2tan (problem 3.3.2)

Percentage Accurate: 62.2% → 99.6%

Time: 19.1s

Alternatives: 14

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: 62.2% 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: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(1.0 + fma(eps, Float64(fma(eps, Float64(Float64(Float64(0.3333333333333333 + t_0) - Float64(t_0 * -0.3333333333333333)) + Float64((sin(x) ^ 4.0) * (cos(x) ^ -4.0))), tan(x)) + (tan(x) ^ 3.0)), t_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[(1.0 + N[(eps * N[(N[(eps * N[(N[(N[(0.3333333333333333 + t$95$0), $MachinePrecision] - N[(t$95$0 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.0%

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

   1. tan-sum 62.2%

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

   2. div-inv 62.2%

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

   3. fmm-def 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 100.0%

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

7. Simplified 100.0%

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

9. Applied egg-rr 100.0%

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

    1. associate-*r/ 100.0%

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

    2. *-commutative 100.0%

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

    3. remove-double-neg 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in x around inf 100.0%

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

13. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(1.0 + fma(eps, Float64(fma(eps, Float64(0.3333333333333333 + Float64(Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - Float64(t_0 * -0.3333333333333333))), Float64(sin(x) / cos(x))) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))), t_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[(1.0 + N[(eps * N[(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 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.0%

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

   1. tan-sum 62.2%

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

   2. div-inv 62.2%

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

   3. fmm-def 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. tan-sum 62.2%

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

   2. div-inv 62.2%

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

   3. fmm-def 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 100.0%

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

7. Simplified 100.0%

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

9. Applied egg-rr 100.0%

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

    1. associate-*r/ 100.0%

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

    2. *-commutative 100.0%

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

    3. remove-double-neg 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in x around 0 99.8%

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

13. Final simplification 99.8%

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

Alternative 4: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * (1.0 + (t_0 + (eps * ((eps * 0.3333333333333333) + ((sin(x) * (1.0 + t_0)) / cos(x))))));
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[(1.0 + N[(t$95$0 + N[(eps * N[(N[(eps * 0.3333333333333333), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in eps around 0 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (1.0 + ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + (eps * ((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 * (1.0d0 + (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + (eps * ((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 * (1.0 + ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + (eps * ((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 * (1.0 + ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + (eps * ((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(1.0 + Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + Float64(eps * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. tan-sum 62.2%

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

   2. div-inv 62.2%

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

   3. fmm-def 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 100.0%

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

7. Simplified 100.0%

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

9. Applied egg-rr 100.0%

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

    1. associate-*r/ 100.0%

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

    2. *-commutative 100.0%

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

    3. remove-double-neg 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in eps around 0 99.7%

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

13. Final simplification 99.7%

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

Alternative 6: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in eps around inf 98.9%

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

10. Final simplification 98.9%

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

Alternative 7: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

Alternative 8: 98.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in x around 0 97.4%

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

Alternative 9: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. tan-sum 62.2%

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

   2. div-inv 62.2%

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

   3. fmm-def 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 97.4%

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

9. Taylor expanded in eps around 0 97.4%

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

    1. +-commutative 97.4%

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

11. Simplified 97.4%

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

12. Final simplification 97.4%

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

Alternative 10: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in x around 0 97.4%

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

Alternative 11: 97.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. tan-sum 62.2%

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

   2. div-inv 62.2%

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

   3. fmm-def 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 97.3%

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

Alternative 12: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in x around 0 97.2%

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

4. Taylor expanded in eps around 0 97.2%

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

Alternative 13: 97.9% 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 62.0%

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

3. Taylor expanded in x around 0 97.2%

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

   1. tan-quot 97.2%

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

   2. *-un-lft-identity 97.2%

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

5. Applied egg-rr 97.2%

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

   1. *-lft-identity 97.2%

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

7. Simplified 97.2%

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

Alternative 14: 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 62.0%

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

3. Taylor expanded in x around 0 97.2%

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

4. Taylor expanded in eps around 0 97.2%

   \[\leadsto \color{blue}{\varepsilon} \]
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 2024150 
(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)))