2tan (problem 3.3.2)

Percentage Accurate: 62.1% → 99.4%

Time: 27.0s

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: 62.1% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(\varepsilon \cdot 0.6666666666666666 + x \cdot 1.3333333333333333\right), \left(1 + \frac{{\sin x}^{2}}{t\_0}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{t\_0}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)))
   (*
    eps
    (+
     1.0
     (fma
      eps
      (fma
       eps
       (+
        0.3333333333333333
        (* x (+ (* eps 0.6666666666666666) (* x 1.3333333333333333))))
       (* (+ 1.0 (/ (pow (sin x) 2.0) t_0)) (/ (sin x) (cos x))))
      (/ (- 0.5 (/ (cos (* x 2.0)) 2.0)) t_0))))))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	return eps * (1.0 + fma(eps, fma(eps, (0.3333333333333333 + (x * ((eps * 0.6666666666666666) + (x * 1.3333333333333333)))), ((1.0 + (pow(sin(x), 2.0) / t_0)) * (sin(x) / cos(x)))), ((0.5 - (cos((x * 2.0)) / 2.0)) / t_0)));
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	return Float64(eps * Float64(1.0 + fma(eps, fma(eps, Float64(0.3333333333333333 + Float64(x * Float64(Float64(eps * 0.6666666666666666) + Float64(x * 1.3333333333333333)))), Float64(Float64(1.0 + Float64((sin(x) ^ 2.0) / t_0)) * Float64(sin(x) / cos(x)))), Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 2.0)) / t_0))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(1.0 + N[(eps * N[(eps * N[(0.3333333333333333 + N[(x * N[(N[(eps * 0.6666666666666666), $MachinePrecision] + N[(x * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $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]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right)}, \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
7. Step-by-step derivation

   1. unpow2 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)\right) \]

   2. sin-mult 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right) \]

8. Applied egg-rr 99.3%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right) \]
9. Step-by-step derivation

   1. div-sub 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{\color{blue}{\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right) \]

   2. +-inverses 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right) \]

   3. cos-0 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right) \]

   4. metadata-eval 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right) \]

   5. count-2 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right)\right) \]

   6. *-commutative 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right)\right) \]

10. Simplified 99.3%

    \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{\color{blue}{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}}{{\cos x}^{2}}\right)\right) \]

11. Final simplification 99.3%

    \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(\varepsilon \cdot 0.6666666666666666 + x \cdot 1.3333333333333333\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(\varepsilon \cdot 0.6666666666666666 + x \cdot 1.3333333333333333\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), {\tan x}^{2}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (fma
    eps
    (fma
     eps
     (+
      0.3333333333333333
      (* x (+ (* eps 0.6666666666666666) (* x 1.3333333333333333))))
     (* (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))) (/ (sin x) (cos x))))
    (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	return eps * (1.0 + fma(eps, fma(eps, (0.3333333333333333 + (x * ((eps * 0.6666666666666666) + (x * 1.3333333333333333)))), ((1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0))) * (sin(x) / cos(x)))), pow(tan(x), 2.0)));
}

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + fma(eps, fma(eps, Float64(0.3333333333333333 + Float64(x * Float64(Float64(eps * 0.6666666666666666) + Float64(x * 1.3333333333333333)))), Float64(Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))) * Float64(sin(x) / cos(x)))), (tan(x) ^ 2.0))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(eps * N[(eps * N[(0.3333333333333333 + N[(x * N[(N[(eps * 0.6666666666666666), $MachinePrecision] + N[(x * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right)}, \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
7. Step-by-step derivation

   1. *-un-lft-identity 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \color{blue}{1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)\right) \]

   2. unpow2 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), 1 \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)\right) \]

   3. unpow2 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), 1 \cdot \frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}}\right)\right) \]

   4. frac-times 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), 1 \cdot \color{blue}{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)}\right)\right) \]

   5. tan-quot 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), 1 \cdot \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}\right)\right)\right) \]

   6. tan-quot 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), 1 \cdot \left(\tan x \cdot \color{blue}{\tan x}\right)\right)\right) \]

   7. pow2 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), 1 \cdot \color{blue}{{\tan x}^{2}}\right)\right) \]

8. Applied egg-rr 99.3%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \color{blue}{1 \cdot {\tan x}^{2}}\right)\right) \]
9. Step-by-step derivation

   1. *-lft-identity 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \color{blue}{{\tan x}^{2}}\right)\right) \]

10. Simplified 99.3%

    \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \color{blue}{{\tan x}^{2}}\right)\right) \]

11. Final simplification 99.3%

    \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(\varepsilon \cdot 0.6666666666666666 + x \cdot 1.3333333333333333\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), {\tan x}^{2}\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(\varepsilon \cdot 0.6666666666666666 + x \cdot 1.3333333333333333\right), \frac{\sin x}{\cos x} \cdot \left(1 + {x}^{2}\right)\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (fma
    eps
    (fma
     eps
     (+
      0.3333333333333333
      (* x (+ (* eps 0.6666666666666666) (* x 1.3333333333333333))))
     (* (/ (sin x) (cos x)) (+ 1.0 (pow x 2.0))))
    (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	return eps * (1.0 + fma(eps, fma(eps, (0.3333333333333333 + (x * ((eps * 0.6666666666666666) + (x * 1.3333333333333333)))), ((sin(x) / cos(x)) * (1.0 + pow(x, 2.0)))), (pow(sin(x), 2.0) / pow(cos(x), 2.0))));
}

