2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.8%

Time: 15.2s

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: 62.3% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \varepsilon\right)\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (sin eps)
  (*
   (cos x)
   (fma
    (cos x)
    (cos eps)
    (*
     (- (sin x))
     (fma
      eps
      (*
       (* eps eps)
       (fma
        (* eps eps)
        (fma (* eps eps) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666))
      eps))))))

C

double code(double x, double eps) {
	return sin(eps) / (cos(x) * fma(cos(x), cos(eps), (-sin(x) * fma(eps, ((eps * eps) * fma((eps * eps), fma((eps * eps), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), eps))));
}

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(cos(x) * fma(cos(x), cos(eps), Float64(Float64(-sin(x)) * fma(eps, Float64(Float64(eps * eps) * fma(Float64(eps * eps), fma(Float64(eps * eps), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), eps)))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[((-N[Sin[x], $MachinePrecision]) * N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

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

   1. lift-+.f64 N/A

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

   2. tan-quot N/A

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

   3. tan-quot N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 59.4

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

4. Applied egg-rr 59.4%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. +-commutative N/A

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

   11. remove-double-neg N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sub-neg N/A

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

   17. mul-1-neg N/A

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

   18. remove-double-neg N/A

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

   19. lower-cos.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 99.9

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

7. Simplified 99.9%

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

   1. cos-sum N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. lift-cos.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 100.0

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

9. Applied egg-rr 100.0%

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

10. Taylor expanded in eps around 0

    \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right)} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)\right)} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) + \varepsilon \cdot 1\right)}\right)} \]

    3. *-rgt-identity N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) + \color{blue}{\varepsilon}\right)\right)} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right), \varepsilon\right)}\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}, \varepsilon\right)\right)} \]

    6. unpow2 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right), \varepsilon\right)\right)} \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right), \varepsilon\right)\right)} \]

    8. sub-neg N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, \varepsilon\right)\right)} \]

    9. metadata-eval N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) + \color{blue}{\frac{-1}{6}}\right), \varepsilon\right)\right)} \]

    10. lower-fma.f64 N/A

        \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}, \frac{-1}{6}\right)}, \varepsilon\right)\right)} \]

    11. unpow2 N/A

        \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}, \frac{-1}{6}\right), \varepsilon\right)\right)} \]

    12. lower-*.f64 N/A

        \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}, \frac{-1}{6}\right), \varepsilon\right)\right)} \]

    13. +-commutative N/A

        \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{-1}{5040} \cdot {\varepsilon}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), \varepsilon\right)\right)} \]

    14. *-commutative N/A

        \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), \varepsilon\right)\right)} \]

    15. lower-fma.f64 N/A

        \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), \varepsilon\right)\right)} \]

    16. unpow2 N/A

        \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), \varepsilon\right)\right)} \]

    17. lower-*.f64 100.0

        \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \varepsilon\right)\right)} \]

12. Simplified 100.0%

    \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \varepsilon\right)}\right)} \]
13. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.041666666666666664, -0.5\right), 1\right), -\sin \varepsilon \cdot \sin x\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (sin eps)
  (*
   (cos x)
   (fma
    (cos x)
    (fma (* eps eps) (fma (* eps eps) 0.041666666666666664 -0.5) 1.0)
    (- (* (sin eps) (sin x)))))))

C

double code(double x, double eps) {
	return sin(eps) / (cos(x) * fma(cos(x), fma((eps * eps), fma((eps * eps), 0.041666666666666664, -0.5), 1.0), -(sin(eps) * sin(x))));
}

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(cos(x) * fma(cos(x), fma(Float64(eps * eps), fma(Float64(eps * eps), 0.041666666666666664, -0.5), 1.0), Float64(-Float64(sin(eps) * sin(x))))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + (-N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

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

   1. lift-+.f64 N/A

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

   2. tan-quot N/A

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

   3. tan-quot N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 59.4

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

4. Applied egg-rr 59.4%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. +-commutative N/A

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

   11. remove-double-neg N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sub-neg N/A

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

   17. mul-1-neg N/A

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

   18. remove-double-neg N/A

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

   19. lower-cos.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 99.9

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

7. Simplified 99.9%

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

   1. cos-sum N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. lift-cos.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 100.0

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

9. Applied egg-rr 100.0%

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

10. Taylor expanded in eps around 0

    \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \color{blue}{1 + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)}, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + 1}, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}, 1\right)}, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

    3. unpow2 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}, 1\right), \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}, 1\right), \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

    5. sub-neg N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{1}{24} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right), \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

    6. *-commutative N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right), \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

    7. metadata-eval N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, {\varepsilon}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right), \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

    8. lower-fma.f64 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right), \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

    9. unpow2 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{24}, \frac{-1}{2}\right), 1\right), \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

    10. lower-*.f64 99.9

        \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, 0.041666666666666664, -0.5\right), 1\right), \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]

12. Simplified 99.9%

    \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.041666666666666664, -0.5\right), 1\right)}, \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]

13. Final simplification 99.9%

    \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.041666666666666664, -0.5\right), 1\right), -\sin \varepsilon \cdot \sin x\right)} \]
14. Add Preprocessing

Alternative 3: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \varepsilon\right)}{\cos x \cdot \cos \left(\varepsilon + x\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (fma
   eps
   (*
    (* eps eps)
    (fma
     eps
     (* eps (fma (* eps eps) -0.0001984126984126984 0.008333333333333333))
     -0.16666666666666666))
   eps)
  (* (cos x) (cos (+ eps x)))))

C

double code(double x, double eps) {
	return fma(eps, ((eps * eps) * fma(eps, (eps * fma((eps * eps), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), eps) / (cos(x) * cos((eps + x)));
}

