2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.7%

Time: 25.6s

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: 62.4% 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: 99.7% accurate, 0.1× speedup

Math

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

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)))
   (*
    eps
    (+
     t_2
     (+
      (*
       eps
       (-
        (/ (* (sin x) t_3) (cos x))
        (*
         eps
         (-
          0.16666666666666666
          (-
           (/ (* t_0 t_3) t_1)
           (+ (* t_3 -0.5) (* 0.16666666666666666 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;
	return eps * (t_2 + ((eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666 - (((t_0 * t_3) / t_1) - ((t_3 * -0.5) + (0.16666666666666666 * t_2))))))) + 1.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = sin(x) ** 2.0d0
    t_1 = cos(x) ** 2.0d0
    t_2 = t_0 / t_1
    t_3 = t_2 + 1.0d0
    code = eps * (t_2 + ((eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666d0 - (((t_0 * t_3) / t_1) - ((t_3 * (-0.5d0)) + (0.16666666666666666d0 * t_2))))))) + 1.0d0))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = Math.pow(Math.cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	return eps * (t_2 + ((eps * (((Math.sin(x) * t_3) / Math.cos(x)) - (eps * (0.16666666666666666 - (((t_0 * t_3) / t_1) - ((t_3 * -0.5) + (0.16666666666666666 * t_2))))))) + 1.0));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = math.pow(math.cos(x), 2.0)
	t_2 = t_0 / t_1
	t_3 = t_2 + 1.0
	return eps * (t_2 + ((eps * (((math.sin(x) * t_3) / math.cos(x)) - (eps * (0.16666666666666666 - (((t_0 * t_3) / t_1) - ((t_3 * -0.5) + (0.16666666666666666 * 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)
	return Float64(eps * Float64(t_2 + Float64(Float64(eps * Float64(Float64(Float64(sin(x) * t_3) / cos(x)) - Float64(eps * Float64(0.16666666666666666 - Float64(Float64(Float64(t_0 * t_3) / t_1) - Float64(Float64(t_3 * -0.5) + Float64(0.16666666666666666 * t_2))))))) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	t_1 = cos(x) ^ 2.0;
	t_2 = t_0 / t_1;
	t_3 = t_2 + 1.0;
	tmp = eps * (t_2 + ((eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666 - (((t_0 * t_3) / t_1) - ((t_3 * -0.5) + (0.16666666666666666 * 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]}, N[(eps * N[(t$95$2 + N[(N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.16666666666666666 - N[(N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(t$95$3 * -0.5), $MachinePrecision] + N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* eps (+ (pow (tan x) 2.0) (* eps (+ (tan x) (pow (tan x) 3.0)))))))

C

double code(double x, double eps) {
	return eps + (eps * (pow(tan(x), 2.0) + (eps * (tan(x) + pow(tan(x), 3.0)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * ((tan(x) ** 2.0d0) + (eps * (tan(x) + (tan(x) ** 3.0d0)))))
end function

Java

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

Python

def code(x, eps):
	return eps + (eps * (math.pow(math.tan(x), 2.0) + (eps * (math.tan(x) + math.pow(math.tan(x), 3.0)))))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * Float64((tan(x) ^ 2.0) + Float64(eps * Float64(tan(x) + (tan(x) ^ 3.0))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * ((tan(x) ^ 2.0) + (eps * (tan(x) + (tan(x) ^ 3.0)))));
end

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. tan-sum 63.9%

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

   2. div-inv 63.9%

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

   3. fma-neg 63.9%

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

4. Applied egg-rr 63.9%

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

5. Taylor expanded in eps around 0 99.3%

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

   1. associate--l+ 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-rgt-identity 99.3%

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

   4. associate-*r* 99.3%

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

   5. fma-neg 99.3%

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

7. Simplified 99.3%

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

   1. *-commutative 99.3%

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

9. Applied egg-rr 99.3%

   \[\leadsto \varepsilon + \color{blue}{\left(\left(0 - \varepsilon\right) \cdot \left(-1 \cdot \tan x - {\tan x}^{3}\right) + {\tan x}^{2}\right) \cdot \varepsilon} \]
10. Step-by-step derivation

    1. +-commutative 99.3%

       \[\leadsto \varepsilon + \color{blue}{\left({\tan x}^{2} + \left(0 - \varepsilon\right) \cdot \left(-1 \cdot \tan x - {\tan x}^{3}\right)\right)} \cdot \varepsilon \]

    2. add-sqr-sqrt 0.0%

       \[\leadsto \varepsilon + \left({\tan x}^{2} + \color{blue}{\left(\sqrt{0 - \varepsilon} \cdot \sqrt{0 - \varepsilon}\right)} \cdot \left(-1 \cdot \tan x - {\tan x}^{3}\right)\right) \cdot \varepsilon \]

