2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.7%

Time: 25.2s

Alternatives: 7

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: 62.4% 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, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (eps * (cos(x) + ((pow(eps, 2.0) * (0.3333333333333333 * (cos(x) + (pow(sin(x), 2.0) / cos(x))))) + ((0.5 - (cos((x * 2.0)) / 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) + (((eps ** 2.0d0) * (0.3333333333333333d0 * (cos(x) + ((sin(x) ** 2.0d0) / cos(x))))) + ((0.5d0 - (cos((x * 2.0d0)) / 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(eps, 2.0) * (0.3333333333333333 * (Math.cos(x) + (Math.pow(Math.sin(x), 2.0) / Math.cos(x))))) + ((0.5 - (Math.cos((x * 2.0)) / 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(eps, 2.0) * (0.3333333333333333 * (math.cos(x) + (math.pow(math.sin(x), 2.0) / math.cos(x))))) + ((0.5 - (math.cos((x * 2.0)) / 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(Float64((eps ^ 2.0) * Float64(0.3333333333333333 * Float64(cos(x) + Float64((sin(x) ^ 2.0) / cos(x))))) + Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 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) + (((eps ^ 2.0) * (0.3333333333333333 * (cos(x) + ((sin(x) ^ 2.0) / cos(x))))) + ((0.5 - (cos((x * 2.0)) / 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[(N[Power[eps, 2.0], $MachinePrecision] * N[(0.3333333333333333 * N[(N[Cos[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $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 63.1%

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

   1. tan-sum 63.2%

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

   2. tan-quot 63.2%

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

   3. frac-sub 63.2%

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

4. Applied egg-rr 63.2%

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

5. Taylor expanded in eps around 0 100.0%

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

   1. associate--l+ 100.0%

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

   2. cancel-sign-sub-inv 100.0%

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

   3. metadata-eval 100.0%

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

   4. mul-1-neg 100.0%

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

7. Simplified 100.0%

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

   1. unpow2 100.0%

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

   2. sin-mult 100.0%

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

9. Applied egg-rr 100.0%

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

    1. +-inverses 100.0%

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

    2. cos-0 100.0%

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

    3. count-2 100.0%

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

11. Simplified 100.0%

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

    1. sub-neg 100.0%

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

    2. distribute-lft-out 100.0%

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

    3. distribute-neg-frac2 100.0%

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

    4. div-sub 100.0%

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

    5. metadata-eval 100.0%

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

13. Applied egg-rr 100.0%

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

14. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \left(0.5 \cdot \frac{\cos \left(x \cdot 2\right) + -1}{\cos x} - \cos x\right)}{\cos x \cdot \left(\tan x \cdot \tan \varepsilon + -1\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* eps (- (* 0.5 (/ (+ (cos (* x 2.0)) -1.0) (cos x))) (cos x)))
  (* (cos x) (+ (* (tan x) (tan eps)) -1.0))))

C

double code(double x, double eps) {
	return (eps * ((0.5 * ((cos((x * 2.0)) + -1.0) / cos(x))) - cos(x))) / (cos(x) * ((tan(x) * tan(eps)) + -1.0));
}

Fortran

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

Java

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

Python

def code(x, eps):
	return (eps * ((0.5 * ((math.cos((x * 2.0)) + -1.0) / math.cos(x))) - math.cos(x))) / (math.cos(x) * ((math.tan(x) * math.tan(eps)) + -1.0))

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(Float64(0.5 * Float64(Float64(cos(Float64(x * 2.0)) + -1.0) / cos(x))) - cos(x))) / Float64(cos(x) * Float64(Float64(tan(x) * tan(eps)) + -1.0)))
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[(0.5 * N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.1%

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

   1. tan-sum 63.2%

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

   2. tan-quot 63.2%

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

   3. frac-sub 63.2%

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

4. Applied egg-rr 63.2%

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

5. Taylor expanded in eps around 0 100.0%

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

   1. associate--l+ 100.0%

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

   2. cancel-sign-sub-inv 100.0%

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

   3. metadata-eval 100.0%

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

   4. mul-1-neg 100.0%

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

7. Simplified 100.0%

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

   1. unpow2 100.0%

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

   2. sin-mult 100.0%

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

9. Applied egg-rr 100.0%

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

    1. +-inverses 100.0%

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

    2. cos-0 100.0%

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

    3. count-2 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in eps around 0 99.9%

