2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 99.6%

Time: 14.5s

Alternatives: 9

Speedup: 17.3×

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: 62.7% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\varepsilon, -0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x, \cos x\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (fma eps (* -0.16666666666666666 (* eps eps)) eps)
  (* (cos x) (fma eps (- (* eps (* (cos x) -0.5)) (sin x)) (cos x)))))

C

double code(double x, double eps) {
	return fma(eps, (-0.16666666666666666 * (eps * eps)), eps) / (cos(x) * fma(eps, ((eps * (cos(x) * -0.5)) - sin(x)), cos(x)));
}

Julia

function code(x, eps)
	return Float64(fma(eps, Float64(-0.16666666666666666 * Float64(eps * eps)), eps) / Float64(cos(x) * fma(eps, Float64(Float64(eps * Float64(cos(x) * -0.5)) - sin(x)), cos(x))))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(-0.16666666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(eps * N[(N[(eps * N[(N[Cos[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \varepsilon \cdot 1}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{\varepsilon \cdot \left(\frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \color{blue}{\varepsilon}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot {\varepsilon}^{2}, \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{6} \cdot {\varepsilon}^{2}}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   7. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right) + \cos x\right)}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x, \cos x\right)}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \cos x\right) \cdot \frac{-1}{2}} - \sin x, \cos x\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\cos x \cdot \frac{-1}{2}\right)} - \sin x, \cos x\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos x\right)} - \sin x, \cos x\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) - \sin x}, \cos x\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)} - \sin x, \cos x\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\left(\cos x \cdot \frac{-1}{2}\right)} - \sin x, \cos x\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\left(\cos x \cdot \frac{-1}{2}\right)} - \sin x, \cos x\right)} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\color{blue}{\cos x} \cdot \frac{-1}{2}\right) - \sin x, \cos x\right)} \]

   11. lower-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\cos x \cdot \frac{-1}{2}\right) - \color{blue}{\sin x}, \cos x\right)} \]

   12. lower-cos.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\varepsilon, -0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (fma eps (* -0.16666666666666666 (* eps eps)) eps)
  (* (cos x) (cos (+ x eps)))))

C

double code(double x, double eps) {
	return fma(eps, (-0.16666666666666666 * (eps * eps)), eps) / (cos(x) * cos((x + eps)));
}

Julia

function code(x, eps)
	return Float64(fma(eps, Float64(-0.16666666666666666 * Float64(eps * eps)), eps) / Float64(cos(x) * cos(Float64(x + eps))))
end

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \varepsilon \cdot 1}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{\varepsilon \cdot \left(\frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \color{blue}{\varepsilon}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot {\varepsilon}^{2}, \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{6} \cdot {\varepsilon}^{2}}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \frac{-1}{6} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   7. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, -0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
8. Add Preprocessing

Alternative 3: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{1}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (/ 1.0 (+ 0.5 (* 0.5 (cos (+ x x)))))
  eps
  (* (* eps eps) (fma eps 0.3333333333333333 x))))

C

double code(double x, double eps) {
	return fma((1.0 / (0.5 + (0.5 * cos((x + x))))), eps, ((eps * eps) * fma(eps, 0.3333333333333333, x)));
}

Julia

function code(x, eps)
	return fma(Float64(1.0 / Float64(0.5 + Float64(0.5 * cos(Float64(x + x))))), eps, Float64(Float64(eps * eps) * fma(eps, 0.3333333333333333, x)))
end

Wolfram

code[x_, eps_] := N[(N[(1.0 / N[(0.5 + N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(eps * eps), $MachinePrecision] * N[(eps * 0.3333333333333333 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}\right) + \frac{1}{{\cos x}^{2}}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}\right) + \frac{1}{{\cos x}^{2}}\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}, \frac{1}{{\cos x}^{2}}\right)} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \frac{0.5}{{\cos x}^{2}} - \left(\frac{0.16666666666666666}{{\cos x}^{2}} - \frac{{\sin x}^{2}}{{\cos x}^{4}}\right), \frac{\sin x}{{\cos x}^{3}}\right), \frac{1}{{\cos x}^{2}}\right)} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.2%

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

12. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\frac{1}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, \color{blue}{\varepsilon}, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)\right) \]
13. Add Preprocessing

Alternative 4: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), \frac{1}{0.5 + 0.5 \cdot \cos \left(x + x\right)}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (fma eps 0.3333333333333333 x)
   (/ 1.0 (+ 0.5 (* 0.5 (cos (+ x x))))))))

C

double code(double x, double eps) {
	return eps * fma(eps, fma(eps, 0.3333333333333333, x), (1.0 / (0.5 + (0.5 * cos((x + x))))));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, fma(eps, 0.3333333333333333, x), Float64(1.0 / Float64(0.5 + Float64(0.5 * cos(Float64(x + x)))))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(eps * 0.3333333333333333 + x), $MachinePrecision] + N[(1.0 / N[(0.5 + N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}\right) + \frac{1}{{\cos x}^{2}}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}\right) + \frac{1}{{\cos x}^{2}}\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}, \frac{1}{{\cos x}^{2}}\right)} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \frac{0.5}{{\cos x}^{2}} - \left(\frac{0.16666666666666666}{{\cos x}^{2}} - \frac{{\sin x}^{2}}{{\cos x}^{4}}\right), \frac{\sin x}{{\cos x}^{3}}\right), \frac{1}{{\cos x}^{2}}\right)} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.2%

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

12. Applied rewrites 99.2%

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

13. Final simplification 99.2%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), \frac{1}{0.5 + 0.5 \cdot \cos \left(x + x\right)}\right) \]
14. Add Preprocessing

