}{\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot \left(\varepsilon + \frac{\sin x}{\cos x} \cdot {\varepsilon}^{2}\right) + \left(-\left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot \left(-0.5 + \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(\sin x \cdot \frac{0.16666666666666666 + \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot \left(-0.5 + \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x} + \left(\frac{\sin x}{\cos x} \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right)\right) \cdot -0.3333333333333333\right) \cdot {\varepsilon}^{4}\right)\right)} \]
      Proof

      [Start] 0.3

      \[ \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + \left(-1 \cdot \left(\left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{\left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \cdot \sin x}{\cos x} + 0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) \cdot {\varepsilon}^{4}\right) + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   if 0.0179999999999999986 < eps

   1. Initial program 28.4

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

   2. Taylor expanded in x around 0 27.1

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} - \tan x \]
3. Recombined 3 regimes into one program.

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0075:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{elif}\;\varepsilon \leq 0.018:\\ \;\;\;\;\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot \left(\varepsilon + \frac{\sin x}{\cos x} \cdot {\varepsilon}^{2}\right) + \left(-\left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot \left(-0.5 + \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(\sin x \cdot \frac{0.16666666666666666 + \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot \left(-0.5 + \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x} + \left(\frac{\sin x}{\cos x} \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right)\right) \cdot -0.3333333333333333\right) \cdot {\varepsilon}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Error	13.5
Cost	131272

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_2 := t_1 - -1\\ \mathbf{if}\;\varepsilon \leq -0.007:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0033:\\ \;\;\;\;t_2 \cdot \left(\varepsilon + \frac{\sin x}{\cos x} \cdot {\varepsilon}^{2}\right) + {\varepsilon}^{3} \cdot \left(-\left(0.16666666666666666 + \left(0.16666666666666666 \cdot t_1 + t_2 \cdot \left(-0.5 + \left(-t_1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \tan x\\ \end{array} \]

Alternative 2
Error	13.6
Cost	46088

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -0.007:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.01:\\ \;\;\;\;\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot \left(\varepsilon + \frac{\sin x}{\cos x} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \tan x\\ \end{array} \]

Alternative 3
Error	13.7
Cost	26440

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -0.007:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \tan x\\ \end{array} \]

Alternative 4
Error	13.7
Cost	26440

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -0.007:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \tan x\\ \end{array} \]

Alternative 5
Error	26.7
Cost	12992

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

Alternative 6
Error	27.9
Cost	7112

\[\begin{array}{l} t_0 := \left(\tan \left(x + \varepsilon\right) + 1\right) + -1\\ \mathbf{if}\;\varepsilon \leq -0.007:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 235000000000:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	41.4
Cost	6592

\[1 \cdot \sin \varepsilon \]

Alternative 8
Error	43.8
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023064 
(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)))