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

13. Final simplification 99.9%

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

Alternative 3: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left({\left({\left(\sqrt[3]{\tan x}\right)}^{2}\right)}^{3} + 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (pow (pow (cbrt (tan x)) 2.0) 3.0) 1.0)))

C

double code(double x, double eps) {
	return eps * (pow(pow(cbrt(tan(x)), 2.0), 3.0) + 1.0);
}

Java

public static double code(double x, double eps) {
	return eps * (Math.pow(Math.pow(Math.cbrt(Math.tan(x)), 2.0), 3.0) + 1.0);
}

Julia

function code(x, eps)
	return Float64(eps * Float64(((cbrt(tan(x)) ^ 2.0) ^ 3.0) + 1.0))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[Power[N[Power[N[Power[N[Tan[x], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. sub-neg 99.5%

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

   2. mul-1-neg 99.5%

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

   3. remove-double-neg 99.5%

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

5. Simplified 99.5%

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

   1. add-cube-cbrt 99.5%

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

   2. pow3 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left({\left({\left(\sqrt[3]{\tan x}\right)}^{2}\right)}^{3} + 1\right) \]
9. Add Preprocessing

Alternative 4: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(e^{\log \left({\tan x}^{2}\right)} + 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (exp (log (pow (tan x) 2.0))) 1.0)))

C

double code(double x, double eps) {
	return eps * (exp(log(pow(tan(x), 2.0))) + 1.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (exp(log((tan(x) ** 2.0d0))) + 1.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps * (Math.exp(Math.log(Math.pow(Math.tan(x), 2.0))) + 1.0);
}

Python

def code(x, eps):
	return eps * (math.exp(math.log(math.pow(math.tan(x), 2.0))) + 1.0)

Julia

function code(x, eps)
	return Float64(eps * Float64(exp(log((tan(x) ^ 2.0))) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (exp(log((tan(x) ^ 2.0))) + 1.0);
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[Exp[N[Log[N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. sub-neg 99.5%

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

   2. mul-1-neg 99.5%

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

   3. remove-double-neg 99.5%

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

5. Simplified 99.5%

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

   1. expm1-log1p-u 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right) \]

   2. log1p-define 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

   3. add-exp-log 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{e^{\log \left(\mathsf{expm1}\left(\log \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}}\right) \]

   4. log1p-define 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + e^{\log \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right)}\right) \]

   5. expm1-log1p-u 99.5%

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

   6. unpow2 99.5%

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

   7. unpow2 99.5%

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

   8. frac-times 99.5%

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

   9. tan-quot 99.5%

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

   10. tan-quot 99.5%

       \[\leadsto \varepsilon \cdot \left(1 + e^{\log \left(\tan x \cdot \color{blue}{\tan x}\right)}\right) \]

   11. pow2 99.5%

       \[\leadsto \varepsilon \cdot \left(1 + e^{\log \color{blue}{\left({\tan x}^{2}\right)}}\right) \]

7. Applied egg-rr 99.5%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{e^{\log \left({\tan x}^{2}\right)}}\right) \]

8. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(e^{\log \left({\tan x}^{2}\right)} + 1\right) \]
9. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* eps (+ (pow (tan x) 2.0) 1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. sub-neg 99.5%

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

   2. mul-1-neg 99.5%

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

   3. remove-double-neg 99.5%

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

5. Simplified 99.5%

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

   1. expm1-log1p-u 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right) \]

   2. log1p-define 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

   3. expm1-undefine 99.5%

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

   4. add-exp-log 99.5%

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

   5. +-commutative 99.5%

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

   6. unpow2 99.5%

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

   7. unpow2 99.5%

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

   8. frac-times 99.5%

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

   9. tan-quot 99.5%

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

   10. tan-quot 99.5%

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

   11. pow2 99.5%

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

7. Applied egg-rr 99.5%

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

   1. associate--l+ 99.5%

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

   2. metadata-eval 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + \left({\tan x}^{2} + \color{blue}{0}\right)\right) \]

   3. +-rgt-identity 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

    \[\leadsto \varepsilon \cdot \left({\tan x}^{2} + 1\right) \]
11. Add Preprocessing

Alternative 6: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon} \]
5. Add Preprocessing

Alternative 7: 97.7% 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 63.1%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. sub-neg 99.5%

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

   2. mul-1-neg 99.5%

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

   3. remove-double-neg 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0 98.7%

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

7. Final simplification 98.7%

   \[\leadsto \varepsilon \]
8. 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 2024076 
(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)))