Alternative 5: 98.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.044444444444444446, 0.3333333333333333\right), -1\right), 1\right)}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (fma eps 0.3333333333333333 x)
   (/
    1.0
    (fma
     (* x x)
     (fma (* x x) (fma (* x x) -0.044444444444444446 0.3333333333333333) -1.0)
     1.0)))))

C

double code(double x, double eps) {
	return eps * fma(eps, fma(eps, 0.3333333333333333, x), (1.0 / fma((x * x), fma((x * x), fma((x * x), -0.044444444444444446, 0.3333333333333333), -1.0), 1.0)));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, fma(eps, 0.3333333333333333, x), Float64(1.0 / fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.044444444444444446, 0.3333333333333333), -1.0), 1.0))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(eps * 0.3333333333333333 + x), $MachinePrecision] + N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.044444444444444446 + 0.3333333333333333), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}\right) + \frac{1}{{\cos x}^{2}}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}\right) + \frac{1}{{\cos x}^{2}}\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}, \frac{1}{{\cos x}^{2}}\right)} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \frac{0.5}{{\cos x}^{2}} - \left(\frac{0.16666666666666666}{{\cos x}^{2}} - \frac{{\sin x}^{2}}{{\cos x}^{4}}\right), \frac{\sin x}{{\cos x}^{3}}\right), \frac{1}{{\cos x}^{2}}\right)} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.2%

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

11. Taylor expanded in x around 0

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \frac{1}{3}, x\right), \frac{1}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{-2}{45} \cdot {x}^{2}\right) - 1\right)}\right) \]
12. Step-by-step derivation

13. Applied rewrites 98.8%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.044444444444444446, 0.3333333333333333\right), -1\right), 1\right)}\right) \]
14. Add Preprocessing

Alternative 6: 98.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.3333333333333333, x \cdot x, -1\right), 1\right)}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (fma eps 0.3333333333333333 x)
   (/ 1.0 (fma (* x x) (fma 0.3333333333333333 (* x x) -1.0) 1.0)))))

C

double code(double x, double eps) {
	return eps * fma(eps, fma(eps, 0.3333333333333333, x), (1.0 / fma((x * x), fma(0.3333333333333333, (x * x), -1.0), 1.0)));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, fma(eps, 0.3333333333333333, x), Float64(1.0 / fma(Float64(x * x), fma(0.3333333333333333, Float64(x * x), -1.0), 1.0))))
end

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}\right) + \frac{1}{{\cos x}^{2}}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}\right) + \frac{1}{{\cos x}^{2}}\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\frac{1}{2} \cdot \frac{1}{{\cos x}^{2}} - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{4}} + \frac{1}{6} \cdot \frac{1}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{\sin x}{{\cos x}^{3}}, \frac{1}{{\cos x}^{2}}\right)} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \frac{0.5}{{\cos x}^{2}} - \left(\frac{0.16666666666666666}{{\cos x}^{2}} - \frac{{\sin x}^{2}}{{\cos x}^{4}}\right), \frac{\sin x}{{\cos x}^{3}}\right), \frac{1}{{\cos x}^{2}}\right)} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.2%

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

11. Taylor expanded in x around 0

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \frac{1}{3}, x\right), \frac{1}{1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot {x}^{2} - 1\right)}\right) \]
12. Step-by-step derivation

13. Applied rewrites 98.7%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.3333333333333333, x \cdot x, -1\right), 1\right)}\right) \]
14. Add Preprocessing

Alternative 7: 98.3% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma eps (fma eps (fma eps 0.3333333333333333 x) (* x x)) eps))

C

double code(double x, double eps) {
	return fma(eps, fma(eps, fma(eps, 0.3333333333333333, x), (x * x)), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(eps, fma(eps, 0.3333333333333333, x), Float64(x * x)), eps)
end

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \mathsf{fma}\left({\sin x}^{2}, \frac{0.16666666666666666}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right) + -0.16666666666666666, \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{1}{3} \cdot {\varepsilon}^{2} + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), \varepsilon\right)}, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right) \]

9. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(x + \varepsilon \cdot \left(\frac{1}{3} + \frac{4}{3} \cdot {x}^{2}\right)\right) + {x}^{\color{blue}{2}}, \varepsilon\right) \]
10. Step-by-step derivation

11. Applied rewrites 98.3%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 0.3333333333333333\right)}, x\right), x \cdot x\right), \varepsilon\right) \]

12. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \frac{1}{3}, x\right), x \cdot x\right), \varepsilon\right) \]
13. Step-by-step derivation

14. Applied rewrites 98.4%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), x \cdot x\right), \varepsilon\right) \]
15. Add Preprocessing

Alternative 8: 98.3% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \mathsf{fma}\left({\sin x}^{2}, \frac{0.16666666666666666}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right) + -0.16666666666666666, \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{1}{3} \cdot {\varepsilon}^{2} + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), \varepsilon\right)}, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.3%

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

12. Final simplification 98.3%

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

Alternative 9: 98.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \mathsf{fma}\left({\sin x}^{2}, \frac{0.16666666666666666}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right) + -0.16666666666666666, \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{1}{3} \cdot {\varepsilon}^{2} + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), \varepsilon\right)}, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right) \]

9. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right) \]
10. Step-by-step derivation

11. Applied rewrites 98.2%

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

Developer Target 1: 99.9% accurate, 0.6× 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]

Developer Target 2: 62.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024234 
(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)))))

  :alt
  (! :herbie-platform default (- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x)))

  :alt
  (! :herbie-platform default (+ eps (* eps (tan x) (tan x))))

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