2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.4%

Time: 22.5s

Alternatives: 9

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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ \varepsilon \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - -1.3333333333333333 \cdot \left(x \cdot x\right)\right) - \frac{\sin x \cdot \left(-1 - \frac{{\sin x}^{2}}{t\_0}\right)}{\cos x}\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{t\_0}\right) + 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)))
   (*
    eps
    (+
     (+
      (*
       eps
       (-
        (* eps (- 0.3333333333333333 (* -1.3333333333333333 (* x x))))
        (/ (* (sin x) (- -1.0 (/ (pow (sin x) 2.0) t_0))) (cos x))))
      (/ (- 0.5 (/ (cos (* x 2.0)) 2.0)) t_0))
     1.0))))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	return eps * (((eps * ((eps * (0.3333333333333333 - (-1.3333333333333333 * (x * x)))) - ((sin(x) * (-1.0 - (pow(sin(x), 2.0) / t_0))) / cos(x)))) + ((0.5 - (cos((x * 2.0)) / 2.0)) / t_0)) + 1.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = cos(x) ** 2.0d0
    code = eps * (((eps * ((eps * (0.3333333333333333d0 - ((-1.3333333333333333d0) * (x * x)))) - ((sin(x) * ((-1.0d0) - ((sin(x) ** 2.0d0) / t_0))) / cos(x)))) + ((0.5d0 - (cos((x * 2.0d0)) / 2.0d0)) / t_0)) + 1.0d0)
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.cos(x), 2.0);
	return eps * (((eps * ((eps * (0.3333333333333333 - (-1.3333333333333333 * (x * x)))) - ((Math.sin(x) * (-1.0 - (Math.pow(Math.sin(x), 2.0) / t_0))) / Math.cos(x)))) + ((0.5 - (Math.cos((x * 2.0)) / 2.0)) / t_0)) + 1.0);
}

Python

def code(x, eps):
	t_0 = math.pow(math.cos(x), 2.0)
	return eps * (((eps * ((eps * (0.3333333333333333 - (-1.3333333333333333 * (x * x)))) - ((math.sin(x) * (-1.0 - (math.pow(math.sin(x), 2.0) / t_0))) / math.cos(x)))) + ((0.5 - (math.cos((x * 2.0)) / 2.0)) / t_0)) + 1.0)

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	return Float64(eps * Float64(Float64(Float64(eps * Float64(Float64(eps * Float64(0.3333333333333333 - Float64(-1.3333333333333333 * Float64(x * x)))) - Float64(Float64(sin(x) * Float64(-1.0 - Float64((sin(x) ^ 2.0) / t_0))) / cos(x)))) + Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 2.0)) / t_0)) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	t_0 = cos(x) ^ 2.0;
	tmp = eps * (((eps * ((eps * (0.3333333333333333 - (-1.3333333333333333 * (x * x)))) - ((sin(x) * (-1.0 - ((sin(x) ^ 2.0) / t_0))) / cos(x)))) + ((0.5 - (cos((x * 2.0)) / 2.0)) / t_0)) + 1.0);
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(N[(N[(eps * N[(N[(eps * N[(0.3333333333333333 - N[(-1.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[x], $MachinePrecision] * N[(-1.0 - N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

   1. unpow2 100.0%

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

   2. sin-mult 100.0%

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

8. Applied egg-rr 100.0%

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

   1. div-sub 100.0%

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

   2. +-inverses 100.0%

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

   3. cos-0 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-1.3333333333333333 \cdot {x}^{2} - 0.3333333333333333\right) - \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) - \left(-\frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)\right) \]

   5. count-2 100.0%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-1.3333333333333333 \cdot {x}^{2} - 0.3333333333333333\right) - \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) - \left(-\frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right)\right)\right) \]

   6. *-commutative 100.0%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-1.3333333333333333 \cdot {x}^{2} - 0.3333333333333333\right) - \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) - \left(-\frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right)\right)\right) \]

