2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.7%

Time: 21.4s

Alternatives: 14

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: 62.6% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ \varepsilon \cdot \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, \left(t\_0 \cdot {\cos x}^{-2}\right) \cdot 1.3333333333333333 + 0.3333333333333333\right), \tan x + {\tan x}^{3}\right) + 1\right) + \frac{t\_0}{{\cos x}^{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0)))
   (*
    eps
    (+
     (+
      (*
       eps
       (fma
        eps
        (fma
         (pow (sin x) 4.0)
         (pow (cos x) -4.0)
         (+
          (* (* t_0 (pow (cos x) -2.0)) 1.3333333333333333)
          0.3333333333333333))
        (+ (tan x) (pow (tan x) 3.0))))
      1.0)
     (/ t_0 (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	return eps * (((eps * fma(eps, fma(pow(sin(x), 4.0), pow(cos(x), -4.0), (((t_0 * pow(cos(x), -2.0)) * 1.3333333333333333) + 0.3333333333333333)), (tan(x) + pow(tan(x), 3.0)))) + 1.0) + (t_0 / pow(cos(x), 2.0)));
}

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	return Float64(eps * Float64(Float64(Float64(eps * fma(eps, fma((sin(x) ^ 4.0), (cos(x) ^ -4.0), Float64(Float64(Float64(t_0 * (cos(x) ^ -2.0)) * 1.3333333333333333) + 0.3333333333333333)), Float64(tan(x) + (tan(x) ^ 3.0)))) + 1.0) + Float64(t_0 / (cos(x) ^ 2.0))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(N[(N[(eps * N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], -4.0], $MachinePrecision] + N[(N[(N[(t$95$0 * N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * 1.3333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(t$95$0 / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.0%

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

   1. tan-sum 64.1%

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

   2. div-inv 64.0%

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

   3. fma-neg 64.1%

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

4. Applied egg-rr 64.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
5. Step-by-step derivation

   1. fma-neg 64.0%

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

   2. associate-*r/ 64.1%

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

   3. *-rgt-identity 64.1%

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

6. Simplified 64.1%

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

7. Taylor expanded in eps around 0 99.3%

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

9. Applied egg-rr 99.3%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(0.3333333333333333 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right) - \left(\left(-\tan x\right) + {\left(-\tan x\right)}^{3}\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
10. Step-by-step derivation

    1. fma-neg 99.3%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \left(0.3333333333333333 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right), -\left(\left(-\tan x\right) + {\left(-\tan x\right)}^{3}\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

11. Simplified 99.3%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \left(\left(0.3333333333333333 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\sin x}^{4} \cdot {\cos x}^{-4}, \tan x + {\tan x}^{3}\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
12. Step-by-step derivation

13. Applied egg-rr 99.3%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, \mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 0.3333333333333333\right)\right)\right), \tan x + {\tan x}^{3}\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
14. Step-by-step derivation

15. Simplified 99.3%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) \cdot 1.3333333333333333 + 0.3333333333333333\right), \tan x + {\tan x}^{3}\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

16. Final simplification 99.3%

    \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) \cdot 1.3333333333333333 + 0.3333333333333333\right), \tan x + {\tan x}^{3}\right) + 1\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
17. Add Preprocessing

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \left(-1 - \varepsilon \cdot \left(\varepsilon \cdot 0.3333333333333333 + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
   (-
    -1.0
    (*
     eps
     (+
      (* eps 0.3333333333333333)
      (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))))))

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) - (-1.0 - (eps * ((eps * 0.3333333333333333) + ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0)))))));
}

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) - (eps * ((eps * 0.3333333333333333d0) + ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)))))))
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 - (eps * ((eps * 0.3333333333333333) + ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0)))))));
}

Python

def code(x, eps):
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) - (-1.0 - (eps * ((eps * 0.3333333333333333) + ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0)))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) - Float64(-1.0 - Float64(eps * Float64(Float64(eps * 0.3333333333333333) + Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) - (-1.0 - (eps * ((eps * 0.3333333333333333) + ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))))));
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] - N[(-1.0 - N[(eps * N[(N[(eps * 0.3333333333333333), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.0%