    3. sqrt-unprod 98.7%

       \[\leadsto \varepsilon + \left({\tan x}^{2} + \color{blue}{\sqrt{\left(0 - \varepsilon\right) \cdot \left(0 - \varepsilon\right)}} \cdot \left(-1 \cdot \tan x - {\tan x}^{3}\right)\right) \cdot \varepsilon \]

    4. sub0-neg 98.7%

       \[\leadsto \varepsilon + \left({\tan x}^{2} + \sqrt{\color{blue}{\left(-\varepsilon\right)} \cdot \left(0 - \varepsilon\right)} \cdot \left(-1 \cdot \tan x - {\tan x}^{3}\right)\right) \cdot \varepsilon \]

    5. sub0-neg 98.7%

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

    6. sqr-neg 98.7%

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

    7. sqrt-unprod 98.7%

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

    8. add-sqr-sqrt 98.7%

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

    9. sub-neg 98.7%

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

    10. add-sqr-sqrt 51.5%

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

    11. sqrt-unprod 98.7%

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

    12. mul-1-neg 98.7%

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

    13. mul-1-neg 98.7%

        \[\leadsto \varepsilon + \left({\tan x}^{2} + \varepsilon \cdot \left(\sqrt{\left(-\tan x\right) \cdot \color{blue}{\left(-\tan x\right)}} + \left(-{\tan x}^{3}\right)\right)\right) \cdot \varepsilon \]

    14. sqr-neg 98.7%

        \[\leadsto \varepsilon + \left({\tan x}^{2} + \varepsilon \cdot \left(\sqrt{\color{blue}{\tan x \cdot \tan x}} + \left(-{\tan x}^{3}\right)\right)\right) \cdot \varepsilon \]

    15. sqrt-prod 47.3%

        \[\leadsto \varepsilon + \left({\tan x}^{2} + \varepsilon \cdot \left(\color{blue}{\sqrt{\tan x} \cdot \sqrt{\tan x}} + \left(-{\tan x}^{3}\right)\right)\right) \cdot \varepsilon \]

    16. add-sqr-sqrt 98.9%

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

    17. cube-neg 98.9%

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

    18. mul-1-neg 98.9%

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

11. Applied egg-rr 99.3%

    \[\leadsto \varepsilon + \color{blue}{\left({\tan x}^{2} + \varepsilon \cdot \left(\tan x + {\tan x}^{3}\right)\right)} \cdot \varepsilon \]

12. Final simplification 99.3%

    \[\leadsto \varepsilon + \varepsilon \cdot \left({\tan x}^{2} + \varepsilon \cdot \left(\tan x + {\tan x}^{3}\right)\right) \]
13. Add Preprocessing

Alternative 3: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0 99.4%

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

4. Taylor expanded in x around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. *-commutative 98.8%

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

   3. fma-define 98.8%

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

   4. unpow2 98.8%

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

6. Simplified 98.8%

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

   1. cancel-sign-sub-inv 98.8%

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

   2. fma-undefine 98.8%

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

   3. associate-*r* 98.8%

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

   4. metadata-eval 98.8%

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

   5. *-un-lft-identity 98.8%

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

   6. add-sqr-sqrt 98.8%

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

   7. pow2 98.8%

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

   8. sqrt-div 98.8%

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

   9. sqrt-pow1 98.8%

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

   10. metadata-eval 98.8%

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

   11. pow1 98.8%

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

   12. sqrt-pow1 98.8%

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

   13. metadata-eval 98.8%

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

   14. pow1 98.8%

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

   15. tan-quot 98.8%

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

8. Applied egg-rr 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. associate-+l+ 98.8%

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

   3. distribute-lft-in 98.8%

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

   4. *-rgt-identity 98.8%

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

   5. fma-define 98.8%

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

   6. +-commutative 98.8%

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

   7. *-commutative 98.8%

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

   8. fma-define 98.8%

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

10. Simplified 98.8%

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

Alternative 4: 99.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0 99.4%

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

4. Taylor expanded in x around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. *-commutative 98.8%

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

   3. fma-define 98.8%

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

   4. unpow2 98.8%

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

6. Simplified 98.8%

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

   1. *-commutative 98.8%

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

8. Applied egg-rr 98.8%

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

9. Final simplification 98.8%

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

Alternative 5: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. cancel-sign-sub-inv 98.7%

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

   2. distribute-lft-in 98.7%

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

   3. *-rgt-identity 98.7%

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

   4. metadata-eval 98.7%

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

   5. associate-*r/ 98.7%

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

   6. *-lft-identity 98.7%

      \[\leadsto \varepsilon + \varepsilon \cdot \frac{\color{blue}{{\sin x}^{2}}}{{\cos x}^{2}} \]

5. Simplified 98.7%

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

   1. associate-*r/ 98.7%

      \[\leadsto \varepsilon + \color{blue}{\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}} \]

