2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.7%

Time: 20.1s

Alternatives: 10

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: 62.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (+ t_2 1.0)))
   (*
    eps
    (+
     (+
      t_2
      (*
       eps
       (-
        (/ (* (sin x) t_3) (cos x))
        (*
         eps
         (+
          0.16666666666666666
          (fma
           -1.0
           (/ (* t_0 t_3) t_1)
           (fma -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 + fma(-1.0, ((t_0 * t_3) / t_1), fma(-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(Float64(t_2 + Float64(eps * Float64(Float64(Float64(sin(x) * t_3) / cos(x)) - Float64(eps * Float64(0.16666666666666666 + fma(-1.0, Float64(Float64(t_0 * t_3) / t_1), fma(-0.5, t_3, Float64(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[(N[(t$95$2 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.16666666666666666 + N[(-1.0 * N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(-0.5 * t$95$3 + N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * ((t_0 - (eps * ((eps * -0.3333333333333333) + ((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 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    code = eps * ((t_0 - (eps * ((eps * (-0.3333333333333333d0)) + ((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.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	return eps * ((t_0 - (eps * ((eps * -0.3333333333333333) + ((Math.sin(x) * (-1.0 - t_0)) / Math.cos(x))))) + 1.0);
}

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * ((t_0 - (eps * ((eps * -0.3333333333333333) + ((sin(x) * (-1.0 - t_0)) / cos(x))))) + 1.0);
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.7%

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

7. Taylor expanded in eps around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-/l* 99.6%

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

   3. associate-/l* 99.6%

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

9. Simplified 99.6%

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

    1. +-commutative 99.6%

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

    2. *-commutative 99.6%

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

    3. fma-define 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 4: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.4%

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

7. Final simplification 99.4%

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

Alternative 5: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.7%

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

7. Taylor expanded in eps around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-/l* 99.6%

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

   3. associate-/l* 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in x around 0 99.4%

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

11. Final simplification 99.4%

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

Alternative 6: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.3%

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

   1. sub-neg 99.3%

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

   2. mul-1-neg 99.3%

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

   3. remove-double-neg 99.3%

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

5. Simplified 99.3%

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

   1. distribute-rgt-in 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. unpow2 99.3%

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

   4. unpow2 99.3%

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

   5. frac-times 99.3%

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

   6. tan-quot 99.3%

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

   7. tan-quot 99.3%

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

   8. pow2 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 7: 98.6% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.7%

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

7. Taylor expanded in eps around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-/l* 99.6%

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

   3. associate-/l* 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in x around 0 98.9%

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

    1. associate--l+ 98.9%

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

    2. *-commutative 98.9%

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

    3. distribute-rgt-out-- 98.9%

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

    4. metadata-eval 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

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

Alternative 8: 98.5% 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 62.6%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.7%

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

7. Taylor expanded in eps around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-/l* 99.6%

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

   3. associate-/l* 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in x around 0 98.9%

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

11. Final simplification 98.9%

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

Alternative 9: 98.1% 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 62.6%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.7%

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

7. Taylor expanded in eps around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-/l* 99.6%

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

   3. associate-/l* 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in x around 0 98.2%

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

11. Final simplification 98.2%

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

Alternative 10: 98.1% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.6%

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

   1. flip3-- 23.7%

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

   2. clear-num 23.7%

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

   3. +-commutative 23.7%

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

   4. distribute-rgt-out 23.7%

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

   5. +-commutative 23.7%

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

   6. fma-define 23.7%

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

   7. pow2 23.7%

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

4. Applied egg-rr 23.7%

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

5. Taylor expanded in x around 0 97.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{\cos \varepsilon}{\sin \varepsilon}}} \]

6. Taylor expanded in eps around 0 98.2%

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

Developer target: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024085 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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