2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.9%

Time: 15.1s

Alternatives: 10

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: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. sin-diff N/A

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

   8. lower-sin.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.0

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

4. Applied rewrites 64.0%

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

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. cancel-sign-sub-inv N/A

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

   17. metadata-eval N/A

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

   18. *-lft-identity 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. lower-+.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right) \cdot \frac{1}{\cos x \cdot \cos \left(\varepsilon + x\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (fma (* eps eps) 0.008333333333333333 -0.16666666666666666)
   (* eps (* eps eps))
   eps)
  (/ 1.0 (* (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) * (1.0 / (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(1.0 / 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[(1.0 / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. sin-diff N/A

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

   8. lower-sin.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.0

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

4. Applied rewrites 64.0%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. pow-plus N/A

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

   7. metadata-eval N/A

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

   8. cube-unmult N/A

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

   9. unpow2 N/A

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

   10. *-rgt-identity N/A

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

   11. lower-fma.f64 N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 99.6

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon\right) \cdot \frac{1}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)} \cdot \frac{1}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

8. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right) \cdot \frac{1}{\cos x \cdot \cos \left(\varepsilon + x\right)} \]
9. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\cos x \cdot \cos \left(\varepsilon + x\right)} \cdot \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (/ 1.0 (* (cos x) (cos (+ eps x))))
  (fma eps (* (* eps eps) -0.16666666666666666) eps)))

C

double code(double x, double eps) {
	return (1.0 / (cos(x) * cos((eps + x)))) * fma(eps, ((eps * eps) * -0.16666666666666666), eps);
}

Julia

function code(x, eps)
	return Float64(Float64(1.0 / Float64(cos(x) * cos(Float64(eps + x)))) * fma(eps, Float64(Float64(eps * eps) * -0.16666666666666666), eps))
end

Wolfram

code[x_, eps_] := N[(N[(1.0 / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. sin-diff N/A

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

   8. lower-sin.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.0

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

4. Applied rewrites 64.0%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 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 64.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. sin-diff N/A

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

   8. lower-sin.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.0

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

4. Applied rewrites 64.0%

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

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 98.9

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

7. Applied rewrites 98.9%

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

Alternative 5: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.5083333333333333, 0.8333333333333334\right), x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.19682539682539682, 0.37777777777777777\right), 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma (* x x) (* x (fma (* x x) 0.5083333333333333 0.8333333333333334)) x)
   (/ eps (cos x))
   (*
    (* x x)
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) 0.19682539682539682 0.37777777777777777)
      0.6666666666666666)
     1.0)))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma((x * x), (x * fma((x * x), 0.5083333333333333, 0.8333333333333334)), x), (eps / cos(x)), ((x * x) * fma((x * x), fma((x * x), fma((x * x), 0.19682539682539682, 0.37777777777777777), 0.6666666666666666), 1.0))), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(Float64(x * x), Float64(x * fma(Float64(x * x), 0.5083333333333333, 0.8333333333333334)), x), Float64(eps / cos(x)), Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.19682539682539682, 0.37777777777777777), 0.6666666666666666), 1.0))), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5083333333333333 + 0.8333333333333334), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(eps / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.19682539682539682 + 0.37777777777777777), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right)\right)}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) + 1\right)}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right)\right) \cdot x + 1 \cdot x}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x\right)} + 1 \cdot x, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2} \cdot \left(\left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{x}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x, x\right)}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x}, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{61}{120} \cdot {x}^{2} + \frac{5}{6}\right)} \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \left(\color{blue}{{x}^{2} \cdot \frac{61}{120}} + \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{61}{120}, \frac{5}{6}\right)} \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   13. lower-*.f64 99.1

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5083333333333333, 0.8333333333333334\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

10. Applied rewrites 99.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.5083333333333333, 0.8333333333333334\right) \cdot x, x\right)}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)}\right), \varepsilon, \varepsilon\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)}\right), \varepsilon, \varepsilon\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, \varepsilon\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, \varepsilon\right) \]

    4. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right) + 1\right)}\right), \varepsilon, \varepsilon\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right), 1\right)}\right), \varepsilon, \varepsilon\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

    7. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right) + \frac{2}{3}}, 1\right)\right), \varepsilon, \varepsilon\right) \]

    9. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{17}{45} + \frac{62}{315} \cdot {x}^{2}, \frac{2}{3}\right)}, 1\right)\right), \varepsilon, \varepsilon\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{17}{45} + \frac{62}{315} \cdot {x}^{2}, \frac{2}{3}\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

    11. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{17}{45} + \frac{62}{315} \cdot {x}^{2}, \frac{2}{3}\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

    12. +-commutative N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{62}{315} \cdot {x}^{2} + \frac{17}{45}}, \frac{2}{3}\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{62}{315}} + \frac{17}{45}, \frac{2}{3}\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

    14. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{62}{315}, \frac{17}{45}\right)}, \frac{2}{3}\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

    15. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{62}{315}, \frac{17}{45}\right), \frac{2}{3}\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

    16. lower-*.f64 98.9

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.5083333333333333, 0.8333333333333334\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.19682539682539682, 0.37777777777777777\right), 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

13. Applied rewrites 98.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.5083333333333333, 0.8333333333333334\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.19682539682539682, 0.37777777777777777\right), 0.6666666666666666\right), 1\right)}\right), \varepsilon, \varepsilon\right) \]

