2tan (problem 3.3.2)

Percentage Accurate: 62.8% → 99.5%

Time: 29.8s

Alternatives: 9

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := t\_2 + 1\\ t_4 := \mathsf{fma}\left(-0.5, t\_3, 0.16666666666666666 \cdot t\_2\right)\\ t_5 := \frac{\sin x}{\cos x}\\ t_6 := t\_3 \cdot t\_5\\ \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(0.16666666666666666 + \left(t\_4 + t\_0 \cdot \frac{-1 - t\_2}{t\_1}\right)\right) \cdot t\_5 + t\_6 \cdot -0.3333333333333333, -0.16666666666666666\right) + \left(t\_0 \cdot \frac{t\_3}{t\_1} - t\_4\right), t\_6\right), t\_2\right) + 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (+ t_2 1.0))
        (t_4 (fma -0.5 t_3 (* 0.16666666666666666 t_2)))
        (t_5 (/ (sin x) (cos x)))
        (t_6 (* t_3 t_5)))
   (*
    eps
    (+
     (fma
      eps
      (fma
       eps
       (+
        (fma
         (- eps)
         (+
          (* (+ 0.16666666666666666 (+ t_4 (* t_0 (/ (- -1.0 t_2) t_1)))) t_5)
          (* t_6 -0.3333333333333333))
         -0.16666666666666666)
        (- (* t_0 (/ t_3 t_1)) t_4))
       t_6)
      t_2)
     1.0))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	double t_4 = fma(-0.5, t_3, (0.16666666666666666 * t_2));
	double t_5 = sin(x) / cos(x);
	double t_6 = t_3 * t_5;
	return eps * (fma(eps, fma(eps, (fma(-eps, (((0.16666666666666666 + (t_4 + (t_0 * ((-1.0 - t_2) / t_1)))) * t_5) + (t_6 * -0.3333333333333333)), -0.16666666666666666) + ((t_0 * (t_3 / t_1)) - t_4)), t_6), t_2) + 1.0);
}

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(t_2 + 1.0)
	t_4 = fma(-0.5, t_3, Float64(0.16666666666666666 * t_2))
	t_5 = Float64(sin(x) / cos(x))
	t_6 = Float64(t_3 * t_5)
	return Float64(eps * Float64(fma(eps, fma(eps, Float64(fma(Float64(-eps), Float64(Float64(Float64(0.16666666666666666 + Float64(t_4 + Float64(t_0 * Float64(Float64(-1.0 - t_2) / t_1)))) * t_5) + Float64(t_6 * -0.3333333333333333)), -0.16666666666666666) + Float64(Float64(t_0 * Float64(t_3 / t_1)) - t_4)), t_6), t_2) + 1.0))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * t$95$3 + N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 * t$95$5), $MachinePrecision]}, N[(eps * N[(N[(eps * N[(eps * N[(N[((-eps) * N[(N[(N[(0.16666666666666666 + N[(t$95$4 + N[(t$95$0 * N[(N[(-1.0 - t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] + N[(t$95$6 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + N[(N[(t$95$0 * N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $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 99.4%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(N[(N[(t$95$1 / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(eps * N[(N[(0.3333333333333333 + N[(t$95$1 * t$95$0 + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.3333333333333333 * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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. fma-neg 62.2%

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

4. Applied egg-rr 62.2%

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

5. Taylor expanded in eps around 0 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in eps around 0 99.4%

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

10. Applied egg-rr 99.4%

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

    1. unpow1 99.4%

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

    2. associate-+r- 99.4%

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

    3. *-commutative 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

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)))) * (exp(sin(x)) ** (eps / cos(x)))
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)))) * Math.pow(Math.exp(Math.sin(x)), (eps / Math.cos(x)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(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] * N[Power[N[Exp[N[Sin[x], $MachinePrecision]], $MachinePrecision], N[(eps / N[Cos[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. add-exp-log 61.1%

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in eps around 0 89.8%

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

   1. mul-1-neg 89.8%

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

   2. *-commutative 89.8%

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

7. Simplified 89.8%

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

   1. associate-+r+ 89.8%

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

   2. exp-sum 89.8%

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

   3. sum-log 89.8%

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

   4. distribute-neg-frac 89.8%

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

   5. associate-/l* 89.8%

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

   6. exp-prod 89.8%

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

9. Applied egg-rr 89.8%

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

    1. rem-exp-log 99.2%

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

    2. sub-neg 99.2%

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

    3. distribute-neg-frac 99.2%

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

    4. remove-double-neg 99.2%

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

    5. +-commutative 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 4: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return pow(exp(sin(x)), (eps / cos(x))) * (eps * (((0.5 - (cos((x * 2.0)) / 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 = (exp(sin(x)) ** (eps / cos(x))) * (eps * (((0.5d0 - (cos((x * 2.0d0)) / 2.0d0)) / (cos(x) ** 2.0d0)) + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[Power[N[Exp[N[Sin[x], $MachinePrecision]], $MachinePrecision], N[(eps / N[Cos[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(eps * N[(N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $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. add-exp-log 61.1%

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in eps around 0 89.8%

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

   1. mul-1-neg 89.8%

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

   2. *-commutative 89.8%

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

7. Simplified 89.8%

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

   1. associate-+r+ 89.8%

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

   2. exp-sum 89.8%

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

   3. sum-log 89.8%

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

   4. distribute-neg-frac 89.8%

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

   5. associate-/l* 89.8%

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

   6. exp-prod 89.8%

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

9. Applied egg-rr 89.8%

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

    1. rem-exp-log 99.2%

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

    2. sub-neg 99.2%

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

    3. distribute-neg-frac 99.2%

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

    4. remove-double-neg 99.2%

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

    5. +-commutative 99.2%

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

11. Simplified 99.2%

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

    1. unpow2 99.2%

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

    2. sin-mult 99.2%

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

13. Applied egg-rr 99.2%

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

    1. div-sub 99.2%

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

    2. +-inverses 99.2%

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

    3. cos-0 99.2%

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

    4. metadata-eval 99.2%

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

    5. count-2 99.2%

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

    6. *-commutative 99.2%

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

15. Simplified 99.2%

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

16. Final simplification 99.2%

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

Alternative 5: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

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))) * exp((eps * (sin(x) / cos(x)))))
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))) * Math.exp((eps * (Math.sin(x) / Math.cos(x)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(eps * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $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. add-exp-log 61.1%

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in eps around 0 89.8%

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

   1. mul-1-neg 89.8%

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

   2. *-commutative 89.8%

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

7. Simplified 89.8%

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

   1. associate-+r+ 89.8%

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

   2. exp-sum 89.8%

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

   3. sum-log 89.8%

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

   4. distribute-neg-frac 89.8%

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

   5. associate-/l* 89.8%

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

   6. exp-prod 89.8%

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

9. Applied egg-rr 89.8%

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

    1. rem-exp-log 99.2%

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

    2. sub-neg 99.2%

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

    3. distribute-neg-frac 99.2%

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

    4. remove-double-neg 99.2%

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

    5. +-commutative 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in eps around inf 99.2%

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

    1. *-commutative 99.2%

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

    2. +-commutative 99.2%

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

    3. associate-/l* 99.2%

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

14. Simplified 99.2%

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

15. Final simplification 99.2%

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

Alternative 6: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0 98.8%

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

   1. sub-neg 98.8%

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

   2. mul-1-neg 98.8%

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

   3. remove-double-neg 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 7: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon - x \cdot \left(\varepsilon \cdot \left(x \cdot \left(-1 - 0.5 \cdot {\varepsilon}^{2}\right)\right) - {\varepsilon}^{2}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (- eps (* x (- (* eps (* x (- -1.0 (* 0.5 (pow eps 2.0))))) (pow eps 2.0)))))

C

double code(double x, double eps) {
	return eps - (x * ((eps * (x * (-1.0 - (0.5 * pow(eps, 2.0))))) - pow(eps, 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps - (x * ((eps * (x * ((-1.0d0) - (0.5d0 * (eps ** 2.0d0))))) - (eps ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps - (x * ((eps * (x * (-1.0 - (0.5 * Math.pow(eps, 2.0))))) - Math.pow(eps, 2.0)));
}

Python

def code(x, eps):
	return eps - (x * ((eps * (x * (-1.0 - (0.5 * math.pow(eps, 2.0))))) - math.pow(eps, 2.0)))

Julia

function code(x, eps)
	return Float64(eps - Float64(x * Float64(Float64(eps * Float64(x * Float64(-1.0 - Float64(0.5 * (eps ^ 2.0))))) - (eps ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps - (x * ((eps * (x * (-1.0 - (0.5 * (eps ^ 2.0))))) - (eps ^ 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps - N[(x * N[(N[(eps * N[(x * N[(-1.0 - N[(0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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. Step-by-step derivation

   1. add-exp-log 61.1%

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in eps around 0 89.8%

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

   1. mul-1-neg 89.8%

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

   2. *-commutative 89.8%

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

7. Simplified 89.8%

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

8. Taylor expanded in x around 0 98.1%

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

9. Final simplification 98.1%

   \[\leadsto \varepsilon - x \cdot \left(\varepsilon \cdot \left(x \cdot \left(-1 - 0.5 \cdot {\varepsilon}^{2}\right)\right) - {\varepsilon}^{2}\right) \]
10. Add Preprocessing

Alternative 8: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon + 0.3333333333333333 \cdot {\varepsilon}^{3} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (0.3333333333333333d0 * (eps ** 3.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps + (0.3333333333333333 * Math.pow(eps, 3.0));
}

Python

def code(x, eps):
	return eps + (0.3333333333333333 * math.pow(eps, 3.0))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps + (0.3333333333333333 * (eps ^ 3.0));
end

Wolfram

code[x_, eps_] := N[(eps + N[(0.3333333333333333 * N[Power[eps, 3.0], $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 98.0%

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

4. Taylor expanded in eps around 0 98.1%

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

   1. distribute-lft-in 98.1%

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

   2. *-rgt-identity 98.1%

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

   3. *-commutative 98.1%

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

   4. associate-*r* 98.1%

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

   5. unpow2 98.1%

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

   6. cube-mult 98.1%

      \[\leadsto \varepsilon + \color{blue}{{\varepsilon}^{3}} \cdot 0.3333333333333333 \]

6. Simplified 98.1%

   \[\leadsto \color{blue}{\varepsilon + {\varepsilon}^{3} \cdot 0.3333333333333333} \]

7. Final simplification 98.1%

   \[\leadsto \varepsilon + 0.3333333333333333 \cdot {\varepsilon}^{3} \]
8. Add Preprocessing

Alternative 9: 97.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in x around 0 98.0%

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

4. Taylor expanded in eps around 0 98.1%

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

5. Final simplification 98.1%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024080 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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