2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.7%

Time: 25.8s

Alternatives: 10

Speedup: 205.0×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{\cos x}\\ \frac{\varepsilon \cdot \left(t\_0 + \left(\cos x + {\varepsilon}^{2} \cdot \left(\left(\cos x \cdot 0.3333333333333333 + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + t\_0\right)\right)\right) + -0.3333333333333333 \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{2} - 0.5}{\cos x}\right)\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (cos x))))
   (/
    (*
     eps
     (+
      t_0
      (+
       (cos x)
       (*
        (pow eps 2.0)
        (+
         (+
          (* (cos x) 0.3333333333333333)
          (* (pow eps 2.0) (* 0.13333333333333333 (+ (cos x) t_0))))
         (*
          -0.3333333333333333
          (/ (- (/ (cos (* x 2.0)) 2.0) 0.5) (cos x))))))))
    (* (cos x) (- 1.0 (* (tan x) (tan eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / cos(x);
	return (eps * (t_0 + (cos(x) + (pow(eps, 2.0) * (((cos(x) * 0.3333333333333333) + (pow(eps, 2.0) * (0.13333333333333333 * (cos(x) + t_0)))) + (-0.3333333333333333 * (((cos((x * 2.0)) / 2.0) - 0.5) / cos(x)))))))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(x) ** 2.0d0) / cos(x)
    code = (eps * (t_0 + (cos(x) + ((eps ** 2.0d0) * (((cos(x) * 0.3333333333333333d0) + ((eps ** 2.0d0) * (0.13333333333333333d0 * (cos(x) + t_0)))) + ((-0.3333333333333333d0) * (((cos((x * 2.0d0)) / 2.0d0) - 0.5d0) / cos(x)))))))) / (cos(x) * (1.0d0 - (tan(x) * tan(eps))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0) / Math.cos(x);
	return (eps * (t_0 + (Math.cos(x) + (Math.pow(eps, 2.0) * (((Math.cos(x) * 0.3333333333333333) + (Math.pow(eps, 2.0) * (0.13333333333333333 * (Math.cos(x) + t_0)))) + (-0.3333333333333333 * (((Math.cos((x * 2.0)) / 2.0) - 0.5) / Math.cos(x)))))))) / (Math.cos(x) * (1.0 - (Math.tan(x) * Math.tan(eps))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.cos(x)
	return (eps * (t_0 + (math.cos(x) + (math.pow(eps, 2.0) * (((math.cos(x) * 0.3333333333333333) + (math.pow(eps, 2.0) * (0.13333333333333333 * (math.cos(x) + t_0)))) + (-0.3333333333333333 * (((math.cos((x * 2.0)) / 2.0) - 0.5) / math.cos(x)))))))) / (math.cos(x) * (1.0 - (math.tan(x) * math.tan(eps))))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / cos(x))
	return Float64(Float64(eps * Float64(t_0 + Float64(cos(x) + Float64((eps ^ 2.0) * Float64(Float64(Float64(cos(x) * 0.3333333333333333) + Float64((eps ^ 2.0) * Float64(0.13333333333333333 * Float64(cos(x) + t_0)))) + Float64(-0.3333333333333333 * Float64(Float64(Float64(cos(Float64(x * 2.0)) / 2.0) - 0.5) / cos(x)))))))) / Float64(cos(x) * Float64(1.0 - Float64(tan(x) * tan(eps)))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / cos(x);
	tmp = (eps * (t_0 + (cos(x) + ((eps ^ 2.0) * (((cos(x) * 0.3333333333333333) + ((eps ^ 2.0) * (0.13333333333333333 * (cos(x) + t_0)))) + (-0.3333333333333333 * (((cos((x * 2.0)) / 2.0) - 0.5) / cos(x)))))))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(t$95$0 + N[(N[Cos[x], $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(0.13333333333333333 * N[(N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 59.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tan-sum 59.7%

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

   2. tan-quot 59.7%

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

   3. frac-sub 59.8%

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

4. Applied egg-rr 59.8%

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

5. Taylor expanded in eps around 0 100.0%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. pow1 100.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + \color{blue}{{\left({\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)}^{1}}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

