2tan (problem 3.3.2)

Average Error: 37.9 → 14.9

Time: 11.9s

Precision: binary64

Cost: 72200

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{if}\;\varepsilon \leq -0.00025:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot t_1 + \frac{{\varepsilon}^{2} \cdot \left(t_1 \cdot \sin x\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps)))
        (t_1 (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))
   (if (<= eps -0.00025)
     t_0
     (if (<= eps 2.8e-13)
       (+ (* eps t_1) (/ (* (pow eps 2.0) (* t_1 (sin x))) (cos x)))
       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) / cos(eps);
	double t_1 = 1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0));
	double tmp;
	if (eps <= -0.00025) {
		tmp = t_0;
	} else if (eps <= 2.8e-13) {
		tmp = (eps * t_1) + ((pow(eps, 2.0) * (t_1 * sin(x))) / cos(x));
	} 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) :: t_1
    real(8) :: tmp
    t_0 = sin(eps) / cos(eps)
    t_1 = 1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0))
    if (eps <= (-0.00025d0)) then
        tmp = t_0
    else if (eps <= 2.8d-13) then
        tmp = (eps * t_1) + (((eps ** 2.0d0) * (t_1 * sin(x))) / cos(x))
    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) / Math.cos(eps);
	double t_1 = 1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0));
	double tmp;
	if (eps <= -0.00025) {
		tmp = t_0;
	} else if (eps <= 2.8e-13) {
		tmp = (eps * t_1) + ((Math.pow(eps, 2.0) * (t_1 * Math.sin(x))) / Math.cos(x));
	} 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) / math.cos(eps)
	t_1 = 1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0))
	tmp = 0
	if eps <= -0.00025:
		tmp = t_0
	elif eps <= 2.8e-13:
		tmp = (eps * t_1) + ((math.pow(eps, 2.0) * (t_1 * math.sin(x))) / math.cos(x))
	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(sin(eps) / cos(eps))
	t_1 = Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))
	tmp = 0.0
	if (eps <= -0.00025)
		tmp = t_0;
	elseif (eps <= 2.8e-13)
		tmp = Float64(Float64(eps * t_1) + Float64(Float64((eps ^ 2.0) * Float64(t_1 * sin(x))) / cos(x)));
	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) / cos(eps);
	t_1 = 1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0));
	tmp = 0.0;
	if (eps <= -0.00025)
		tmp = t_0;
	elseif (eps <= 2.8e-13)
		tmp = (eps * t_1) + (((eps ^ 2.0) * (t_1 * sin(x))) / cos(x));
	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[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00025], t$95$0, If[LessEqual[eps, 2.8e-13], N[(N[(eps * t$95$1), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Target

Original	37.9
Target	15.6
Herbie	14.9

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

Derivation

1. Split input into 2 regimes

2. if eps < -2.5000000000000001e-4 or 2.8000000000000002e-13 < eps

   1. Initial program 30.7

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

   2. Taylor expanded in x around 0 32.8

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

   3. Taylor expanded in x around 0 28.8

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

   if -2.5000000000000001e-4 < eps < 2.8000000000000002e-13

   1. Initial program 45.5

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

   2. Taylor expanded in eps around 0 45.4

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

   3. Simplified 45.4

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

      [Start] 45.4

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

      rational.json-simplify-33 [=>] 45.4

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

   4. Taylor expanded in eps around 0 0.3

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

4. Final simplification 14.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00025:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.0
Cost	26440

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -6.4 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	27.4
Cost	12992

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

Alternative 3
Error	29.5
Cost	6984

\[\begin{array}{l} t_0 := \tan \left(x + \varepsilon\right) - x\\ \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	41.6
Cost	6464

\[\sin \varepsilon \]

Alternative 5
Error	44.1
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023053 
(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)))