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

   1. tan-sum 64.1%

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

   2. div-inv 64.0%

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

   3. fma-neg 64.1%

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

4. Applied egg-rr 64.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
5. Step-by-step derivation

   1. fma-neg 64.0%

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

   2. associate-*r/ 64.1%

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

   3. *-rgt-identity 64.1%

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

6. Simplified 64.1%

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

7. Taylor expanded in eps around 0 99.3%

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

8. Taylor expanded in x around 0 99.2%

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

9. Final simplification 99.2%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \left(-1 - \varepsilon \cdot \left(\varepsilon \cdot 0.3333333333333333 + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \left(-1 - \varepsilon \cdot \left(t\_0 + {t\_0}^{3}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x))))
   (*
    eps
    (-
     (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
     (- -1.0 (* eps (+ t_0 (pow t_0 3.0))))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	return eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) - (-1.0 - (eps * (t_0 + pow(t_0, 3.0)))));
}

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) / cos(x)
    code = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) - ((-1.0d0) - (eps * (t_0 + (t_0 ** 3.0d0)))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(x) / Math.cos(x);
	return eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) - (-1.0 - (eps * (t_0 + Math.pow(t_0, 3.0)))));
}

Python

def code(x, eps):
	t_0 = math.sin(x) / math.cos(x)
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) - (-1.0 - (eps * (t_0 + math.pow(t_0, 3.0)))))

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) - Float64(-1.0 - Float64(eps * Float64(t_0 + (t_0 ^ 3.0))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) / cos(x);
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) - (-1.0 - (eps * (t_0 + (t_0 ^ 3.0)))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(eps * N[(t$95$0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.0%

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

   1. tan-sum 64.1%

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

   2. div-inv 64.0%

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

   3. fma-neg 64.1%

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

4. Applied egg-rr 64.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
5. Step-by-step derivation

   1. fma-neg 64.0%

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

   2. associate-*r/ 64.1%

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

   3. *-rgt-identity 64.1%

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

6. Simplified 64.1%

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

7. Taylor expanded in eps around 0 99.3%

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

8. Taylor expanded in eps around 0 99.0%

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

   1. mul-1-neg 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. distribute-lft-out 99.0%

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

   4. mul-1-neg 99.0%

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

   5. remove-double-neg 99.0%

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

   6. +-commutative 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

    \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \left(-1 - \varepsilon \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)\right) \]
12. Add Preprocessing

Alternative 4: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right) + 1\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (+
    (+
     (* 0.3333333333333333 (pow eps 2.0))
     (*
      x
      (+
       eps
       (*
        x
        (+
         (* 1.3333333333333333 (* eps x))
         (* 1.3333333333333333 (pow eps 2.0)))))))
    1.0)
   (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))

C

double code(double x, double eps) {
	return eps * ((((0.3333333333333333 * pow(eps, 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (1.3333333333333333 * pow(eps, 2.0))))))) + 1.0) + (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 * ((((0.3333333333333333d0 * (eps ** 2.0d0)) + (x * (eps + (x * ((1.3333333333333333d0 * (eps * x)) + (1.3333333333333333d0 * (eps ** 2.0d0))))))) + 1.0d0) + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * ((((0.3333333333333333 * Math.pow(eps, 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (1.3333333333333333 * Math.pow(eps, 2.0))))))) + 1.0) + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
}

Python

def code(x, eps):
	return eps * ((((0.3333333333333333 * math.pow(eps, 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (1.3333333333333333 * math.pow(eps, 2.0))))))) + 1.0) + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(Float64(Float64(0.3333333333333333 * (eps ^ 2.0)) + Float64(x * Float64(eps + Float64(x * Float64(Float64(1.3333333333333333 * Float64(eps * x)) + Float64(1.3333333333333333 * (eps ^ 2.0))))))) + 1.0) + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((((0.3333333333333333 * (eps ^ 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (1.3333333333333333 * (eps ^ 2.0))))))) + 1.0) + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[(N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps + N[(x * N[(N[(1.3333333333333333 * N[(eps * x), $MachinePrecision]), $MachinePrecision] + N[(1.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 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 64.0%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.9%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