   2. cancel-sign-sub-inv 100.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\left({\varepsilon}^{2} \cdot \color{blue}{\left(0.13333333333333333 \cdot \cos x + \left(--0.13333333333333333\right) \cdot \frac{{\sin x}^{2}}{\cos x}\right)}\right)}^{1}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

   3. *-commutative 100.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\left({\varepsilon}^{2} \cdot \left(\color{blue}{\cos x \cdot 0.13333333333333333} + \left(--0.13333333333333333\right) \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)}^{1}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

   4. fma-define 100.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\left({\varepsilon}^{2} \cdot \color{blue}{\mathsf{fma}\left(\cos x, 0.13333333333333333, \left(--0.13333333333333333\right) \cdot \frac{{\sin x}^{2}}{\cos x}\right)}\right)}^{1}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

   5. metadata-eval 100.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\left({\varepsilon}^{2} \cdot \mathsf{fma}\left(\cos x, 0.13333333333333333, \color{blue}{0.13333333333333333} \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)}^{1}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

7. Applied egg-rr 100.0%

   \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + \color{blue}{{\left({\varepsilon}^{2} \cdot \mathsf{fma}\left(\cos x, 0.13333333333333333, 0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)}^{1}}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. unpow1 100.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + \color{blue}{{\varepsilon}^{2} \cdot \mathsf{fma}\left(\cos x, 0.13333333333333333, 0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

   2. fma-undefine 100.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \color{blue}{\left(\cos x \cdot 0.13333333333333333 + 0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

   3. *-commutative 100.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(\color{blue}{0.13333333333333333 \cdot \cos x} + 0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

   4. distribute-lft-out 100.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \color{blue}{\left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

9. Simplified 100.0%

   \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + \color{blue}{{\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
10. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right) - -0.3333333333333333 \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    2. sin-mult 100.0%

       \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right) - -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right) - -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
12. Step-by-step derivation

    1. div-sub 100.0%

       \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right) - -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    2. +-inverses 100.0%

       \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right) - -0.3333333333333333 \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    3. cos-0 100.0%

       \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right) - -0.3333333333333333 \cdot \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    4. metadata-eval 100.0%

       \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right) - -0.3333333333333333 \cdot \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    5. count-2 100.0%

       \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right) - -0.3333333333333333 \cdot \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

13. Simplified 100.0%

    \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right) - -0.3333333333333333 \cdot \frac{\color{blue}{0.5 - \frac{\cos \left(2 \cdot x\right)}{2}}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

