2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.9%

Time: 16.8s

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: 62.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   6. sin-diff N/A

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

   7. lower-sin.f64 N/A

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

   8. lower--.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 63.8

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

4. Applied egg-rr 63.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 63.8

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

   12. lift--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. associate--l+ N/A

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

   16. +-inverses N/A

       \[\leadsto \frac{1}{\cos x \cdot \cos \left(x + \varepsilon\right)} \cdot \sin \left(\varepsilon + \color{blue}{0}\right) \]

   17. lower-+.f64 99.8

       \[\leadsto \frac{1}{\cos x \cdot \cos \left(x + \varepsilon\right)} \cdot \sin \color{blue}{\left(\varepsilon + 0\right)} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{1}{\cos x \cdot \cos \left(x + \varepsilon\right)} \cdot \sin \left(\varepsilon + 0\right)} \]

7. Final simplification 99.8%

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

Alternative 2: 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]

Derivation

1. Initial program 63.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   6. sin-diff N/A

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

   7. lower-sin.f64 N/A

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

   8. lower--.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 63.8

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

4. Applied egg-rr 63.8%

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

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. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 99.8

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

7. Simplified 99.8%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma (pow (tan x) 2.0) eps eps))

C

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

Julia

function code(x, eps)
	return fma((tan(x) ^ 2.0), eps, eps)
end

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.3

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

5. Simplified 98.3%

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

   1. lift-sin.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 98.3

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

7. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)} \]
8. Add Preprocessing

Alternative 4: 98.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 1.3333333333333333, \varepsilon \cdot 0.6666666666666666\right), 1\right), x\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (fma
   x
   (fma
    eps
    (fma eps (fma x 1.3333333333333333 (* eps 0.6666666666666666)) 1.0)
    x)
   (* 0.3333333333333333 (* eps eps)))
  eps))

C

double code(double x, double eps) {
	return fma(eps, fma(x, fma(eps, fma(eps, fma(x, 1.3333333333333333, (eps * 0.6666666666666666)), 1.0), x), (0.3333333333333333 * (eps * eps))), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(x, fma(eps, fma(eps, fma(x, 1.3333333333333333, