2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 99.5%

Time: 28.9s

Alternatives: 11

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.7% 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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{-2}\\ t_1 := {\sin x}^{2}\\ \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \mathsf{fma}\left(t\_1, -t\_0, -0.3333333333333333 \cdot \left(t\_1 \cdot t\_0\right) - {\sin x}^{4} \cdot {\cos x}^{-4}\right), \tan x + {\tan x}^{3}\right)\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) -2.0)) (t_1 (pow (sin x) 2.0)))
   (*
    eps
    (+
     (+
      1.0
      (*
       eps
       (fma
        eps
        (-
         0.3333333333333333
         (fma
          t_1
          (- t_0)
          (-
           (* -0.3333333333333333 (* t_1 t_0))
           (* (pow (sin x) 4.0) (pow (cos x) -4.0)))))
        (+ (tan x) (pow (tan x) 3.0)))))
     (/ (- 0.5 (/ (cos (* x 2.0)) 2.0)) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), -2.0);
	double t_1 = pow(sin(x), 2.0);
	return eps * ((1.0 + (eps * fma(eps, (0.3333333333333333 - fma(t_1, -t_0, ((-0.3333333333333333 * (t_1 * t_0)) - (pow(sin(x), 4.0) * pow(cos(x), -4.0))))), (tan(x) + pow(tan(x), 3.0))))) + ((0.5 - (cos((x * 2.0)) / 2.0)) / pow(cos(x), 2.0)));
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ -2.0
	t_1 = sin(x) ^ 2.0
	return Float64(eps * Float64(Float64(1.0 + Float64(eps * fma(eps, Float64(0.3333333333333333 - fma(t_1, Float64(-t_0), Float64(Float64(-0.3333333333333333 * Float64(t_1 * t_0)) - Float64((sin(x) ^ 4.0) * (cos(x) ^ -4.0))))), Float64(tan(x) + (tan(x) ^ 3.0))))) + Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 2.0)) / (cos(x) ^ 2.0))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(N[(1.0 + N[(eps * N[(eps * N[(0.3333333333333333 - N[(t$95$1 * (-t$95$0) + N[(N[(-0.3333333333333333 * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 65.7%

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

   1. tan-sum 66.0%

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

   2. div-inv 66.0%

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

   3. fma-neg 66.0%

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

4. Applied egg-rr 66.0%

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

5. Taylor expanded in eps around 0 99.7%

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

   1. unpow2 99.7%

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

   2. sin-mult 99.7%

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

7. Applied egg-rr 99.7%

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

   1. div-sub 99.7%

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

   2. +-inverses 99.7%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right) \]

   3. cos-0 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right) \]

   5. count-2 99.7%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right) \]

   6. *-commutative 99.7%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right) \]

9. Simplified 99.7%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{\color{blue}{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}}{{\cos x}^{2}}\right) \]

11. Applied egg-rr 99.7%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{{\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right), -\mathsf{fma}\left(-1, {\tan x}^{3}, -\tan x\right)\right)\right)}^{1}}\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]
12. Step-by-step derivation

    1. unpow1 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right), -\mathsf{fma}\left(-1, {\tan x}^{3}, -\tan x\right)\right)}\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]

    2. fma-neg 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right)\right) - \mathsf{fma}\left(-1, {\tan x}^{3}, -\tan x\right)\right)}\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]

    3. fma-undefine 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right)\right) - \color{blue}{\left(-1 \cdot {\tan x}^{3} + \left(-\tan x\right)\right)}\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]

    4. neg-mul-1 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right)\right) - \left(-1 \cdot {\tan x}^{3} + \color{blue}{-1 \cdot \tan x}\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]

    5. distribute-lft-in 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right)\right) - \color{blue}{-1 \cdot \left({\tan x}^{3} + \tan x\right)}\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]

    6. +-commutative 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right)\right) - -1 \cdot \color{blue}{\left(\tan x + {\tan x}^{3}\right)}\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]

    7. *-commutative 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right)\right) - \color{blue}{\left(\tan x + {\tan x}^{3}\right) \cdot -1}\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]

13. Simplified 99.7%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \mathsf{fma}\left({\sin x}^{2}, -{\cos x}^{-2}, -0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - {\sin x}^{4} \cdot {\cos x}^{-4}\right), \tan x + {\tan x}^{3}\right)}\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]

14. Final simplification 99.7%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \mathsf{fma}\left({\sin x}^{2}, -{\cos x}^{-2}, -0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - {\sin x}^{4} \cdot {\cos x}^{-4}\right), \tan x + {\tan x}^{3}\right)\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]
15. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}} + \left(1 + \varepsilon \cdot \left(\varepsilon \cdot 0.3333333333333333 + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (/ (- 0.5 (/ (cos (* x 2.0)) 2.0)) (pow (cos x) 2.0))
   (+
    1.0
    (*
     eps
     (+
      (* eps 0.3333333333333333)
      (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))))))