14. Final simplification 100.0%

    \[\leadsto \frac{\varepsilon \cdot \left(\frac{{\sin x}^{2}}{\cos x} + \left(\cos x + {\varepsilon}^{2} \cdot \left(\left(\cos x \cdot 0.3333333333333333 + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right) + -0.3333333333333333 \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{2} - 0.5}{\cos x}\right)\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
15. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{\cos x}\\ \frac{\varepsilon \cdot \left(t\_0 - \left({\varepsilon}^{2} \cdot \left(t\_0 \cdot -0.3333333333333333 - \left(\cos x \cdot 0.3333333333333333 + {\varepsilon}^{2} \cdot 0.13333333333333333\right)\right) - \cos x\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (cos x))))
   (/
    (*
     eps
     (-
      t_0
      (-
       (*
        (pow eps 2.0)
        (-
         (* t_0 -0.3333333333333333)
         (+
          (* (cos x) 0.3333333333333333)
          (* (pow eps 2.0) 0.13333333333333333))))
       (cos x))))
    (* (cos x) (- 1.0 (* (tan x) (tan eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / cos(x);
	return (eps * (t_0 - ((pow(eps, 2.0) * ((t_0 * -0.3333333333333333) - ((cos(x) * 0.3333333333333333) + (pow(eps, 2.0) * 0.13333333333333333)))) - cos(x)))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(x) ** 2.0d0) / cos(x)
    code = (eps * (t_0 - (((eps ** 2.0d0) * ((t_0 * (-0.3333333333333333d0)) - ((cos(x) * 0.3333333333333333d0) + ((eps ** 2.0d0) * 0.13333333333333333d0)))) - cos(x)))) / (cos(x) * (1.0d0 - (tan(x) * tan(eps))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0) / Math.cos(x);
	return (eps * (t_0 - ((Math.pow(eps, 2.0) * ((t_0 * -0.3333333333333333) - ((Math.cos(x) * 0.3333333333333333) + (Math.pow(eps, 2.0) * 0.13333333333333333)))) - Math.cos(x)))) / (Math.cos(x) * (1.0 - (Math.tan(x) * Math.tan(eps))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.cos(x)
	return (eps * (t_0 - ((math.pow(eps, 2.0) * ((t_0 * -0.3333333333333333) - ((math.cos(x) * 0.3333333333333333) + (math.pow(eps, 2.0) * 0.13333333333333333)))) - math.cos(x)))) / (math.cos(x) * (1.0 - (math.tan(x) * math.tan(eps))))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / cos(x))
	return Float64(Float64(eps * Float64(t_0 - Float64(Float64((eps ^ 2.0) * Float64(Float64(t_0 * -0.3333333333333333) - Float64(Float64(cos(x) * 0.3333333333333333) + Float64((eps ^ 2.0) * 0.13333333333333333)))) - cos(x)))) / Float64(cos(x) * Float64(1.0 - Float64(tan(x) * tan(eps)))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / cos(x);
	tmp = (eps * (t_0 - (((eps ^ 2.0) * ((t_0 * -0.3333333333333333) - ((cos(x) * 0.3333333333333333) + ((eps ^ 2.0) * 0.13333333333333333)))) - cos(x)))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(t$95$0 - N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(t$95$0 * -0.3333333333333333), $MachinePrecision] - N[(N[(N[Cos[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 59.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tan-sum 59.7%

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

   2. tan-quot 59.7%

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

   3. frac-sub 59.8%

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

4. Applied egg-rr 59.8%

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

5. Taylor expanded in eps around 0 100.0%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

6. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + \color{blue}{0.13333333333333333 \cdot {\varepsilon}^{2}}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
7. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + \color{blue}{{\varepsilon}^{2} \cdot 0.13333333333333333}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

8. Simplified 100.0%

   \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + \color{blue}{{\varepsilon}^{2} \cdot 0.13333333333333333}\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

9. Final simplification 100.0%

   \[\leadsto \frac{\varepsilon \cdot \left(\frac{{\sin x}^{2}}{\cos x} - \left({\varepsilon}^{2} \cdot \left(\frac{{\sin x}^{2}}{\cos x} \cdot -0.3333333333333333 - \left(\cos x \cdot 0.3333333333333333 + {\varepsilon}^{2} \cdot 0.13333333333333333\right)\right) - \cos x\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
10. Add Preprocessing

Alternative 3: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{\cos x}\\ \frac{\varepsilon \cdot \left(t\_0 + \left(\cos x + {\varepsilon}^{2} \cdot \left(0.3333333333333333 \cdot \left(\cos x + t\_0\right)\right)\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (cos x))))
   (/
    (*
     eps
     (+
      t_0
      (+ (cos x) (* (pow eps 2.0) (* 0.3333333333333333 (+ (cos x) t_0))))))
    (* (cos x) (- 1.0 (* (tan x) (tan eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / cos(x);
	return (eps * (t_0 + (cos(x) + (pow(eps, 2.0) * (0.3333333333333333 * (cos(x) + t_0)))))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(x) ** 2.0d0) / cos(x)
    code = (eps * (t_0 + (cos(x) + ((eps ** 2.0d0) * (0.3333333333333333d0 * (cos(x) + t_0)))))) / (cos(x) * (1.0d0 - (tan(x) * tan(eps))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0) / Math.cos(x);
	return (eps * (t_0 + (Math.cos(x) + (Math.pow(eps, 2.0) * (0.3333333333333333 * (Math.cos(x) + t_0)))))) / (Math.cos(x) * (1.0 - (Math.tan(x) * Math.tan(eps))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.cos(x)
	return (eps * (t_0 + (math.cos(x) + (math.pow(eps, 2.0) * (0.3333333333333333 * (math.cos(x) + t_0)))))) / (math.cos(x) * (1.0 - (math.tan(x) * math.tan(eps))))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / cos(x))
	return Float64(Float64(eps * Float64(t_0 + Float64(cos(x) + Float64((eps ^ 2.0) * Float64(0.3333333333333333 * Float64(cos(x) + t_0)))))) / Float64(cos(x) * Float64(1.0 - Float64(tan(x) * tan(eps)))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / cos(x);
	tmp = (eps * (t_0 + (cos(x) + ((eps ^ 2.0) * (0.3333333333333333 * (cos(x) + t_0)))))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(t$95$0 + N[(N[Cos[x], $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(0.3333333333333333 * N[(N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 59.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tan-sum 59.7%