   2. unpow2 98.7%

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

   3. sqr-sin-a 98.7%

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

   4. unpow2 98.7%

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

   5. sqr-cos-a 98.7%

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

7. Applied egg-rr 98.7%

   \[\leadsto \varepsilon + \color{blue}{\frac{\varepsilon \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}} \]
8. Step-by-step derivation

   1. associate-/l* 98.7%

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

   2. sqr-sin-a 98.7%

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

   3. unpow2 98.7%

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

   4. sqr-cos-a 98.7%

      \[\leadsto \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{\color{blue}{\cos x \cdot \cos x}} \]

   5. unpow2 98.7%

      \[\leadsto \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{\color{blue}{{\cos x}^{2}}} \]

   6. add-sqr-sqrt 98.7%

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

   7. pow2 98.7%

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

   8. sqrt-div 98.7%

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

   9. sqrt-pow1 98.7%

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

   10. metadata-eval 98.7%

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

   11. pow1 98.7%

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

   12. sqrt-pow1 98.7%

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

   13. metadata-eval 98.7%

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

   14. pow1 98.7%

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

   15. tan-quot 98.7%

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

9. Applied egg-rr 98.7%

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

Alternative 6: 98.6% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot \left(x \cdot \left(\varepsilon + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot 1.3333333333333333\right) + 1\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  eps
  (*
   eps
   (*
    x
    (+
     eps
     (*
      x
      (+
       (* x (+ (* x 0.6666666666666666) (* eps 1.3333333333333333)))
       1.0)))))))

C

double code(double x, double eps) {
	return eps + (eps * (x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))) + 1.0)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * (x * (eps + (x * ((x * ((x * 0.6666666666666666d0) + (eps * 1.3333333333333333d0))) + 1.0d0)))))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * (x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))) + 1.0)))));
}

Python

def code(x, eps):
	return eps + (eps * (x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))) + 1.0)))))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * Float64(x * Float64(eps + Float64(x * Float64(Float64(x * Float64(Float64(x * 0.6666666666666666) + Float64(eps * 1.3333333333333333))) + 1.0))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))) + 1.0)))));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[(x * N[(eps + N[(x * N[(N[(x * N[(N[(x * 0.6666666666666666), $MachinePrecision] + N[(eps * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. tan-sum 63.9%

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

   2. div-inv 63.9%

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

   3. fma-neg 63.9%

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

4. Applied egg-rr 63.9%

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

5. Taylor expanded in eps around 0 99.3%

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

   1. associate--l+ 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-rgt-identity 99.3%

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

   4. associate-*r* 99.3%

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

   5. fma-neg 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in x around 0 98.1%

   \[\leadsto \varepsilon + \varepsilon \cdot \color{blue}{\left(x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(0.6666666666666666 \cdot x + 1.3333333333333333 \cdot \varepsilon\right)\right)\right)\right)} \]

9. Final simplification 98.1%

   \[\leadsto \varepsilon + \varepsilon \cdot \left(x \cdot \left(\varepsilon + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot 1.3333333333333333\right) + 1\right)\right)\right) \]
10. Add Preprocessing

Alternative 7: 98.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps + (x * (eps * (eps + x)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. tan-sum 63.9%

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

   2. div-inv 63.9%

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

   3. fma-neg 63.9%

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

4. Applied egg-rr 63.9%

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

5. Taylor expanded in eps around 0 99.3%

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

   1. associate--l+ 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-rgt-identity 99.3%

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

   4. associate-*r* 99.3%

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

   5. fma-neg 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in x around 0 97.9%

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

   1. unpow2 97.9%

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

   2. distribute-lft-out 97.9%

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

10. Simplified 97.9%

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

11. Final simplification 97.9%

    \[\leadsto \varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right) \]
12. Add Preprocessing

Alternative 8: 98.4% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps + (eps * (x * x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. cancel-sign-sub-inv 98.7%

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

   2. distribute-lft-in 98.7%

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

   3. *-rgt-identity 98.7%

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

   4. metadata-eval 98.7%

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

   5. associate-*r/ 98.7%

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

   6. *-lft-identity 98.7%

      \[\leadsto \varepsilon + \varepsilon \cdot \frac{\color{blue}{{\sin x}^{2}}}{{\cos x}^{2}} \]

5. Simplified 98.7%

   \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

6. Taylor expanded in x around 0 97.9%

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

   1. *-commutative 97.9%

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

   2. unpow2 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

   \[\leadsto \varepsilon + \varepsilon \cdot \left(x \cdot x\right) \]
10. Add Preprocessing

Alternative 9: 98.1% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in x around 0 97.4%

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

4. Taylor expanded in eps around 0 97.4%

   \[\leadsto \color{blue}{\varepsilon} \]
5. 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 2024097 
(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)))