2tan (problem 3.3.2)

Percentage Accurate: 62.0% → 99.6%

Time: 18.6s

Alternatives: 14

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: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.1%

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

5. Simplified 99.1%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

7. Applied egg-rr 99.1%

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

   1. unpow-1 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

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 (+ 1.0 t_2)))
   (*
    eps
    (+
     1.0
     (+
      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))))))))))))

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 = 1.0 + t_2;
	return eps * (1.0 + (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)))))))));
}

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(1.0 + t_2)
	return Float64(eps * Float64(1.0 + Float64(t_2 + Float64(eps * Float64(Float64(Float64(sin(x) * t_3) / cos(x)) - Float64(eps * Float64(0.16666666666666666 + fma(-1.0, Float64(t_0 * Float64(t_3 / t_1)), fma(-0.5, t_3, Float64(0.16666666666666666 * t_2))))))))))
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[(1.0 + t$95$2), $MachinePrecision]}, N[(eps * N[(1.0 + 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[(t$95$0 * N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * t$95$3 + N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 3: 99.6% accurate, 0.1× speedup

Math

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

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

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

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 = 1.0d0 + t_2
    code = eps * (t_2 + (1.0d0 + (eps * ((eps * ((((t_0 * t_3) / t_1) + (((-0.5d0) * ((-1.0d0) - t_2)) - (0.16666666666666666d0 * t_2))) - 0.16666666666666666d0)) + ((sin(x) * t_3) / cos(x))))))
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 = 1.0 + t_2;
	return eps * (t_2 + (1.0 + (eps * ((eps * ((((t_0 * t_3) / t_1) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_2))) - 0.16666666666666666)) + ((Math.sin(x) * t_3) / Math.cos(x))))));
}

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 = 1.0 + t_2
	return eps * (t_2 + (1.0 + (eps * ((eps * ((((t_0 * t_3) / t_1) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_2))) - 0.16666666666666666)) + ((math.sin(x) * t_3) / math.cos(x))))))

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(1.0 + t_2)
	return Float64(eps * Float64(t_2 + Float64(1.0 + Float64(eps * Float64(Float64(eps * Float64(Float64(Float64(Float64(t_0 * t_3) / t_1) + Float64(Float64(-0.5 * Float64(-1.0 - t_2)) - Float64(0.16666666666666666 * t_2))) - 0.16666666666666666)) + Float64(Float64(sin(x) * t_3) / cos(x)))))))
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 = 1.0 + t_2;
	tmp = eps * (t_2 + (1.0 + (eps * ((eps * ((((t_0 * t_3) / t_1) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_2))) - 0.16666666666666666)) + ((sin(x) * t_3) / cos(x))))));
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[(1.0 + t$95$2), $MachinePrecision]}, N[(eps * N[(t$95$2 + N[(1.0 + N[(eps * N[(N[(eps * N[(N[(N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(-0.5 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.1%

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

4. Final simplification 99.1%

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

Alternative 4: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(1 + \left(t\_0 + \varepsilon \cdot \left(\frac{\sin x \cdot \left(1 + t\_0\right)}{\cos x} + \varepsilon \cdot \left(0.3333333333333333 - -1.3333333333333333 \cdot {x}^{2}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    eps
    (+
     1.0
     (+
      t_0
      (*
       eps
       (+
        (/ (* (sin x) (+ 1.0 t_0)) (cos x))
        (*
         eps
         (- 0.3333333333333333 (* -1.3333333333333333 (pow x 2.0)))))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * (1.0 + (t_0 + (eps * (((sin(x) * (1.0 + t_0)) / cos(x)) + (eps * (0.3333333333333333 - (-1.3333333333333333 * pow(x, 2.0))))))));
}