5. Final simplification 98.9%

   \[\leadsto \varepsilon \cdot \left(\left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right) + 1\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
6. Add Preprocessing

Alternative 5: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right) + 1\right) + \frac{{\sin x}^{2}}{\frac{\cos \left(x \cdot 2\right) + 1}{2}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (+
    (+
     (* 0.3333333333333333 (pow eps 2.0))
     (*
      x
      (+
       eps
       (*
        x
        (+
         (* 1.3333333333333333 (* eps x))
         (* 1.3333333333333333 (pow eps 2.0)))))))
    1.0)
   (/ (pow (sin x) 2.0) (/ (+ (cos (* x 2.0)) 1.0) 2.0)))))

C

double code(double x, double eps) {
	return eps * ((((0.3333333333333333 * pow(eps, 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (1.3333333333333333 * pow(eps, 2.0))))))) + 1.0) + (pow(sin(x), 2.0) / ((cos((x * 2.0)) + 1.0) / 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((((0.3333333333333333d0 * (eps ** 2.0d0)) + (x * (eps + (x * ((1.3333333333333333d0 * (eps * x)) + (1.3333333333333333d0 * (eps ** 2.0d0))))))) + 1.0d0) + ((sin(x) ** 2.0d0) / ((cos((x * 2.0d0)) + 1.0d0) / 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * ((((0.3333333333333333 * Math.pow(eps, 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (1.3333333333333333 * Math.pow(eps, 2.0))))))) + 1.0) + (Math.pow(Math.sin(x), 2.0) / ((Math.cos((x * 2.0)) + 1.0) / 2.0)));
}

Python

def code(x, eps):
	return eps * ((((0.3333333333333333 * math.pow(eps, 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (1.3333333333333333 * math.pow(eps, 2.0))))))) + 1.0) + (math.pow(math.sin(x), 2.0) / ((math.cos((x * 2.0)) + 1.0) / 2.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(Float64(Float64(0.3333333333333333 * (eps ^ 2.0)) + Float64(x * Float64(eps + Float64(x * Float64(Float64(1.3333333333333333 * Float64(eps * x)) + Float64(1.3333333333333333 * (eps ^ 2.0))))))) + 1.0) + Float64((sin(x) ^ 2.0) / Float64(Float64(cos(Float64(x * 2.0)) + 1.0) / 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((((0.3333333333333333 * (eps ^ 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (1.3333333333333333 * (eps ^ 2.0))))))) + 1.0) + ((sin(x) ^ 2.0) / ((cos((x * 2.0)) + 1.0) / 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[(N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps + N[(x * N[(N[(1.3333333333333333 * N[(eps * x), $MachinePrecision]), $MachinePrecision] + N[(1.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.9%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
5. Step-by-step derivation

   1. unpow2 98.9%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\cos x \cdot \cos x}}\right) \]

   2. cos-mult 98.9%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right) \]

6. Applied egg-rr 98.9%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right) \]
7. Step-by-step derivation

   1. +-commutative 98.9%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\color{blue}{\cos \left(x - x\right) + \cos \left(x + x\right)}}{2}}\right) \]

   2. +-inverses 98.9%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\cos \color{blue}{0} + \cos \left(x + x\right)}{2}}\right) \]

   3. cos-0 98.9%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{\color{blue}{1} + \cos \left(x + x\right)}{2}}\right) \]

   4. count-2 98.9%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(2 \cdot x\right)}}{2}}\right) \]

   5. *-commutative 98.9%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(x \cdot 2\right)}}{2}}\right) \]

8. Simplified 98.9%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\color{blue}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}}\right) \]

9. Final simplification 98.9%

   \[\leadsto \varepsilon \cdot \left(\left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right) + 1\right) + \frac{{\sin x}^{2}}{\frac{\cos \left(x \cdot 2\right) + 1}{2}}\right) \]
10. Add Preprocessing

