2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 99.7%

Time: 21.9s

Alternatives: 10

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

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

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 t_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 * t_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) * t_0)))) - (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 * t_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 * t_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 * t_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 * t_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 * t$95$0), $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 61.1%

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

   1. tan-sum 61.3%

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

   2. tan-quot 61.2%

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

   3. frac-sub 61.2%

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

4. Applied egg-rr 61.2%

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

5. Taylor expanded in eps around 0 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

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) (* 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) - (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) - (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) - (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) - (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(cos(x) * 0.3333333333333333) - 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) - (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[Cos[x], $MachinePrecision] * 0.3333333333333333), $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 61.1%

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

   1. tan-sum 61.3%

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

   2. tan-quot 61.2%

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

   3. frac-sub 61.2%

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

4. Applied egg-rr 61.2%

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

5. Taylor expanded in eps around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (cos x))))
   (/
    (*
     eps
     (+
      (cos x)
      (+
       t_0
       (*
        (pow eps 2.0)
        (+ (* (cos x) 0.3333333333333333) (* 0.3333333333333333 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 * (cos(x) + (t_0 + (pow(eps, 2.0) * ((cos(x) * 0.3333333333333333) + (0.3333333333333333 * 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 * (cos(x) + (t_0 + ((eps ** 2.0d0) * ((cos(x) * 0.3333333333333333d0) + (0.3333333333333333d0 * 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.cos(x) + (t_0 + (Math.pow(eps, 2.0) * ((Math.cos(x) * 0.3333333333333333) + (0.3333333333333333 * 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.cos(x) + (t_0 + (math.pow(eps, 2.0) * ((math.cos(x) * 0.3333333333333333) + (0.3333333333333333 * 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(cos(x) + Float64(t_0 + Float64((eps ^ 2.0) * Float64(Float64(cos(x) * 0.3333333333333333) + Float64(0.3333333333333333 * 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 * (cos(x) + (t_0 + ((eps ^ 2.0) * ((cos(x) * 0.3333333333333333) + (0.3333333333333333 * 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[Cos[x], $MachinePrecision] + N[(t$95$0 + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(0.3333333333333333 * 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 61.1%

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

   1. tan-sum 61.3%

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

   2. tan-quot 61.2%

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

   3. frac-sub 61.2%

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

4. Applied egg-rr 61.2%

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

5. Taylor expanded in eps around 0 99.9%

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

   1. associate--l+ 99.9%

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

   2. cancel-sign-sub-inv 99.9%

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

   3. *-commutative 99.9%

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

   4. metadata-eval 99.9%

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

   5. mul-1-neg 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 4: 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 61.1%

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

   1. tan-sum 61.3%

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

   2. tan-quot 61.2%

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

   3. frac-sub 61.2%

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

4. Applied egg-rr 61.2%

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

5. Taylor expanded in eps around 0 99.6%

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

   1. cancel-sign-sub-inv 99.6%

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

   2. metadata-eval 99.6%

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

   3. *-lft-identity 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 5: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.3333333333333333 + x \cdot \left({\varepsilon}^{2} \cdot 0.6666666666666666 + 1\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (fma
    eps
    (+
     (* eps 0.3333333333333333)
     (* x (+ (* (pow eps 2.0) 0.6666666666666666) 1.0)))
    (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
   1.0)))

C

double code(double x, double eps) {
	return eps * (fma(eps, ((eps * 0.3333333333333333) + (x * ((pow(eps, 2.0) * 0.6666666666666666) + 1.0))), (pow(sin(x), 2.0) / pow(cos(x), 2.0))) + 1.0);
}

Julia

function code(x, eps)
	return Float64(eps * Float64(fma(eps, Float64(Float64(eps * 0.3333333333333333) + Float64(x * Float64(Float64((eps ^ 2.0) * 0.6666666666666666) + 1.0))), Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))) + 1.0))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(N[(eps * 0.3333333333333333), $MachinePrecision] + N[(x * N[(N[(N[Power[eps, 2.0], $MachinePrecision] * 0.6666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in x around 0 98.9%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \color{blue}{0.3333333333333333 \cdot \varepsilon + x \cdot \left(1 + 0.6666666666666666 \cdot {\varepsilon}^{2}\right)}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

7. Final simplification 98.9%

   \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.3333333333333333 + x \cdot \left({\varepsilon}^{2} \cdot 0.6666666666666666 + 1\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 1\right) \]
8. Add Preprocessing