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 * (1.0d0 + (t_0 + (eps * (((sin(x) * (1.0d0 + t_0)) / cos(x)) + (eps * (0.3333333333333333d0 - ((-1.3333333333333333d0) * (x ** 2.0d0))))))))
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 * (1.0 + (t_0 + (eps * (((Math.sin(x) * (1.0 + t_0)) / Math.cos(x)) + (eps * (0.3333333333333333 - (-1.3333333333333333 * Math.pow(x, 2.0))))))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return eps * (1.0 + (t_0 + (eps * (((math.sin(x) * (1.0 + t_0)) / math.cos(x)) + (eps * (0.3333333333333333 - (-1.3333333333333333 * math.pow(x, 2.0))))))))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(1.0 + Float64(t_0 + Float64(eps * Float64(Float64(Float64(sin(x) * Float64(1.0 + t_0)) / cos(x)) + Float64(eps * Float64(0.3333333333333333 - Float64(-1.3333333333333333 * (x ^ 2.0)))))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * (1.0 + (t_0 + (eps * (((sin(x) * (1.0 + t_0)) / cos(x)) + (eps * (0.3333333333333333 - (-1.3333333333333333 * (x ^ 2.0))))))));
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[(1.0 + N[(t$95$0 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(0.3333333333333333 - N[(-1.3333333333333333 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 98.9%

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

7. Final simplification 98.9%

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

Alternative 5: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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
    t_0 = cos(x) ** 2.0d0
    t_1 = sin(x) ** 2.0d0
    code = eps * (1.0d0 + ((1.0d0 / (t_0 / t_1)) - ((eps * (sin(x) * ((-1.0d0) - (t_1 / t_0)))) / cos(x))))
end function

Java

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

Python

def code(x, eps):
	t_0 = math.pow(math.cos(x), 2.0)
	t_1 = math.pow(math.sin(x), 2.0)
	return eps * (1.0 + ((1.0 / (t_0 / t_1)) - ((eps * (math.sin(x) * (-1.0 - (t_1 / t_0)))) / math.cos(x))))

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = cos(x) ^ 2.0;
	t_1 = sin(x) ^ 2.0;
	tmp = eps * (1.0 + ((1.0 / (t_0 / t_1)) - ((eps * (sin(x) * (-1.0 - (t_1 / t_0)))) / cos(x))));
end

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.1%

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

5. Simplified 99.1%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

7. Applied egg-rr 99.1%

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

   1. unpow-1 99.1%

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

9. Simplified 99.1%

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

10. Taylor expanded in eps around 0 98.7%

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

11. Final simplification 98.7%

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

Alternative 6: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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 * (1.0d0 + (t_0 - (eps * (sin(x) * (((-1.0d0) - t_0) / cos(x))))))
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 * (1.0 + (t_0 - (eps * (Math.sin(x) * ((-1.0 - t_0) / Math.cos(x))))));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * (1.0 + (t_0 - (eps * (sin(x) * ((-1.0 - t_0) / cos(x))))));
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[(1.0 + N[(t$95$0 - N[(eps * N[(N[Sin[x], $MachinePrecision] * N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 98.9%

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

7. Taylor expanded in eps around 0 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 7: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 98.3%

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

   1. *-commutative 98.3%

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 8: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(\varepsilon + x\right) - \tan x\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan (+ eps x)) (tan x))))
   (if (<= t_0 5e-15) (+ eps (* eps (pow x 2.0))) t_0)))

C

double code(double x, double eps) {
	double t_0 = tan((eps + x)) - tan(x);
	double tmp;
	if (t_0 <= 5e-15) {
		tmp = eps + (eps * pow(x, 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((eps + x)) - tan(x)
    if (t_0 <= 5d-15) then
        tmp = eps + (eps * (x ** 2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan((eps + x)) - Math.tan(x);
	double tmp;
	if (t_0 <= 5e-15) {
		tmp = eps + (eps * Math.pow(x, 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan((eps + x)) - math.tan(x)
	tmp = 0
	if t_0 <= 5e-15:
		tmp = eps + (eps * math.pow(x, 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(Float64(eps + x)) - tan(x))
	tmp = 0.0
	if (t_0 <= 5e-15)
		tmp = Float64(eps + Float64(eps * (x ^ 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan((eps + x)) - tan(x);
	tmp = 0.0;
	if (t_0 <= 5e-15)
		tmp = eps + (eps * (x ^ 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-15], N[(eps + N[(eps * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x)) < 4.99999999999999999e-15

   1. Initial program 61.0%

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

   3. Taylor expanded in eps around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. remove-double-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

   if 4.99999999999999999e-15 < (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x))

   1. Initial program 73.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(\varepsilon + x\right) - \tan x \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 98.2%

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

   1. sub-neg 98.2%

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

   2. mul-1-neg 98.2%

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

   3. remove-double-neg 98.2%

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

5. Simplified 98.2%

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

   1. log1p-expm1-u 98.2%

      \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{2}\right)\right)}}\right) \]