C

double code(double x, double eps) {
	return eps * (((0.5 - (cos((x * 2.0)) / 2.0)) / pow(cos(x), 2.0)) + (1.0 + (eps * ((eps * 0.3333333333333333) + ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0)))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((0.5d0 - (cos((x * 2.0d0)) / 2.0d0)) / (cos(x) ** 2.0d0)) + (1.0d0 + (eps * ((eps * 0.3333333333333333d0) + ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (((0.5 - (Math.cos((x * 2.0)) / 2.0)) / Math.pow(Math.cos(x), 2.0)) + (1.0 + (eps * ((eps * 0.3333333333333333) + ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0)))))));
}

Python

def code(x, eps):
	return eps * (((0.5 - (math.cos((x * 2.0)) / 2.0)) / math.pow(math.cos(x), 2.0)) + (1.0 + (eps * ((eps * 0.3333333333333333) + ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0)))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 2.0)) / (cos(x) ^ 2.0)) + Float64(1.0 + Float64(eps * Float64(Float64(eps * 0.3333333333333333) + Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((0.5 - (cos((x * 2.0)) / 2.0)) / (cos(x) ^ 2.0)) + (1.0 + (eps * ((eps * 0.3333333333333333) + ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[(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] + N[(1.0 + N[(eps * N[(N[(eps * 0.3333333333333333), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. tan-sum 66.0%

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

   2. div-inv 66.0%

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

   3. fma-neg 66.0%

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

4. Applied egg-rr 66.0%

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

5. Taylor expanded in eps around 0 99.7%

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

   1. unpow2 99.7%

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

   2. sin-mult 99.7%

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

7. Applied egg-rr 99.7%

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

   1. div-sub 99.7%

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

   2. +-inverses 99.7%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right) \]

   3. cos-0 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right) \]

   5. count-2 99.7%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right) \]

   6. *-commutative 99.7%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right) \]

9. Simplified 99.7%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{\color{blue}{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}}{{\cos x}^{2}}\right) \]

10. Taylor expanded in x around 0 99.5%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{0.3333333333333333} - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]

11. Final simplification 99.5%

    \[\leadsto \varepsilon \cdot \left(\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}} + \left(1 + \varepsilon \cdot \left(\varepsilon \cdot 0.3333333333333333 + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. add-log-exp 10.1%

      \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\log \left(e^{\tan x}\right)} \]

4. Applied egg-rr 10.1%

   \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\log \left(e^{\tan x}\right)} \]

5. Taylor expanded in x around inf 65.7%

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

6. Taylor expanded in eps around 0 99.4%

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

8. Simplified 99.4%

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

Alternative 4: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

3. Taylor expanded in eps around 0 99.7%

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

4. Taylor expanded in x around 0 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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)) + (1.0d0 + (eps * (x + (eps * 0.3333333333333333d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. tan-sum 66.0%

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

   2. div-inv 66.0%

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

   3. fma-neg 66.0%

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

4. Applied egg-rr 66.0%

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

5. Taylor expanded in eps around 0 99.7%

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

6. Taylor expanded in x around 0 98.7%

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

   1. +-commutative 98.7%

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

8. Simplified 98.7%

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

   1. associate-*r/ 98.7%

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

10. Applied egg-rr 98.7%

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

    1. associate-*r/ 98.7%

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

    2. mul-1-neg 98.7%

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

    3. unpow2 98.7%

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

    4. unpow2 98.7%

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

    5. times-frac 98.7%

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

    6. *-lft-identity 98.7%

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

    7. associate-*l/ 98.7%

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

    8. associate-/r/ 98.7%

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

    9. unpow-1 98.7%

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

    10. *-lft-identity 98.7%

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

    11. associate-*l/ 98.7%

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

    12. associate-/r/ 98.7%

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

    13. unpow-1 98.7%

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

    14. pow-sqr 98.7%

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

    15. metadata-eval 98.7%

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

12. Simplified 98.7%

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

13. Final simplification 98.7%

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

Alternative 6: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. tan-sum 66.0%

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

   2. div-inv 66.0%

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

   3. fma-neg 66.0%

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

4. Applied egg-rr 66.0%

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

5. Taylor expanded in eps around 0 99.7%

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

6. Taylor expanded in x around 0 98.7%

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

   1. +-commutative 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in x around 0 97.6%

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

10. Final simplification 97.6%

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

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \varepsilon \cdot \left(\varepsilon + x\right), \varepsilon + 0.3333333333333333 \cdot {\varepsilon}^{3}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma x (* eps (+ eps x)) (+ eps (* 0.3333333333333333 (pow eps 3.0)))))

C

double code(double x, double eps) {
	return fma(x, (eps * (eps + x)), (eps + (0.3333333333333333 * pow(eps, 3.0))));
}

Julia

function code(x, eps)
	return fma(x, Float64(eps * Float64(eps + x)), Float64(eps + Float64(0.3333333333333333 * (eps ^ 3.0))))
end

