2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.5%

Time: 18.4s

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: 62.4% 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: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(t_0 * (cos(x) ^ -2.0))
	t_2 = Float64(t_1 + 1.0)
	return Float64(eps * Float64(fma(eps, fma(Float64(-eps), Float64(0.16666666666666666 + fma(-1.0, Float64(t_0 * Float64(t_2 / (cos(x) ^ 2.0))), fma(0.16666666666666666, t_1, Float64(t_2 * -0.5)))), Float64(sin(x) * Float64(t_2 / cos(x)))), 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[(t$95$0 * N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1.0), $MachinePrecision]}, N[(eps * N[(N[(eps * N[((-eps) * N[(0.16666666666666666 + N[(-1.0 * N[(t$95$0 * N[(t$95$2 / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * t$95$1 + N[(t$95$2 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(t$95$2 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

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 (- -1.0 t_2)) t_1)
           (+ (* -0.5 t_3) (* 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 * (-1.0 - t_2)) / t_1) + ((-0.5 * t_3) + (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 * ((-1.0d0) - t_2)) / t_1) + (((-0.5d0) * t_3) + (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 * (-1.0 - t_2)) / t_1) + ((-0.5 * t_3) + (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 * (-1.0 - t_2)) / t_1) + ((-0.5 * t_3) + (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 * Float64(-1.0 - t_2)) / t_1) + Float64(Float64(-0.5 * t_3) + 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 * (-1.0 - t_2)) / t_1) + ((-0.5 * t_3) + (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 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(-0.5 * t$95$3), $MachinePrecision] + N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.3% accurate, 0.3× speedup

Math

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

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(Float64(eps * 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[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in eps around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-*r* 99.8%

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

   3. *-commutative 99.8%

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

10. Simplified 99.8%

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

    1. distribute-rgt-in 99.8%

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

    2. *-un-lft-identity 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 4: 99.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (* eps (+ (- t_0 (* eps (/ (* (sin x) (- -1.0 t_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) * (-1.0 - t_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) * ((-1.0d0) - t_0)) / 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) * (-1.0 - t_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) * (-1.0 - t_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(-1.0 - t_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) * (-1.0 - t_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[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in eps around 0 99.8%

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

   1. fma-undefine 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 5: 98.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

4. Taylor expanded in x around 0 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

4. Taylor expanded in x around 0 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in eps around 0 99.3%

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

   1. *-commutative 99.3%

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

10. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. add-sqr-sqrt 99.3%

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

    3. pow2 99.3%

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

    4. sqrt-div 99.3%

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

    5. sqrt-pow1 99.3%

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

    6. metadata-eval 99.3%

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

    7. pow1 99.3%

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

    8. sqrt-pow1 99.3%

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

    9. metadata-eval 99.3%

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

    10. pow1 99.3%

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

    11. tan-quot 99.3%

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

12. Applied egg-rr 99.3%

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

    1. *-lft-identity 99.3%

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

14. Simplified 99.3%

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

15. Final simplification 99.3%

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

Alternative 8: 98.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))) + 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 * 1.3333333333333333d0))) + 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 * 1.3333333333333333))) + 1.0)))) + 1.0);
}

Python

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))) + 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 * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in eps around 0 99.8%

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

9. Taylor expanded in x around 0 99.0%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\left(0.6666666666666666 \cdot x + 0.8333333333333334 \cdot \varepsilon\right) - -0.5 \cdot \varepsilon\right)\right)\right)}\right) \]
10. Step-by-step derivation

    1. associate--l+ 99.0%

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

    2. *-commutative 99.0%

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

    3. distribute-rgt-out-- 99.0%

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

    4. metadata-eval 99.0%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 9: 98.1% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in eps around 0 99.8%

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

9. Taylor expanded in x around 0 98.9%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)\right)}\right) \]
10. Step-by-step derivation

    1. distribute-rgt-out-- 98.9%

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

    2. metadata-eval 98.9%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 10: 98.1% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in eps around 0 99.8%

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

9. Taylor expanded in x around 0 98.9%

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

    1. +-commutative 98.9%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 11: 97.7% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in eps around 0 99.8%

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

9. Taylor expanded in x around 0 98.1%

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

    1. *-commutative 98.1%

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

11. Simplified 98.1%

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

12. Final simplification 98.1%

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

Alternative 12: 97.7% 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 61.2%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in eps around 0 99.8%

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

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 2024116 
(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)))