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

   2. tan-quot 59.7%

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

   3. frac-sub 59.8%

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

4. Applied egg-rr 59.8%

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

5. Taylor expanded in eps around 0 100.0%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

6. Taylor expanded in eps around 0 99.9%

   \[\leadsto \frac{\varepsilon \cdot \left(\left(\cos x + \color{blue}{{\varepsilon}^{2} \cdot \left(0.3333333333333333 \cdot \cos x - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
7. Step-by-step derivation

   1. cancel-sign-sub-inv 99.9%

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

   2. metadata-eval 99.9%

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

   3. distribute-lft-out 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \frac{\varepsilon \cdot \left(\frac{{\sin x}^{2}}{\cos x} + \left(\cos x + {\varepsilon}^{2} \cdot \left(0.3333333333333333 \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
10. Add Preprocessing

Alternative 4: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* eps (+ (cos x) (/ (pow (sin x) 2.0) (cos x))))
  (* (cos x) (- 1.0 (* (tan x) (tan eps))))))

C

double code(double x, double eps) {
	return (eps * (cos(x) + (pow(sin(x), 2.0) / cos(x)))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * (cos(x) + ((sin(x) ** 2.0d0) / cos(x)))) / (cos(x) * (1.0d0 - (tan(x) * tan(eps))))
end function

Java

public static double code(double x, double eps) {
	return (eps * (Math.cos(x) + (Math.pow(Math.sin(x), 2.0) / Math.cos(x)))) / (Math.cos(x) * (1.0 - (Math.tan(x) * Math.tan(eps))));
}

Python

def code(x, eps):
	return (eps * (math.cos(x) + (math.pow(math.sin(x), 2.0) / math.cos(x)))) / (math.cos(x) * (1.0 - (math.tan(x) * math.tan(eps))))

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(cos(x) + Float64((sin(x) ^ 2.0) / cos(x)))) / Float64(cos(x) * Float64(1.0 - Float64(tan(x) * tan(eps)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (cos(x) + ((sin(x) ^ 2.0) / cos(x)))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tan-sum 59.7%

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

   2. tan-quot 59.7%

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

   3. frac-sub 59.8%

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

4. Applied egg-rr 59.8%

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

5. Taylor expanded in eps around 0 99.6%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\cos x - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 99.6%

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

   2. metadata-eval 99.6%

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

   3. *-lft-identity 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
9. Add Preprocessing

Alternative 5: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0)))

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
}

Python

def code(x, eps):
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) \]
7. Add Preprocessing

Alternative 6: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* eps (* (pow (sin x) 2.0) (pow (cos x) -2.0)))))

C

double code(double x, double eps) {
	return eps + (eps * (pow(sin(x), 2.0) * pow(cos(x), -2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * ((sin(x) ** 2.0d0) * (cos(x) ** (-2.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * (Math.pow(Math.sin(x), 2.0) * Math.pow(Math.cos(x), -2.0)));
}

Python

def code(x, eps):
	return eps + (eps * (math.pow(math.sin(x), 2.0) * math.pow(math.cos(x), -2.0)))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) * (cos(x) ^ -2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * ((sin(x) ^ 2.0) * (cos(x) ^ -2.0)));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 23.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\tan \left(x + \varepsilon\right) - \tan x\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)}} \]

   2. pow3 23.5%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}}} \]