Wolfram

code[x_, eps_] := N[(x * N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] + N[(eps + N[(0.3333333333333333 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. tan-sum 66.0%

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

   2. div-inv 66.0%

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

   3. fma-neg 66.0%

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

4. Applied egg-rr 66.0%

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

5. Taylor expanded in eps around 0 99.7%

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

6. Taylor expanded in x around 0 98.7%

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

   1. +-commutative 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in x around 0 97.4%

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

    1. +-commutative 97.4%

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

    2. fma-define 97.4%

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

    3. unpow2 97.4%

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

    4. distribute-lft-out 97.4%

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

    5. distribute-rgt-in 97.4%

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

    6. *-lft-identity 97.4%

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

    7. associate-*r* 97.4%

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

    8. unpow2 97.4%

       \[\leadsto \mathsf{fma}\left(x, \varepsilon \cdot \left(x + \varepsilon\right), \varepsilon + 0.3333333333333333 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \varepsilon\right)\right) \]

    9. unpow3 97.4%

       \[\leadsto \mathsf{fma}\left(x, \varepsilon \cdot \left(x + \varepsilon\right), \varepsilon + 0.3333333333333333 \cdot \color{blue}{{\varepsilon}^{3}}\right) \]

11. Simplified 97.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \varepsilon \cdot \left(x + \varepsilon\right), \varepsilon + 0.3333333333333333 \cdot {\varepsilon}^{3}\right)} \]

12. Final simplification 97.4%

    \[\leadsto \mathsf{fma}\left(x, \varepsilon \cdot \left(\varepsilon + x\right), \varepsilon + 0.3333333333333333 \cdot {\varepsilon}^{3}\right) \]
13. Add Preprocessing

Alternative 8: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. tan-sum 66.0%

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

   2. div-inv 66.0%

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

   3. fma-neg 66.0%

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

4. Applied egg-rr 66.0%

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

5. Taylor expanded in eps around 0 99.7%

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

6. Taylor expanded in x around 0 98.7%

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

   1. +-commutative 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in x around 0 97.3%

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

    1. mul-1-neg 97.3%

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

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

Alternative 9: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. add-cbrt-cube 28.1%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\tan \left(x + \varepsilon\right) - \tan x\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)}} \]

   2. pow1/3 27.9%

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

   3. pow3 27.9%

      \[\leadsto {\color{blue}{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}}^{0.3333333333333333} \]

4. Applied egg-rr 27.9%

   \[\leadsto \color{blue}{{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}^{0.3333333333333333}} \]

5. Taylor expanded in eps around 0 35.3%

   \[\leadsto {\left({\color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}}^{3}\right)}^{0.3333333333333333} \]
6. Step-by-step derivation

   1. sub-neg 35.3%

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

   2. mul-1-neg 35.3%

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

   3. remove-double-neg 35.3%

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

7. Simplified 35.3%

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

8. Taylor expanded in x around 0 97.2%

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

Alternative 10: 98.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. add-cbrt-cube 28.1%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\tan \left(x + \varepsilon\right) - \tan x\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)}} \]

   2. pow1/3 27.9%

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

   3. pow3 27.9%

      \[\leadsto {\color{blue}{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}}^{0.3333333333333333} \]

4. Applied egg-rr 27.9%

   \[\leadsto \color{blue}{{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}^{0.3333333333333333}} \]

5. Taylor expanded in eps around 0 35.3%

   \[\leadsto {\left({\color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}}^{3}\right)}^{0.3333333333333333} \]
6. Step-by-step derivation

   1. sub-neg 35.3%

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

   2. mul-1-neg 35.3%

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

   3. remove-double-neg 35.3%

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

7. Simplified 35.3%

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

8. Taylor expanded in x around 0 97.2%

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

   1. *-rgt-identity 97.2%

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

   2. distribute-lft-out 97.2%

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

   3. +-commutative 97.2%

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

   4. unpow2 97.2%

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

   5. fma-define 97.2%

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

10. Simplified 97.2%

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

Alternative 11: 97.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 65.7%

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

   1. add-cbrt-cube 28.1%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\tan \left(x + \varepsilon\right) - \tan x\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)\right) \cdot \left(\tan \left(x + \varepsilon\right) - \tan x\right)}} \]

   2. pow1/3 27.9%

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

   3. pow3 27.9%

      \[\leadsto {\color{blue}{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}}^{0.3333333333333333} \]

4. Applied egg-rr 27.9%

   \[\leadsto \color{blue}{{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}^{0.3333333333333333}} \]

5. Taylor expanded in eps around 0 35.3%

   \[\leadsto {\left({\color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}}^{3}\right)}^{0.3333333333333333} \]
6. Step-by-step derivation

   1. sub-neg 35.3%

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

   2. mul-1-neg 35.3%

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

   3. remove-double-neg 35.3%

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

7. Simplified 35.3%

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

8. Taylor expanded in x around 0 96.9%

   \[\leadsto \color{blue}{\varepsilon} \]
9. 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 2024110 
(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)))