2tan (problem 3.3.2)

Percentage Accurate: 62.8% → 99.6%

Time: 19.3s

Alternatives: 12

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: 99.6% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

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_2 0.16666666666666666) (* -0.5 t_3))
           (/ (* t_0 t_3) t_1))))))
      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_2 * 0.16666666666666666) + (-0.5 * t_3)) - ((t_0 * t_3) / t_1)))))) + 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_2 * 0.16666666666666666d0) + ((-0.5d0) * t_3)) - ((t_0 * t_3) / t_1)))))) + 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_2 * 0.16666666666666666) + (-0.5 * t_3)) - ((t_0 * t_3) / t_1)))))) + 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_2 * 0.16666666666666666) + (-0.5 * t_3)) - ((t_0 * t_3) / t_1)))))) + 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_2 * 0.16666666666666666) + Float64(-0.5 * t_3)) - Float64(Float64(t_0 * t_3) / t_1)))))) + 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_2 * 0.16666666666666666) + (-0.5 * t_3)) - ((t_0 * t_3) / t_1)))))) + 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$2 * 0.16666666666666666), $MachinePrecision] + N[(-0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.4%

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

   1. associate-/l* 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-/l* 99.4%

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

8. Simplified 99.4%

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

   1. distribute-rgt-in 99.4%

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

   2. *-un-lft-identity 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = tan(x) ^ 2.0;
	tmp = eps * ((t_0 + (eps * ((sin(x) * (t_0 + 1.0)) / cos(x)))) + 1.0);
end

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.4%

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

   1. associate-/l* 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-/l* 99.4%

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

8. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. fma-define 99.4%

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

   3. unpow2 99.4%

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

   4. unpow2 99.4%

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

   5. frac-times 99.4%

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

   6. tan-quot 99.4%

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

   7. tan-quot 99.4%

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

   8. pow2 99.4%

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

10. Applied egg-rr 99.4%

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

    1. fma-undefine 99.4%

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

    2. *-lft-identity 99.4%

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

    3. associate-*r/ 99.4%

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

    4. +-commutative 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 5: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.4%

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

   1. associate-/l* 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-/l* 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 99.0%

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

    1. +-commutative 99.0%

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

    2. add-sqr-sqrt 99.0%

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

    3. fma-define 99.0%

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

11. Applied egg-rr 71.3%

    \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\mathsf{hypot}\left(\tan x, \sqrt{\varepsilon \cdot \sin x}\right), \mathsf{hypot}\left(\tan x, \sqrt{\varepsilon \cdot \sin x}\right), 1\right)} \]
12. Step-by-step derivation

    1. fma-undefine 71.3%

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\mathsf{hypot}\left(\tan x, \sqrt{\varepsilon \cdot \sin x}\right) \cdot \mathsf{hypot}\left(\tan x, \sqrt{\varepsilon \cdot \sin x}\right) + 1\right)} \]

    2. hypot-undefine 71.3%

       \[\leadsto \varepsilon \cdot \left(\color{blue}{\sqrt{\tan x \cdot \tan x + \sqrt{\varepsilon \cdot \sin x} \cdot \sqrt{\varepsilon \cdot \sin x}}} \cdot \mathsf{hypot}\left(\tan x, \sqrt{\varepsilon \cdot \sin x}\right) + 1\right) \]

    3. hypot-undefine 71.3%

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

    4. rem-square-sqrt 71.3%

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

    5. unpow2 71.3%

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

    6. rem-square-sqrt 99.0%

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

    7. *-commutative 99.0%

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

13. Simplified 99.0%

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

14. Final simplification 99.0%

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

Alternative 6: 98.9% 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 61.5%

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

   1. distribute-rgt-in 98.8%

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

   2. *-un-lft-identity 98.8%

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

   3. unpow2 98.8%

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

   4. unpow2 98.8%

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

   5. frac-times 98.8%

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

   6. tan-quot 98.8%

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

   7. tan-quot 98.8%

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

   8. pow2 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

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

Alternative 7: 98.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot \left(x \cdot \left(\varepsilon + x \cdot \left(x \cdot \left(\left(x \cdot 0.6666666666666666 + \varepsilon \cdot 0.8333333333333334\right) - \varepsilon \cdot -0.5\right) + 1\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  eps
  (*
   eps
   (*
    x
    (+
     eps
     (*
      x
      (+
       (*
        x
        (-
         (+ (* x 0.6666666666666666) (* eps 0.8333333333333334))
         (* eps -0.5)))
       1.0)))))))