Julia

function code(x, eps)
	return Float64(fma(eps, Float64(Float64(eps * eps) * fma(eps, Float64(eps * fma(Float64(eps * eps), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), eps) / Float64(cos(x) * cos(Float64(eps + x))))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * N[(N[(eps * eps), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

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

   1. lift-+.f64 N/A

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

   2. tan-quot N/A

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

   3. tan-quot N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 59.4

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

4. Applied egg-rr 59.4%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. +-commutative N/A

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

   11. remove-double-neg N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sub-neg N/A

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

   17. mul-1-neg N/A

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

   18. remove-double-neg N/A

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

   19. lower-cos.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 99.9

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

7. Simplified 99.9%

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

8. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right) + 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) + \varepsilon \cdot 1}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) + \color{blue}{\varepsilon}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right), \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

10. Simplified 99.9%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

11. Final simplification 99.9%

    \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \varepsilon\right)}{\cos x \cdot \cos \left(\varepsilon + x\right)} \]
12. Add Preprocessing

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \cos \left(\varepsilon + x\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (fma
   (fma (* eps eps) 0.008333333333333333 -0.16666666666666666)
   (* eps (* eps eps))
   eps)
  (* (cos x) (cos (+ eps x)))))

C

double code(double x, double eps) {
	return fma(fma((eps * eps), 0.008333333333333333, -0.16666666666666666), (eps * (eps * eps)), eps) / (cos(x) * cos((eps + x)));
}

Julia

function code(x, eps)
	return Float64(fma(fma(Float64(eps * eps), 0.008333333333333333, -0.16666666666666666), Float64(eps * Float64(eps * eps)), eps) / Float64(cos(x) * cos(Float64(eps + x))))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(eps * eps), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

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

   1. lift-+.f64 N/A

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

   2. tan-quot N/A

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

   3. tan-quot N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 59.4

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

4. Applied egg-rr 59.4%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. +-commutative N/A

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

   11. remove-double-neg N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sub-neg N/A

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

   17. mul-1-neg N/A

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

   18. remove-double-neg N/A

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

   19. lower-cos.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 99.9

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

7. Simplified 99.9%

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

8. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) + 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) \cdot \varepsilon + 1 \cdot \varepsilon}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) \cdot {\varepsilon}^{2}\right)} \cdot \varepsilon + 1 \cdot \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)} + 1 \cdot \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \varepsilon\right) + 1 \cdot \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   6. unpow3 N/A

      \[\leadsto \frac{\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{\varepsilon}^{3}} + 1 \cdot \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   7. *-lft-identity N/A

      \[\leadsto \frac{\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) \cdot {\varepsilon}^{3} + \color{blue}{\varepsilon}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}, {\varepsilon}^{3}, \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {\varepsilon}^{3}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\varepsilon}^{3}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\varepsilon}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {\varepsilon}^{3}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {\varepsilon}^{3}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120}, \frac{-1}{6}\right), {\varepsilon}^{3}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120}, \frac{-1}{6}\right), {\varepsilon}^{3}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   15. cube-mult N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   16. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right), \varepsilon \cdot \color{blue}{{\varepsilon}^{2}}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{\varepsilon \cdot {\varepsilon}^{2}}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   18. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right), \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   19. lower-*.f64 99.8

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

10. Simplified 99.8%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

11. Final simplification 99.8%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)}{\cos x \cdot \cos \left(\varepsilon + x\right)} \]
12. Add Preprocessing

Alternative 5: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.4%

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

   1. lift-+.f64 N/A

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

   2. tan-quot N/A

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

   3. tan-quot N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 59.4

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

4. Applied egg-rr 59.4%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. +-commutative N/A

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

   11. remove-double-neg N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sub-neg N/A

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

   17. mul-1-neg N/A

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

   18. remove-double-neg N/A

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

   19. lower-cos.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 99.9

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

7. Simplified 99.9%

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

8. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
9. 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

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 6: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{{\cos x}^{2}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (pow (cos x) 2.0)))

C

double code(double x, double eps) {
	return eps / pow(cos(x), 2.0);
}

Fortran

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

Java

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

Python

def code(x, eps):
	return eps / math.pow(math.cos(x), 2.0)

Julia

function code(x, eps)
	return Float64(eps / (cos(x) ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (cos(x) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 59.4%

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

   1. lift-+.f64 N/A

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

   2. tan-quot N/A

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

   3. tan-quot N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 59.4

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

4. Applied egg-rr 59.4%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.0

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

7. Simplified 99.0%

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
8. Add Preprocessing

Alternative 7: 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 59.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{4}{3}, 1\right), \varepsilon\right), \frac{1}{3} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), \varepsilon\right) \]

   12. lower-*.f64 98.5

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

8. Simplified 98.5%

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

9. Taylor expanded in eps around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 98.5

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

11. Simplified 98.5%

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

Alternative 8: 98.2% accurate, 17.3× 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 59.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{4}{3}, 1\right), \varepsilon\right), \frac{1}{3} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), \varepsilon\right) \]

   12. lower-*.f64 98.5

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

8. Simplified 98.5%

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

9. Taylor expanded in eps around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 98.5

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

11. Simplified 98.5%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. distribute-lft1-in N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lower-fma.f64 98.5

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

13. Applied egg-rr 98.5%

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

14. Final simplification 98.5%

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

Alternative 9: 6.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{4}{3}, 1\right), \varepsilon\right), \frac{1}{3} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), \varepsilon\right) \]

   12. lower-*.f64 98.5

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

8. Simplified 98.5%

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

9. Taylor expanded in eps around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 98.5

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

11. Simplified 98.5%

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

12. Taylor expanded in x around inf

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

    1. lower-*.f64 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 6.4

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

14. Simplified 6.4%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot x\right)} \]
15. 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: 62.4% 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 2024211 
(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)))