10. Simplified 100.0%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-1.3333333333333333 \cdot {x}^{2} - 0.3333333333333333\right) - \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) - \left(-\frac{\color{blue}{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}}{{\cos x}^{2}}\right)\right)\right) \]
11. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-1.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right) - \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) - \left(-\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right)\right)\right) \]

12. Applied egg-rr 100.0%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-1.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right) - \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) - \left(-\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right)\right)\right) \]

13. Final simplification 100.0%

    \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - -1.3333333333333333 \cdot \left(x \cdot x\right)\right) - \frac{\sin x \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) + 1\right) \]
14. Add Preprocessing

Alternative 2: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + (eps * (x + (eps * 0.3333333333333333)))) + 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)) + (eps * (x + (eps * 0.3333333333333333d0)))) + 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)) + (eps * (x + (eps * 0.3333333333333333)))) + 1.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

7. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 3: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. sub-neg 99.5%

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

   2. mul-1-neg 99.5%

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

   3. remove-double-neg 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (((0.5 - (cos((x * 2.0)) / 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 * (((0.5d0 - (cos((x * 2.0d0)) / 2.0d0)) / (cos(x) ** 2.0d0)) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. sub-neg 99.5%

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

   2. mul-1-neg 99.5%

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

   3. remove-double-neg 99.5%

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

5. Simplified 99.5%

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

   1. unpow2 100.0%

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

   2. sin-mult 100.0%

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

7. Applied egg-rr 99.5%

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

   1. div-sub 100.0%

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

   2. +-inverses 100.0%

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

   3. cos-0 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-1.3333333333333333 \cdot {x}^{2} - 0.3333333333333333\right) - \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) - \left(-\frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)\right) \]

   5. count-2 100.0%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-1.3333333333333333 \cdot {x}^{2} - 0.3333333333333333\right) - \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) - \left(-\frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right)\right)\right) \]

   6. *-commutative 100.0%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-1.3333333333333333 \cdot {x}^{2} - 0.3333333333333333\right) - \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) - \left(-\frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right)\right)\right) \]

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 5: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((pow(x, 2.0) * (((x * x) * 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 * (((x ** 2.0d0) * (((x * x) * 0.6666666666666666d0) + 1.0d0)) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. sub-neg 99.5%

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

   2. mul-1-neg 99.5%

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

   3. remove-double-neg 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0 98.9%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(1 + 0.6666666666666666 \cdot {x}^{2}\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative 98.9%

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

8. Simplified 98.9%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.6666666666666666\right)}\right) \]
9. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-1.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right) - \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) - \left(-\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right)\right)\right) \]

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 6: 98.4% 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 64.3%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. sub-neg 99.5%

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

   2. mul-1-neg 99.5%

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

   3. remove-double-neg 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0 98.8%

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

   1. *-commutative 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 7: 98.4% 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 64.3%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. sub-neg 99.5%

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

   2. mul-1-neg 99.5%

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

   3. remove-double-neg 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0 98.9%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(1 + 0.6666666666666666 \cdot {x}^{2}\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative 98.9%

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

8. Simplified 98.9%

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

9. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {x}^{2}} \]
10. Step-by-step derivation

    1. *-rgt-identity 98.8%

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

    2. distribute-lft-out 98.8%

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

    3. +-commutative 98.8%

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

    4. unpow2 98.8%

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

    5. fma-define 98.8%

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

11. Simplified 98.8%

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

Alternative 8: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in x around 0 98.3%

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

   1. tan-quot 98.3%

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

   2. *-un-lft-identity 98.3%

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

5. Applied egg-rr 98.3%

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

   1. *-lft-identity 98.3%

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

7. Simplified 98.3%

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

Alternative 9: 98.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in x around 0 98.3%

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

4. Taylor expanded in eps around 0 98.3%

   \[\leadsto \color{blue}{\varepsilon} \]
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 2024145 
(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)))