4. Applied egg-rr 23.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}}} \]

5. Taylor expanded in eps around 0 35.5%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}}^{3}} \]
6. Step-by-step derivation

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}}^{3}} \]
8. Step-by-step derivation

   1. rem-cbrt-cube 99.0%

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

   2. distribute-rgt-in 99.0%

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

   3. *-un-lft-identity 99.0%

      \[\leadsto \color{blue}{\varepsilon} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon \]

   4. div-inv 99.0%

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

   5. pow-flip 99.0%

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

   6. metadata-eval 99.0%

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

9. Applied egg-rr 99.0%

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

10. Final simplification 99.0%

    \[\leadsto \varepsilon + \varepsilon \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) \]
11. Add Preprocessing

Alternative 7: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon + {x}^{2} \cdot \left(\varepsilon + {x}^{2} \cdot \left(0.37777777777777777 \cdot \left(\varepsilon \cdot {x}^{2}\right) + \varepsilon \cdot 0.6666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  eps
  (*
   (pow x 2.0)
   (+
    eps
    (*
     (pow x 2.0)
     (+
      (* 0.37777777777777777 (* eps (pow x 2.0)))
      (* eps 0.6666666666666666)))))))

C

double code(double x, double eps) {
	return eps + (pow(x, 2.0) * (eps + (pow(x, 2.0) * ((0.37777777777777777 * (eps * pow(x, 2.0))) + (eps * 0.6666666666666666)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + ((x ** 2.0d0) * (eps + ((x ** 2.0d0) * ((0.37777777777777777d0 * (eps * (x ** 2.0d0))) + (eps * 0.6666666666666666d0)))))
end function

Java

public static double code(double x, double eps) {
	return eps + (Math.pow(x, 2.0) * (eps + (Math.pow(x, 2.0) * ((0.37777777777777777 * (eps * Math.pow(x, 2.0))) + (eps * 0.6666666666666666)))));
}

Python

def code(x, eps):
	return eps + (math.pow(x, 2.0) * (eps + (math.pow(x, 2.0) * ((0.37777777777777777 * (eps * math.pow(x, 2.0))) + (eps * 0.6666666666666666)))))

Julia

function code(x, eps)
	return Float64(eps + Float64((x ^ 2.0) * Float64(eps + Float64((x ^ 2.0) * Float64(Float64(0.37777777777777777 * Float64(eps * (x ^ 2.0))) + Float64(eps * 0.6666666666666666))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + ((x ^ 2.0) * (eps + ((x ^ 2.0) * ((0.37777777777777777 * (eps * (x ^ 2.0))) + (eps * 0.6666666666666666)))));
end

Wolfram

code[x_, eps_] := N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] * N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.37777777777777777 * N[(eps * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 23.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\tan \left(x + \varepsilon\right) - \tan x\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)}} \]

   2. pow3 23.5%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}}} \]

4. Applied egg-rr 23.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}}} \]

5. Taylor expanded in eps around 0 35.5%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}}^{3}} \]
6. Step-by-step derivation

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

8. Taylor expanded in x around 0 97.6%

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\varepsilon + {x}^{2} \cdot \left(0.37777777777777777 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 0.6666666666666666 \cdot \varepsilon\right)\right)} \]

9. Final simplification 97.6%

   \[\leadsto \varepsilon + {x}^{2} \cdot \left(\varepsilon + {x}^{2} \cdot \left(0.37777777777777777 \cdot \left(\varepsilon \cdot {x}^{2}\right) + \varepsilon \cdot 0.6666666666666666\right)\right) \]
10. Add Preprocessing

Alternative 8: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + {x}^{2} \cdot \left(\varepsilon + {x}^{2} \cdot \left(\varepsilon \cdot 0.6666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* (pow x 2.0) (+ eps (* (pow x 2.0) (* eps 0.6666666666666666))))))

C

double code(double x, double eps) {
	return eps + (pow(x, 2.0) * (eps + (pow(x, 2.0) * (eps * 0.6666666666666666))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + ((x ** 2.0d0) * (eps + ((x ** 2.0d0) * (eps * 0.6666666666666666d0))))
end function