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + fma(eps, fma(eps, Float64(0.3333333333333333 + Float64(x * Float64(Float64(eps * 0.6666666666666666) + Float64(x * 1.3333333333333333)))), Float64(Float64(sin(x) / cos(x)) * Float64(1.0 + (x ^ 2.0)))), Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(eps * N[(eps * N[(0.3333333333333333 + N[(x * N[(N[(eps * 0.6666666666666666), $MachinePrecision] + N[(x * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right)}, \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

7. Taylor expanded in x around 0 99.1%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.3333333333333333 \cdot x\right), \left(1 + \color{blue}{{x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

8. Final simplification 99.1%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + x \cdot \left(\varepsilon \cdot 0.6666666666666666 + x \cdot 1.3333333333333333\right), \frac{\sin x}{\cos x} \cdot \left(1 + {x}^{2}\right)\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
9. Add Preprocessing

Alternative 4: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \frac{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.044444444444444446 + {x}^{2} \cdot -0.0031746031746031746\right) - 0.3333333333333333\right)\right)}{{\cos x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (/
    (*
     (pow x 2.0)
     (+
      1.0
      (*
       (pow x 2.0)
       (-
        (*
         (pow x 2.0)
         (+ 0.044444444444444446 (* (pow x 2.0) -0.0031746031746031746)))
        0.3333333333333333))))
    (pow (cos x) 2.0)))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((pow(x, 2.0) * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.044444444444444446 + (pow(x, 2.0) * -0.0031746031746031746))) - 0.3333333333333333)))) / pow(cos(x), 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (((x ** 2.0d0) * (1.0d0 + ((x ** 2.0d0) * (((x ** 2.0d0) * (0.044444444444444446d0 + ((x ** 2.0d0) * (-0.0031746031746031746d0)))) - 0.3333333333333333d0)))) / (cos(x) ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((Math.pow(x, 2.0) * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.044444444444444446 + (Math.pow(x, 2.0) * -0.0031746031746031746))) - 0.3333333333333333)))) / Math.pow(Math.cos(x), 2.0)));
}

Python

def code(x, eps):
	return eps * (1.0 + ((math.pow(x, 2.0) * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.044444444444444446 + (math.pow(x, 2.0) * -0.0031746031746031746))) - 0.3333333333333333)))) / math.pow(math.cos(x), 2.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64((x ^ 2.0) * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.044444444444444446 + Float64((x ^ 2.0) * -0.0031746031746031746))) - 0.3333333333333333)))) / (cos(x) ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (((x ^ 2.0) * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.044444444444444446 + ((x ^ 2.0) * -0.0031746031746031746))) - 0.3333333333333333)))) / (cos(x) ^ 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.044444444444444446 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.0031746031746031746), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. sub-neg 98.6%

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

   2. mul-1-neg 98.6%

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

   3. remove-double-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in x around 0 98.7%

   \[\leadsto \varepsilon \cdot \left(1 + \frac{\color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.044444444444444446 + -0.0031746031746031746 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)}}{{\cos x}^{2}}\right) \]

7. Final simplification 98.7%

   \[\leadsto \varepsilon \cdot \left(1 + \frac{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.044444444444444446 + {x}^{2} \cdot -0.0031746031746031746\right) - 0.3333333333333333\right)\right)}{{\cos x}^{2}}\right) \]
8. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. sub-neg 98.6%

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

   2. mul-1-neg 98.6%

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

   3. remove-double-neg 98.6%

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

5. Simplified 98.6%

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

   1. distribute-rgt-in 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. unpow2 98.6%

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

   4. unpow2 98.6%

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

   5. frac-times 98.6%

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

   6. tan-quot 98.6%

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

   7. tan-quot 98.6%

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

   8. pow2 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 6: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. sub-neg 98.6%

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

   2. mul-1-neg 98.6%

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

   3. remove-double-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in x around 0 98.2%

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

7. Final simplification 98.2%

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

Alternative 7: 98.1% 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 63.8%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. sub-neg 98.6%

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

   2. mul-1-neg 98.6%

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

   3. remove-double-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in x around 0 98.2%

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

7. Final simplification 98.2%

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

Alternative 8: 97.7% 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 63.8%

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

3. Taylor expanded in x around 0 97.8%

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

   1. tan-quot 97.8%

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

   2. *-un-lft-identity 97.8%

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

5. Applied egg-rr 97.8%

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

   1. *-lft-identity 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

   \[\leadsto \tan \varepsilon \]
9. Add Preprocessing

Alternative 9: 97.7% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in x around 0 97.8%

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

4. Taylor expanded in eps around 0 97.8%

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

5. Final simplification 97.8%

   \[\leadsto \varepsilon \]
6. 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 2024052 
(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)))