   2. log1p-undefine 98.2%

      \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left({\cos x}^{2}\right)\right)}}\right) \]

7. Applied egg-rr 98.2%

   \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left({\cos x}^{2}\right)\right)}}\right) \]
8. Step-by-step derivation

   1. +-commutative 98.2%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{{\sin x}^{2}}{\log \left(1 + \mathsf{expm1}\left({\cos x}^{2}\right)\right)} + 1\right)} \]

   2. distribute-lft-in 98.2%

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{{\sin x}^{2}}{\log \left(1 + \mathsf{expm1}\left({\cos x}^{2}\right)\right)} + \varepsilon \cdot 1} \]

   3. log1p-define 98.2%

      \[\leadsto \varepsilon \cdot \frac{{\sin x}^{2}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{2}\right)\right)}} + \varepsilon \cdot 1 \]

   4. log1p-expm1-u 98.2%

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

   5. unpow2 98.2%

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

   6. unpow2 98.2%

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

   7. frac-times 98.2%

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

   8. tan-quot 98.2%

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

   9. tan-quot 98.2%

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

   10. pow2 98.2%

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

   11. *-rgt-identity 98.2%

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

9. Applied egg-rr 98.2%

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

10. Final simplification 98.2%

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

Alternative 10: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 98.2%

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

   1. sub-neg 98.2%

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

   2. mul-1-neg 98.2%

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

   3. remove-double-neg 98.2%

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

5. Simplified 98.2%

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

   1. log1p-expm1-u 98.2%

      \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{2}\right)\right)}}\right) \]

   2. log1p-undefine 98.2%

      \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left({\cos x}^{2}\right)\right)}}\right) \]

7. Applied egg-rr 98.2%

   \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left({\cos x}^{2}\right)\right)}}\right) \]
8. Step-by-step derivation

   1. *-un-lft-identity 98.2%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 \cdot \left(1 + \frac{{\sin x}^{2}}{\log \left(1 + \mathsf{expm1}\left({\cos x}^{2}\right)\right)}\right)\right)} \]

   2. log1p-define 98.2%

      \[\leadsto \varepsilon \cdot \left(1 \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{2}\right)\right)}}\right)\right) \]

   3. log1p-expm1-u 98.2%

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

   4. unpow2 98.2%

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

   5. unpow2 98.2%

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

   6. frac-times 98.2%

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

   7. tan-quot 98.2%

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

   8. tan-quot 98.2%

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

   9. pow2 98.2%

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

9. Applied egg-rr 98.2%

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

    1. *-lft-identity 98.2%

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

11. Simplified 98.2%

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

Alternative 11: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 98.2%

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

   1. sub-neg 98.2%

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

   2. mul-1-neg 98.2%

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

   3. remove-double-neg 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in x around 0 97.3%

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

Alternative 12: 98.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in eps around 0 98.2%

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

   1. sub-neg 98.2%

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

   2. mul-1-neg 98.2%

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

   3. remove-double-neg 98.2%

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

5. Simplified 98.2%

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

   1. log1p-expm1-u 98.2%

      \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{2}\right)\right)}}\right) \]

   2. log1p-undefine 98.2%

      \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left({\cos x}^{2}\right)\right)}}\right) \]

7. Applied egg-rr 98.2%

   \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left({\cos x}^{2}\right)\right)}}\right) \]

8. Taylor expanded in x around 0 97.3%

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

   1. *-commutative 97.3%

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

   2. distribute-rgt1-in 97.3%

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

   3. unpow2 97.3%

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

   4. fma-define 97.3%

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

10. Simplified 97.3%

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

11. Final simplification 97.3%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x, 1\right) \]
12. Add Preprocessing

Alternative 13: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in x around 0 96.9%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 96.9%

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

   2. quot-tan 96.9%

      \[\leadsto 1 \cdot \color{blue}{\tan \varepsilon} \]

5. Applied egg-rr 96.9%

   \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]
6. Step-by-step derivation

   1. *-lft-identity 96.9%

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

7. Simplified 96.9%

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

Alternative 14: 97.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in x around 0 96.9%

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

4. Taylor expanded in eps around 0 96.9%

   \[\leadsto \color{blue}{\varepsilon} \]
5. 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 2024144 
(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)))