Java

public static double code(double x, double eps) {
	return eps + (Math.pow(x, 2.0) * (eps + (Math.pow(x, 2.0) * (eps * 0.6666666666666666))));
}

Python

def code(x, eps):
	return eps + (math.pow(x, 2.0) * (eps + (math.pow(x, 2.0) * (eps * 0.6666666666666666))))

Julia

function code(x, eps)
	return Float64(eps + Float64((x ^ 2.0) * Float64(eps + Float64((x ^ 2.0) * Float64(eps * 0.6666666666666666)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + ((x ^ 2.0) * (eps + ((x ^ 2.0) * (eps * 0.6666666666666666))));
end

Wolfram

code[x_, eps_] := N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] * N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] * N[(eps * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 23.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\tan \left(x + \varepsilon\right) - \tan x\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)}} \]

   2. pow3 23.5%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}}} \]

4. Applied egg-rr 23.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}}} \]

5. Taylor expanded in eps around 0 35.5%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}}^{3}} \]
6. Step-by-step derivation

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

8. Taylor expanded in x around 0 97.6%

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\varepsilon + 0.6666666666666666 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 97.6%

      \[\leadsto \varepsilon + {x}^{2} \cdot \left(\varepsilon + \color{blue}{\left(0.6666666666666666 \cdot \varepsilon\right) \cdot {x}^{2}}\right) \]

   2. *-commutative 97.6%

      \[\leadsto \varepsilon + {x}^{2} \cdot \left(\varepsilon + \color{blue}{\left(\varepsilon \cdot 0.6666666666666666\right)} \cdot {x}^{2}\right) \]

10. Simplified 97.6%

    \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\varepsilon + \left(\varepsilon \cdot 0.6666666666666666\right) \cdot {x}^{2}\right)} \]

11. Final simplification 97.6%

    \[\leadsto \varepsilon + {x}^{2} \cdot \left(\varepsilon + {x}^{2} \cdot \left(\varepsilon \cdot 0.6666666666666666\right)\right) \]
12. Add Preprocessing

Alternative 9: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* eps (pow x 2.0))))

C

double code(double x, double eps) {
	return eps + (eps * pow(x, 2.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * (x ** 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * Math.pow(x, 2.0));
}

Python

def code(x, eps):
	return eps + (eps * math.pow(x, 2.0))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * (x ^ 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (x ^ 2.0));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 23.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\tan \left(x + \varepsilon\right) - \tan x\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)}} \]

   2. pow3 23.5%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}}} \]

4. Applied egg-rr 23.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}}} \]

5. Taylor expanded in eps around 0 35.5%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}}^{3}} \]
6. Step-by-step derivation

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

8. Taylor expanded in x around 0 97.4%

   \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {x}^{2}} \]

9. Final simplification 97.4%

   \[\leadsto \varepsilon + \varepsilon \cdot {x}^{2} \]
10. Add Preprocessing

Alternative 10: 97.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 eps)

C

double code(double x, double eps) {
	return eps;
}

Fortran

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

Java

public static double code(double x, double eps) {
	return eps;
}

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

function tmp = code(x, eps)
	tmp = eps;
end

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 59.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 23.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\tan \left(x + \varepsilon\right) - \tan x\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)}} \]

   2. pow3 23.5%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}}} \]

4. Applied egg-rr 23.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}}} \]

5. Taylor expanded in eps around 0 35.5%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}}^{3}} \]
6. Step-by-step derivation

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

8. Taylor expanded in x around 0 97.1%

   \[\leadsto \color{blue}{\varepsilon} \]

9. Final simplification 97.1%

   \[\leadsto \varepsilon \]
10. Add Preprocessing

Developer target: 99.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

C

double code(double x, double eps) {
	return sin(eps) / (cos(x) * cos((x + eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin(eps) / (cos(x) * cos((x + eps)))
end function

Java

public static double code(double x, double eps) {
	return Math.sin(eps) / (Math.cos(x) * Math.cos((x + eps)));
}

Python

def code(x, eps):
	return math.sin(eps) / (math.cos(x) * math.cos((x + eps)))

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(cos(x) * cos(Float64(x + eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin(eps) / (cos(x) * cos((x + eps)));
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024081 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))