2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.5%

Time: 20.7s

Alternatives: 8

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: 62.5% 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := {\sin x}^{2}\\ t_2 := \frac{t\_1}{t\_0}\\ t_3 := -1 - t\_2\\ t_4 := t\_2 + 1\\ t_5 := \mathsf{fma}\left(-0.5, t\_4, \frac{t\_1 \cdot 0.16666666666666666}{t\_0}\right) + t\_1 \cdot \frac{t\_3}{t\_0}\\ \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(-0.16666666666666666 - t\_5\right) + \varepsilon \cdot \left(-0.3333333333333333 \cdot \left(\sin x \cdot \frac{t\_3}{\cos x}\right) - \left(0.16666666666666666 + t\_5\right) \cdot \frac{\sin x}{\cos x}\right), \sin x \cdot \frac{t\_4}{\cos x}\right), t\_2\right) + 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (/ t_1 t_0))
        (t_3 (- -1.0 t_2))
        (t_4 (+ t_2 1.0))
        (t_5
         (+
          (fma -0.5 t_4 (/ (* t_1 0.16666666666666666) t_0))
          (* t_1 (/ t_3 t_0)))))
   (*
    eps
    (+
     (fma
      eps
      (fma
       eps
       (+
        (- -0.16666666666666666 t_5)
        (*
         eps
         (-
          (* -0.3333333333333333 (* (sin x) (/ t_3 (cos x))))
          (* (+ 0.16666666666666666 t_5) (/ (sin x) (cos x))))))
       (* (sin x) (/ t_4 (cos x))))
      t_2)
     1.0))))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = t_1 / t_0;
	double t_3 = -1.0 - t_2;
	double t_4 = t_2 + 1.0;
	double t_5 = fma(-0.5, t_4, ((t_1 * 0.16666666666666666) / t_0)) + (t_1 * (t_3 / t_0));
	return eps * (fma(eps, fma(eps, ((-0.16666666666666666 - t_5) + (eps * ((-0.3333333333333333 * (sin(x) * (t_3 / cos(x)))) - ((0.16666666666666666 + t_5) * (sin(x) / cos(x)))))), (sin(x) * (t_4 / cos(x)))), t_2) + 1.0);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(t_1 / t_0)
	t_3 = Float64(-1.0 - t_2)
	t_4 = Float64(t_2 + 1.0)
	t_5 = Float64(fma(-0.5, t_4, Float64(Float64(t_1 * 0.16666666666666666) / t_0)) + Float64(t_1 * Float64(t_3 / t_0)))
	return Float64(eps * Float64(fma(eps, fma(eps, Float64(Float64(-0.16666666666666666 - t_5) + Float64(eps * Float64(Float64(-0.3333333333333333 * Float64(sin(x) * Float64(t_3 / cos(x)))) - Float64(Float64(0.16666666666666666 + t_5) * Float64(sin(x) / cos(x)))))), Float64(sin(x) * Float64(t_4 / cos(x)))), t_2) + 1.0))
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]}, Block[{t$95$2 = N[(t$95$1 / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 - t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-0.5 * t$95$4 + N[(N[(t$95$1 * 0.16666666666666666), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(t$95$3 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(eps * N[(eps * N[(N[(-0.16666666666666666 - t$95$5), $MachinePrecision] + N[(eps * N[(N[(-0.3333333333333333 * N[(N[Sin[x], $MachinePrecision] * N[(t$95$3 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.16666666666666666 + t$95$5), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(t$95$4 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(t\_0 + \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - t\_0 \cdot -1.3333333333333333\right), \tan x + {\tan x}^{3}\right) + 1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    eps
    (+
     t_0
     (+
      (*
       eps
       (fma
        eps
        (+
         0.3333333333333333
         (-
          (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
          (* t_0 -1.3333333333333333)))
        (+ (tan x) (pow (tan x) 3.0))))
      1.0)))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * (t_0 + ((eps * fma(eps, (0.3333333333333333 + ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) - (t_0 * -1.3333333333333333))), (tan(x) + pow(tan(x), 3.0)))) + 1.0));
}

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(t_0 + Float64(Float64(eps * fma(eps, Float64(0.3333333333333333 + Float64(Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - Float64(t_0 * -1.3333333333333333))), Float64(tan(x) + (tan(x) ^ 3.0)))) + 1.0)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(t$95$0 + N[(N[(eps * N[(eps * N[(0.3333333333333333 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * -1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.1%

