2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.6%

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: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

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 * (t_0 - ((-1.0d0) - (eps * (eps * (((0.3333333333333333d0 + ((tan(x) + ((sin(x) ** 3.0d0) * (cos(x) ** (-3.0d0)))) / eps)) + t_0) + (((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - (t_0 * (-0.3333333333333333d0))))))))
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 * (t_0 - (-1.0 - (eps * (eps * (((0.3333333333333333 + ((Math.tan(x) + (Math.pow(Math.sin(x), 3.0) * Math.pow(Math.cos(x), -3.0))) / eps)) + t_0) + ((Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0)) - (t_0 * -0.3333333333333333)))))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return eps * (t_0 - (-1.0 - (eps * (eps * (((0.3333333333333333 + ((math.tan(x) + (math.pow(math.sin(x), 3.0) * math.pow(math.cos(x), -3.0))) / eps)) + t_0) + ((math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0)) - (t_0 * -0.3333333333333333)))))))

Julia

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

MATLAB

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

Derivation

1. Initial program 61.5%

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

   1. tan-sum 61.8%

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

   2. div-inv 61.8%

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

   3. fma-neg 61.8%

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

4. Applied egg-rr 61.8%

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

   1. fma-neg 61.8%

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

   2. *-commutative 61.8%

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

   3. associate-*l/ 61.8%

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

   4. *-lft-identity 61.8%

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

6. Simplified 61.8%

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

7. Taylor expanded in eps around 0 99.2%

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

8. Taylor expanded in eps around inf 99.2%

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

   1. associate--r+ 99.2%

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

10. Simplified 99.2%

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

    1. sub-neg 99.2%

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

    2. distribute-frac-neg 99.2%

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

    3. tan-quot 99.2%

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

    4. div-inv 99.2%

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

    5. pow-flip 99.2%

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

    6. metadata-eval 99.2%

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

12. Applied egg-rr 99.2%

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

    1. sub-neg 99.2%

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

14. Simplified 99.2%

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

15. Final simplification 99.2%

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

Alternative 2: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + ((eps * (((sin(x) ^ 3.0) / (cos(x) ^ 3.0)) + (sin(x) / cos(x)))) + 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[(N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

   1. tan-sum 61.8%

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

   2. div-inv 61.8%

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

   3. fma-neg 61.8%

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

4. Applied egg-rr 61.8%

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

   1. fma-neg 61.8%

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

   2. *-commutative 61.8%

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

   3. associate-*l/ 61.8%

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

   4. *-lft-identity 61.8%

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

6. Simplified 61.8%

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

7. Taylor expanded in eps around 0 99.2%

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

8. Taylor expanded in eps around inf 99.2%

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

   1. associate--r+ 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in eps around 0 99.1%

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

    1. cancel-sign-sub-inv 99.1%

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

    2. metadata-eval 99.1%

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

    3. *-lft-identity 99.1%

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

13. Simplified 99.1%

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

14. Final simplification 99.1%

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

Alternative 3: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.2%

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

4. Taylor expanded in x around 0 98.5%

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

   1. *-commutative 98.5%

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

6. Simplified 98.5%

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

7. Final simplification 98.5%

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

Alternative 4: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 98.5%

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

   1. sub-neg 98.5%

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

   2. mul-1-neg 98.5%

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

   3. remove-double-neg 98.5%

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

5. Simplified 98.5%

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

6. Final simplification 98.5%

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

Alternative 5: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.2%

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

4. Taylor expanded in x around 0 97.6%

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

   1. fma-define 97.6%

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

   2. *-commutative 97.6%

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

   3. +-commutative 97.6%

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

   4. unpow2 97.6%

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

   5. distribute-lft-out 97.6%

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

   6. *-commutative 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

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

Alternative 6: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (((0.3333333333333333 * pow(eps, 2.0)) + (x * (eps - (x * (-1.0 - (pow(eps, 2.0) * 1.3333333333333333)))))) + 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 * (eps - (x * ((-1.0d0) - ((eps ** 2.0d0) * 1.3333333333333333d0)))))) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.2%

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

4. Taylor expanded in x around 0 97.6%

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

5. Final simplification 97.6%

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

Alternative 7: 98.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.2%

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

4. Taylor expanded in x around 0 96.9%

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

5. Final simplification 96.9%

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

Alternative 8: 98.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in x around 0 7.6%

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

4. Taylor expanded in eps around 0 7.6%

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

5. Taylor expanded in eps around inf 96.8%

   \[\leadsto \color{blue}{\varepsilon} \]
6. 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 2024139 
(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)))