Alternative 6: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\varepsilon + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(\mathsf{fma}\left(-\varepsilon, \mathsf{fma}\left(\sin x, \frac{\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, \mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right), \left(0.16666666666666666 \cdot {\sin x}^{2}\right) \cdot {\cos x}^{-2}\right)\right) - {\sin x}^{2} \cdot \left(\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot {\cos x}^{-2}\right)}{\cos x}, \mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot \left(\tan x \cdot -0.3333333333333333\right)\right), -0.16666666666666666\right) - \mathsf{fma}\left(-0.5, \mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right), \left(0.16666666666666666 \cdot {\sin x}^{2}\right) \cdot {\cos x}^{-2}\right)\right) + {\sin x}^{2} \cdot \left(\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot {\cos x}^{-2}\right), \mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot \tan x\right), {\sin x}^{2} \cdot {\cos x}^{-2}\right) \cdot \varepsilon} \]

8. Taylor expanded in eps around 0 98.8%

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

Alternative 7: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in eps around 0 98.8%

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

   1. unpow2 98.8%

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

   2. unpow2 98.8%

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

   3. times-frac 98.8%

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

   4. sqr-neg 98.8%

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

   5. distribute-frac-neg 98.8%

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

   6. neg-mul-1 98.8%

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

   7. *-commutative 98.8%

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

   8. associate-*r/ 98.8%

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

   9. distribute-frac-neg 98.8%

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

   10. neg-mul-1 98.8%

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

   11. *-commutative 98.8%

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

   12. associate-*r/ 98.8%

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

   13. unpow1 98.8%

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

   14. pow-plus 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 8: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left({\varepsilon}^{2} \cdot 0.3333333333333333 + \varepsilon \cdot \left(x \cdot \left({\varepsilon}^{2} \cdot 0.6666666666666666 + 1\right)\right)\right) + 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (+
    (* (pow eps 2.0) 0.3333333333333333)
    (* eps (* x (+ (* (pow eps 2.0) 0.6666666666666666) 1.0))))
   1.0)))

C

double code(double x, double eps) {
	return eps * (((pow(eps, 2.0) * 0.3333333333333333) + (eps * (x * ((pow(eps, 2.0) * 0.6666666666666666) + 1.0)))) + 1.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((((eps ** 2.0d0) * 0.3333333333333333d0) + (eps * (x * (((eps ** 2.0d0) * 0.6666666666666666d0) + 1.0d0)))) + 1.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps * (((Math.pow(eps, 2.0) * 0.3333333333333333) + (eps * (x * ((Math.pow(eps, 2.0) * 0.6666666666666666) + 1.0)))) + 1.0);
}

Python

def code(x, eps):
	return eps * (((math.pow(eps, 2.0) * 0.3333333333333333) + (eps * (x * ((math.pow(eps, 2.0) * 0.6666666666666666) + 1.0)))) + 1.0)

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(Float64((eps ^ 2.0) * 0.3333333333333333) + Float64(eps * Float64(x * Float64(Float64((eps ^ 2.0) * 0.6666666666666666) + 1.0)))) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((((eps ^ 2.0) * 0.3333333333333333) + (eps * (x * (((eps ^ 2.0) * 0.6666666666666666) + 1.0)))) + 1.0);
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[(N[Power[eps, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(eps * N[(x * N[(N[(N[Power[eps, 2.0], $MachinePrecision] * 0.6666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in x around 0 96.3%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + 0.6666666666666666 \cdot {\varepsilon}^{2}\right)\right)\right)\right)} \]

7. Final simplification 96.3%

   \[\leadsto \varepsilon \cdot \left(\left({\varepsilon}^{2} \cdot 0.3333333333333333 + \varepsilon \cdot \left(x \cdot \left({\varepsilon}^{2} \cdot 0.6666666666666666 + 1\right)\right)\right) + 1\right) \]
8. Add Preprocessing

Alternative 9: 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 61.1%

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

3. Taylor expanded in x around 0 96.3%

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

   1. tan-quot 96.3%

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

   2. *-un-lft-identity 96.3%

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

5. Applied egg-rr 96.3%

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

   1. *-lft-identity 96.3%

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

7. Simplified 96.3%

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

Alternative 10: 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 61.1%

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

3. Taylor expanded in x around 0 96.3%

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

4. Taylor expanded in eps around 0 96.3%

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

Developer Target 1: 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 2024165 
(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
  (! :herbie-platform default (/ (sin eps) (* (cos x) (cos (+ x eps)))))

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