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

   1. tan-sum 63.3%

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

   2. div-inv 63.3%

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

   3. fma-neg 63.3%

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

4. Applied egg-rr 63.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
5. Step-by-step derivation

   1. fma-neg 63.3%

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

   2. *-commutative 63.3%

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

   3. associate-*l/ 63.3%

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

   4. *-lft-identity 63.3%

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

6. Simplified 63.3%

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

7. Taylor expanded in eps around 0 99.7%

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

9. Applied egg-rr 99.7%

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

    1. unpow1 99.7%

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

    2. fma-neg 99.7%

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

    3. *-rgt-identity 99.7%

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

    4. cancel-sign-sub-inv 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in x around inf 99.7%

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

    1. distribute-rgt-out 99.7%

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

    2. metadata-eval 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \color{blue}{-1.3333333333333333} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \tan x + {\tan x}^{3}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

14. Simplified 99.7%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \color{blue}{\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -1.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)}, \tan x + {\tan x}^{3}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

15. Final simplification 99.7%

    \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -1.3333333333333333\right), \tan x + {\tan x}^{3}\right) + 1\right)\right) \]
16. Add Preprocessing

Alternative 3: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[(eps * 0.3333333333333333 + N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.1%

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

   1. tan-sum 63.3%

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

   2. div-inv 63.3%

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

   3. fma-neg 63.3%

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

4. Applied egg-rr 63.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
5. Step-by-step derivation

   1. fma-neg 63.3%

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

   2. *-commutative 63.3%

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

   3. associate-*l/ 63.3%

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

   4. *-lft-identity 63.3%

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

6. Simplified 63.3%

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

7. Taylor expanded in eps around 0 99.7%

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

9. Applied egg-rr 99.7%

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

    1. unpow1 99.7%

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

    2. fma-neg 99.7%

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

    3. *-rgt-identity 99.7%

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

    4. cancel-sign-sub-inv 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in x around 0 99.5%

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

13. Final simplification 99.5%

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

Alternative 4: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. sub-neg 98.9%

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

   2. mul-1-neg 98.9%

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

   3. remove-double-neg 98.9%

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Alternative 5: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} - x \cdot \left(x \cdot \left(-1 - {\varepsilon}^{2} \cdot 1.3333333333333333\right) - \varepsilon\right)\right) + 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (-
    (* 0.3333333333333333 (pow eps 2.0))
    (* x (- (* x (- -1.0 (* (pow eps 2.0) 1.3333333333333333))) eps)))
   1.0)))

C

double code(double x, double eps) {
	return eps * (((0.3333333333333333 * pow(eps, 2.0)) - (x * ((x * (-1.0 - (pow(eps, 2.0) * 1.3333333333333333))) - eps))) + 1.0);
}

Fortran

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

Java

public static double code(double x, double eps) {
	return eps * (((0.3333333333333333 * Math.pow(eps, 2.0)) - (x * ((x * (-1.0 - (Math.pow(eps, 2.0) * 1.3333333333333333))) - eps))) + 1.0);
}

Python

def code(x, eps):
	return eps * (((0.3333333333333333 * math.pow(eps, 2.0)) - (x * ((x * (-1.0 - (math.pow(eps, 2.0) * 1.3333333333333333))) - eps))) + 1.0)

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(Float64(0.3333333333333333 * (eps ^ 2.0)) - Float64(x * Float64(Float64(x * Float64(-1.0 - Float64((eps ^ 2.0) * 1.3333333333333333))) - eps))) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((0.3333333333333333 * (eps ^ 2.0)) - (x * ((x * (-1.0 - ((eps ^ 2.0) * 1.3333333333333333))) - eps))) + 1.0);
end

Wolfram

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

Derivation

1. Initial program 63.1%

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

   1. tan-sum 63.3%

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

   2. div-inv 63.3%

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

   3. fma-neg 63.3%

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

4. Applied egg-rr 63.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
5. Step-by-step derivation

   1. fma-neg 63.3%

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

   2. *-commutative 63.3%

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

   3. associate-*l/ 63.3%

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

   4. *-lft-identity 63.3%

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

6. Simplified 63.3%

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

7. Taylor expanded in eps around 0 99.7%

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

8. Taylor expanded in x around 0 98.7%

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

9. Final simplification 98.7%

   \[\leadsto \varepsilon \cdot \left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} - x \cdot \left(x \cdot \left(-1 - {\varepsilon}^{2} \cdot 1.3333333333333333\right) - \varepsilon\right)\right) + 1\right) \]
10. Add Preprocessing

Alternative 6: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 98.3%

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

   1. *-commutative 98.3%

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 7: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in x around 0 98.3%

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

   1. tan-quot 98.3%

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

   2. *-un-lft-identity 98.3%

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

5. Applied egg-rr 98.3%

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

   1. *-lft-identity 98.3%

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

7. Simplified 98.3%

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

Alternative 8: 97.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in x around 0 98.3%

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

4. Taylor expanded in eps around 0 98.3%

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

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024146 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (/ (sin eps) (* (cos x) (cos (+ x eps)))))

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