2tan (problem 3.3.2)

Average Error: 36.6 → 0.3

Time: 2.1min

Precision: binary64

Cost: 33416

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin \varepsilon}{\left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) \cdot {\cos x}^{2} - \left(\varepsilon \cdot \cos x\right) \cdot \sin x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))))
   (if (<= eps -8e-5)
     t_0
     (if (<= eps 6.5e-7)
       (/
        (sin eps)
        (-
         (* (+ (* -0.5 (* eps eps)) 1.0) (pow (cos x) 2.0))
         (* (* eps (cos x)) (sin 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 = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	double tmp;
	if (eps <= -8e-5) {
		tmp = t_0;
	} else if (eps <= 6.5e-7) {
		tmp = sin(eps) / ((((-0.5 * (eps * eps)) + 1.0) * pow(cos(x), 2.0)) - ((eps * cos(x)) * sin(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) :: tmp
    t_0 = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    if (eps <= (-8d-5)) then
        tmp = t_0
    else if (eps <= 6.5d-7) then
        tmp = sin(eps) / (((((-0.5d0) * (eps * eps)) + 1.0d0) * (cos(x) ** 2.0d0)) - ((eps * cos(x)) * sin(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.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	double tmp;
	if (eps <= -8e-5) {
		tmp = t_0;
	} else if (eps <= 6.5e-7) {
		tmp = Math.sin(eps) / ((((-0.5 * (eps * eps)) + 1.0) * Math.pow(Math.cos(x), 2.0)) - ((eps * Math.cos(x)) * Math.sin(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.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	tmp = 0
	if eps <= -8e-5:
		tmp = t_0
	elif eps <= 6.5e-7:
		tmp = math.sin(eps) / ((((-0.5 * (eps * eps)) + 1.0) * math.pow(math.cos(x), 2.0)) - ((eps * math.cos(x)) * math.sin(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(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x))
	tmp = 0.0
	if (eps <= -8e-5)
		tmp = t_0;
	elseif (eps <= 6.5e-7)
		tmp = Float64(sin(eps) / Float64(Float64(Float64(Float64(-0.5 * Float64(eps * eps)) + 1.0) * (cos(x) ^ 2.0)) - Float64(Float64(eps * cos(x)) * sin(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 = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	tmp = 0.0;
	if (eps <= -8e-5)
		tmp = t_0;
	elseif (eps <= 6.5e-7)
		tmp = sin(eps) / ((((-0.5 * (eps * eps)) + 1.0) * (cos(x) ^ 2.0)) - ((eps * cos(x)) * sin(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[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -8e-5], t$95$0, If[LessEqual[eps, 6.5e-7], N[(N[Sin[eps], $MachinePrecision] / N[(N[(N[(N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	36.6
Target	15.3
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if eps < -8.00000000000000065e-5 or 6.50000000000000024e-7 < eps

   1. Initial program 29.7

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

   2. Applied egg-rr 0.4

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

   if -8.00000000000000065e-5 < eps < 6.50000000000000024e-7

   1. Initial program 44.0

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

   2. Applied egg-rr 0.5

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

   3. Applied egg-rr 0.2

      \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \left(-\varepsilon\right)\right)} \cdot \cos x} \]

   4. Taylor expanded in eps around 0 0.2

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

   5. Simplified 0.2

      \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) \cdot {\cos x}^{2} - \left(\varepsilon \cdot \cos x\right) \cdot \sin x}} \]
      Proof
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	0.4
Cost	45440

\[\frac{\sin \varepsilon}{\mathsf{fma}\left(\sin \left(-\varepsilon\right), \sin x, \cos x \cdot \cos \varepsilon\right) \cdot \cos x} \]

Alternative 2
Error	0.4
Cost	45440

\[\frac{\sin \varepsilon}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \left(-\varepsilon\right)\right) \cdot \cos x} \]

Alternative 3
Error	0.4
Cost	39104

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

Alternative 4
Error	0.5
Cost	33216

\[\frac{\sin \varepsilon}{\frac{\cos \varepsilon \cdot \left(1 + \cos \left(x + x\right)\right)}{2} - \frac{\sin \varepsilon \cdot \sin \left(x + x\right)}{2}} \]

Alternative 5
Error	0.3
Cost	32968

\[\begin{array}{l} t_0 := \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{if}\;\varepsilon \leq -9.6 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin \varepsilon}{\left(\left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) \cdot \cos x - \varepsilon \cdot \sin x\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	14.7
Cost	19912

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0074:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\varepsilon}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1} - \tan x\\ \end{array} \]

Alternative 7
Error	14.8
Cost	19904

\[\frac{\sin \varepsilon}{\frac{\cos \left(\left(x + \varepsilon\right) + x\right) + \cos \varepsilon}{2}} \]

Alternative 8
Error	14.8
Cost	19904

\[\frac{\frac{\sin \varepsilon}{\cos \left(\varepsilon + \left(x + x\right)\right) + \cos \varepsilon}}{0.5} \]

Alternative 9
Error	15.3
Cost	19648

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

Alternative 10
Error	14.7
Cost	13320

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.002:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\varepsilon}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 11
Error	41.5
Cost	6464

\[\sin \varepsilon \]

Alternative 12
Error	26.5
Cost	6464

\[\tan \varepsilon \]

Alternative 13
Error	61.3
Cost	64

\[0 \]

Alternative 14
Error	44.0
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023033 
(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)))