2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.7%

Time: 24.0s

Alternatives: 11

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(t$95$0 + N[(N[Cos[x], $MachinePrecision] - N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(t$95$0 * -0.3333333333333333), $MachinePrecision] - N[(N[(N[Cos[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(0.13333333333333333 * N[(N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 62.8%

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

   1. tan-sum 63.0%

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

   2. tan-quot 62.9%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 100.0%

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

   1. pow1 100.0%

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

   2. cancel-sign-sub-inv 100.0%

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

   3. *-commutative 100.0%

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

   4. fma-define 100.0%

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

   5. metadata-eval 100.0%

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

7. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

   2. fma-undefine 100.0%

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

   3. *-commutative 100.0%

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

   4. distribute-lft-out 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. tan-sum 63.0%

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

   2. tan-quot 62.9%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 100.0%

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

6. Taylor expanded in x around 0 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(t$95$0 * -0.3333333333333333), $MachinePrecision] - N[(N[(N[Cos[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $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 62.8%

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

   1. tan-sum 63.0%

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

   2. tan-quot 62.9%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 100.0%

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

6. Taylor expanded in x around 0 99.9%

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

   1. *-commutative 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 4: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. tan-sum 63.0%

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

   2. tan-quot 62.9%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 100.0%

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

6. Taylor expanded in eps around 0 99.8%

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

   1. cancel-sign-sub-inv 99.8%

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

   2. metadata-eval 99.8%

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

   3. distribute-lft-out 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (eps * ((pow(sin(x), 2.0) / cos(x)) + (cos(x) + (pow(eps, 2.0) * (0.3333333333333333 + (pow(eps, 2.0) * 0.13333333333333333)))))) / (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 * (((sin(x) ** 2.0d0) / cos(x)) + (cos(x) + ((eps ** 2.0d0) * (0.3333333333333333d0 + ((eps ** 2.0d0) * 0.13333333333333333d0)))))) / (cos(x) * (1.0d0 - (tan(x) * tan(eps))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. tan-sum 63.0%

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

   2. tan-quot 62.9%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 100.0%

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

6. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 6: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. tan-sum 63.0%

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

   2. tan-quot 62.9%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 99.4%

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

   1. cancel-sign-sub-inv 99.4%

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

   2. metadata-eval 99.4%

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

   3. *-lft-identity 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 7: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0 98.8%

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

   1. sub-neg 98.8%

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

   2. mul-1-neg 98.8%

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

   3. remove-double-neg 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 8: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. tan-sum 63.0%

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

   2. tan-quot 62.9%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in x around 0 98.1%

   \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(-0.5 \cdot \frac{\sin \varepsilon}{\cos \varepsilon} - -1 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \frac{\sin \varepsilon}{\cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. fma-define 98.1%

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

   2. distribute-rgt-out-- 98.1%

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

   3. metadata-eval 98.1%

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

7. Simplified 98.1%

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

8. Taylor expanded in eps around 0 98.0%

   \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left(1 + 0.5 \cdot {x}^{2}\right)}{\cos x}} \]
9. Step-by-step derivation

   1. associate-/l* 98.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{1 + 0.5 \cdot {x}^{2}}{\cos x}} \]

   2. +-commutative 98.0%

      \[\leadsto \varepsilon \cdot \frac{\color{blue}{0.5 \cdot {x}^{2} + 1}}{\cos x} \]

   3. *-commutative 98.0%

      \[\leadsto \varepsilon \cdot \frac{\color{blue}{{x}^{2} \cdot 0.5} + 1}{\cos x} \]

   4. fma-define 98.0%

      \[\leadsto \varepsilon \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, 0.5, 1\right)}}{\cos x} \]

10. Simplified 98.0%

    \[\leadsto \color{blue}{\varepsilon \cdot \frac{\mathsf{fma}\left({x}^{2}, 0.5, 1\right)}{\cos x}} \]

11. Taylor expanded in x around 0 98.1%

    \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(0.5 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)} \]
12. Step-by-step derivation

    1. distribute-rgt-out-- 98.1%

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

    2. metadata-eval 98.1%

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

13. Simplified 98.1%

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

14. Final simplification 98.1%

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

Alternative 9: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(x, x, 1\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (fma x x 1.0)))

C

double code(double x, double eps) {
	return eps * fma(x, x, 1.0);
}

Julia

function code(x, eps)
	return Float64(eps * fma(x, x, 1.0))
end

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. tan-sum 63.0%

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

   2. tan-quot 62.9%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in x around 0 98.1%

   \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(-0.5 \cdot \frac{\sin \varepsilon}{\cos \varepsilon} - -1 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \frac{\sin \varepsilon}{\cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. fma-define 98.1%

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

   2. distribute-rgt-out-- 98.1%

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

   3. metadata-eval 98.1%

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

7. Simplified 98.1%

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

8. Taylor expanded in eps around 0 98.0%

   \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left(1 + 0.5 \cdot {x}^{2}\right)}{\cos x}} \]
9. Step-by-step derivation

   1. associate-/l* 98.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{1 + 0.5 \cdot {x}^{2}}{\cos x}} \]

   2. +-commutative 98.0%

      \[\leadsto \varepsilon \cdot \frac{\color{blue}{0.5 \cdot {x}^{2} + 1}}{\cos x} \]

   3. *-commutative 98.0%

      \[\leadsto \varepsilon \cdot \frac{\color{blue}{{x}^{2} \cdot 0.5} + 1}{\cos x} \]

   4. fma-define 98.0%

      \[\leadsto \varepsilon \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, 0.5, 1\right)}}{\cos x} \]

10. Simplified 98.0%

    \[\leadsto \color{blue}{\varepsilon \cdot \frac{\mathsf{fma}\left({x}^{2}, 0.5, 1\right)}{\cos x}} \]

11. Taylor expanded in x around 0 98.1%

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

    1. +-commutative 98.1%

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

    2. unpow2 98.1%

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

    3. fma-define 98.1%

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

13. Simplified 98.1%

    \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \]
14. Add Preprocessing

Alternative 10: 97.8% 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 62.8%

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

3. Taylor expanded in x around 0 97.7%

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

   1. tan-quot 97.7%

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

   2. *-un-lft-identity 97.7%

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

5. Applied egg-rr 97.7%

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

   1. *-lft-identity 97.7%

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

7. Simplified 97.7%

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

Alternative 11: 97.8% 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 62.8%

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

3. Taylor expanded in x around 0 97.7%

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

4. Taylor expanded in eps around 0 97.7%

   \[\leadsto \color{blue}{\varepsilon} \]
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 2024093 
(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)))