C

double code(double x, double eps) {
	return eps + (eps * (x * (eps + (x * ((x * (((x * 0.6666666666666666) + (eps * 0.8333333333333334)) - (eps * -0.5))) + 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 * 0.8333333333333334d0)) - (eps * (-0.5d0)))) + 1.0d0)))))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * (x * (eps + (x * ((x * (((x * 0.6666666666666666) + (eps * 0.8333333333333334)) - (eps * -0.5))) + 1.0)))));
}

Python

def code(x, eps):
	return eps + (eps * (x * (eps + (x * ((x * (((x * 0.6666666666666666) + (eps * 0.8333333333333334)) - (eps * -0.5))) + 1.0)))))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * Float64(x * Float64(eps + Float64(x * Float64(Float64(x * Float64(Float64(Float64(x * 0.6666666666666666) + Float64(eps * 0.8333333333333334)) - Float64(eps * -0.5))) + 1.0))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (x * (eps + (x * ((x * (((x * 0.6666666666666666) + (eps * 0.8333333333333334)) - (eps * -0.5))) + 1.0)))));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[(x * N[(eps + N[(x * N[(N[(x * N[(N[(N[(x * 0.6666666666666666), $MachinePrecision] + N[(eps * 0.8333333333333334), $MachinePrecision]), $MachinePrecision] - N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.4%

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

   1. associate-/l* 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-/l* 99.4%

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

8. Simplified 99.4%

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

   1. distribute-rgt-in 99.4%

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

   2. *-un-lft-identity 99.4%

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

10. Applied egg-rr 99.4%

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

11. Taylor expanded in x around 0 98.6%

    \[\leadsto \varepsilon + \color{blue}{\left(x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\left(0.6666666666666666 \cdot x + 0.8333333333333334 \cdot \varepsilon\right) - -0.5 \cdot \varepsilon\right)\right)\right)\right)} \cdot \varepsilon \]

12. Final simplification 98.6%

    \[\leadsto \varepsilon + \varepsilon \cdot \left(x \cdot \left(\varepsilon + x \cdot \left(x \cdot \left(\left(x \cdot 0.6666666666666666 + \varepsilon \cdot 0.8333333333333334\right) - \varepsilon \cdot -0.5\right) + 1\right)\right)\right) \]
13. Add Preprocessing

Alternative 8: 98.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(x \cdot \left(\varepsilon + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot -0.16666666666666666\right) + 1\right)\right) + 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (*
    x
    (+
     eps
     (*
      x
      (+
       (* x (+ (* x 0.6666666666666666) (* eps -0.16666666666666666)))
       1.0))))
   1.0)))

C

double code(double x, double eps) {
	return eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * -0.16666666666666666))) + 1.0)))) + 1.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666d0) + (eps * (-0.16666666666666666d0)))) + 1.0d0)))) + 1.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * -0.16666666666666666))) + 1.0)))) + 1.0);
}

Python

def code(x, eps):
	return eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * -0.16666666666666666))) + 1.0)))) + 1.0)

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(x * Float64(eps + Float64(x * Float64(Float64(x * Float64(Float64(x * 0.6666666666666666) + Float64(eps * -0.16666666666666666))) + 1.0)))) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * -0.16666666666666666))) + 1.0)))) + 1.0);
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(x * N[(eps + N[(x * N[(N[(x * N[(N[(x * 0.6666666666666666), $MachinePrecision] + N[(eps * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.4%

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

   1. associate-/l* 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-/l* 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 99.0%

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

10. Taylor expanded in x around 0 98.5%

    \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(-0.16666666666666666 \cdot \varepsilon + 0.6666666666666666 \cdot x\right)\right)\right)}\right) \]

11. Final simplification 98.5%

    \[\leadsto \varepsilon \cdot \left(x \cdot \left(\varepsilon + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot -0.16666666666666666\right) + 1\right)\right) + 1\right) \]
12. Add Preprocessing

Alternative 9: 98.2% 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 61.5%

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

3. Taylor expanded in eps around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.4%

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

   1. associate-/l* 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-/l* 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 98.5%

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

    1. unpow2 98.5%

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

    2. distribute-lft-out 98.5%

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

    3. +-commutative 98.5%

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

11. Simplified 98.5%

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

Alternative 10: 98.2% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.4%

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

   1. associate-/l* 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-/l* 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 98.4%

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

    1. +-commutative 98.4%

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

11. Simplified 98.4%

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

12. Final simplification 98.4%

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

Alternative 11: 97.8% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.4%

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

   1. associate-/l* 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-/l* 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 98.1%

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

    1. +-commutative 98.1%

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

11. Simplified 98.1%

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

12. Final simplification 98.1%

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

Alternative 12: 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 61.5%

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

3. Taylor expanded in eps around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.4%

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

   1. associate-/l* 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-/l* 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 98.1%

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

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024121 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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