Alternative 6: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (((eps * (x + (eps * 0.3333333333333333))) + 1.0) + (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 * (x + (eps * 0.3333333333333333d0))) + 1.0d0) + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[(eps * N[(x + N[(eps * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 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 64.0%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.9%

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

5. Taylor expanded in eps around 0 98.9%

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

6. Final simplification 98.9%

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

Alternative 7: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (((eps * x) + 1.0) + (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 * x) + 1.0d0) + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[(eps * x), $MachinePrecision] + 1.0), $MachinePrecision] + 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 64.0%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.9%

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

5. Taylor expanded in eps around 0 98.8%

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

6. Final simplification 98.8%

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

Alternative 8: 98.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (pow (sin x) 2.0)
   (+ (+ (* 0.3333333333333333 (pow eps 2.0)) (* eps x)) 1.0))))

C

double code(double x, double eps) {
	return eps * (pow(sin(x), 2.0) + (((0.3333333333333333 * pow(eps, 2.0)) + (eps * x)) + 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) + (((0.3333333333333333d0 * (eps ** 2.0d0)) + (eps * x)) + 1.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps * (Math.pow(Math.sin(x), 2.0) + (((0.3333333333333333 * Math.pow(eps, 2.0)) + (eps * x)) + 1.0));
}

Python

def code(x, eps):
	return eps * (math.pow(math.sin(x), 2.0) + (((0.3333333333333333 * math.pow(eps, 2.0)) + (eps * x)) + 1.0))

Julia

function code(x, eps)
	return Float64(eps * Float64((sin(x) ^ 2.0) + Float64(Float64(Float64(0.3333333333333333 * (eps ^ 2.0)) + Float64(eps * x)) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((sin(x) ^ 2.0) + (((0.3333333333333333 * (eps ^ 2.0)) + (eps * x)) + 1.0));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.9%

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

5. Taylor expanded in x around 0 98.7%

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

6. Final simplification 98.7%

   \[\leadsto \varepsilon \cdot \left({\sin x}^{2} + \left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right) + 1\right)\right) \]
7. Add Preprocessing

Alternative 9: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.9%

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

5. Taylor expanded in eps around 0 98.8%

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

6. Taylor expanded in x around 0 98.6%

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

7. Final simplification 98.6%

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

Alternative 10: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (((eps * x) + 1.0) + 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) + 1.0d0) + (x ** 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.9%

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

5. Taylor expanded in eps around 0 98.8%

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

6. Taylor expanded in x around 0 98.5%

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

7. Final simplification 98.5%

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

Alternative 11: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \varepsilon \cdot \left(\varepsilon + x\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma x (* eps (+ eps x)) eps))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.9%

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

5. Taylor expanded in eps around 0 98.8%

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

6. Taylor expanded in x around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. fma-define 98.5%

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

   3. +-commutative 98.5%

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

   4. unpow2 98.5%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \varepsilon} + \varepsilon \cdot x, \varepsilon\right) \]

   5. distribute-lft-out 98.5%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \left(\varepsilon + x\right)}, \varepsilon\right) \]

8. Simplified 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \varepsilon \cdot \left(\varepsilon + x\right), \varepsilon\right)} \]
9. Add Preprocessing

Alternative 12: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.9%

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

5. Taylor expanded in eps around 0 98.8%

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

6. Taylor expanded in x around 0 98.1%

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

7. Final simplification 98.1%

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

Alternative 13: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in x around 0 98.1%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. tan-quot 98.1%

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

   2. *-un-lft-identity 98.1%

      \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]

5. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]
6. Step-by-step derivation

   1. *-lft-identity 98.1%

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

7. Simplified 98.1%

   \[\leadsto \color{blue}{\tan \varepsilon} \]
8. Add Preprocessing

Alternative 14: 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 64.0%

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

3. Taylor expanded in x around 0 98.1%

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

4. Taylor expanded in eps around 0 98.0%

   \[\leadsto \color{blue}{\varepsilon} \]
5. 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 2024096 
(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)))