14. Final simplification 98.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.5083333333333333, 0.8333333333333334\right), x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.19682539682539682, 0.37777777777777777\right), 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \]
15. Add Preprocessing

Alternative 6: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.5083333333333333, 0.8333333333333334\right), x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.37777777777777777, 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma (* x x) (* x (fma (* x x) 0.5083333333333333 0.8333333333333334)) x)
   (/ eps (cos x))
   (*
    (* x x)
    (fma x (* x (fma (* x x) 0.37777777777777777 0.6666666666666666)) 1.0)))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma((x * x), (x * fma((x * x), 0.5083333333333333, 0.8333333333333334)), x), (eps / cos(x)), ((x * x) * fma(x, (x * fma((x * x), 0.37777777777777777, 0.6666666666666666)), 1.0))), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(Float64(x * x), Float64(x * fma(Float64(x * x), 0.5083333333333333, 0.8333333333333334)), x), Float64(eps / cos(x)), Float64(Float64(x * x) * fma(x, Float64(x * fma(Float64(x * x), 0.37777777777777777, 0.6666666666666666)), 1.0))), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5083333333333333 + 0.8333333333333334), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(eps / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.37777777777777777 + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right)\right)}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) + 1\right)}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right)\right) \cdot x + 1 \cdot x}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x\right)} + 1 \cdot x, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2} \cdot \left(\left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{x}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x, x\right)}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{5}{6} + \frac{61}{120} \cdot {x}^{2}\right) \cdot x}, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{61}{120} \cdot {x}^{2} + \frac{5}{6}\right)} \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \left(\color{blue}{{x}^{2} \cdot \frac{61}{120}} + \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{61}{120}, \frac{5}{6}\right)} \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

   13. lower-*.f64 99.1

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5083333333333333, 0.8333333333333334\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

10. Applied rewrites 99.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.5083333333333333, 0.8333333333333334\right) \cdot x, x\right)}, \frac{\varepsilon}{\cos x}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)}\right), \varepsilon, \varepsilon\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)}\right), \varepsilon, \varepsilon\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right), \varepsilon, \varepsilon\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right), \varepsilon, \varepsilon\right) \]

    4. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) + 1\right)}\right), \varepsilon, \varepsilon\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) + 1\right)\right), \varepsilon, \varepsilon\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)} + 1\right)\right), \varepsilon, \varepsilon\right) \]

    7. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right), 1\right)}\right), \varepsilon, \varepsilon\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)}, 1\right)\right), \varepsilon, \varepsilon\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{17}{45} \cdot {x}^{2} + \frac{2}{3}\right)}, 1\right)\right), \varepsilon, \varepsilon\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{17}{45}} + \frac{2}{3}\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

    11. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{17}{45}, \frac{2}{3}\right)}, 1\right)\right), \varepsilon, \varepsilon\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{61}{120}, \frac{5}{6}\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{17}{45}, \frac{2}{3}\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

    13. lower-*.f64 98.8

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.5083333333333333, 0.8333333333333334\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.37777777777777777, 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

13. Applied rewrites 98.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.5083333333333333, 0.8333333333333334\right) \cdot x, x\right), \frac{\varepsilon}{\cos x}, \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.37777777777777777, 0.6666666666666666\right), 1\right)}\right), \varepsilon, \varepsilon\right) \]

14. Final simplification 98.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.5083333333333333, 0.8333333333333334\right), x\right), \frac{\varepsilon}{\cos x}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.37777777777777777, 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \]
15. Add Preprocessing

Alternative 7: 98.4% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)}, \varepsilon\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)}, \varepsilon\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\left(x \cdot \left(1 + x \cdot \left(\left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right) + \varepsilon\right)}, \varepsilon\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(\left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right)}, \varepsilon\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right) + 1}, \varepsilon\right), \varepsilon\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon, 1\right)}, \varepsilon\right), \varepsilon\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{2}{3} \cdot x + \left(\frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right)}, 1\right), \varepsilon\right), \varepsilon\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{3}} + \left(\frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right), 1\right), \varepsilon\right), \varepsilon\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{3}, \frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right)}, 1\right), \varepsilon\right), \varepsilon\right) \]

   9. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{2}{3}, \color{blue}{\varepsilon \cdot \left(\frac{5}{6} - \frac{-1}{2}\right)}\right), 1\right), \varepsilon\right), \varepsilon\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{2}{3}, \color{blue}{\varepsilon \cdot \left(\frac{5}{6} - \frac{-1}{2}\right)}\right), 1\right), \varepsilon\right), \varepsilon\right) \]

   11. metadata-eval 98.6

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

8. Applied rewrites 98.6%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.6666666666666666, \varepsilon \cdot 1.3333333333333333\right), 1\right), \varepsilon\right)}, \varepsilon\right) \]
9. Add Preprocessing

Alternative 8: 98.2% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 98.4

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

8. Applied rewrites 98.4%

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

Alternative 9: 97.7% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 98.0

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

8. Applied rewrites 98.0%

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

Alternative 10: 5.7% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 98.0

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in eps around inf

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

    1. lower-*.f64 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 5.7

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

11. Applied rewrites 5.7%

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

12. Final simplification 5.7%

    \[\leadsto x \cdot \left(\varepsilon \cdot \varepsilon\right) \]
13. 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: 62.6% 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 2024214 
(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)))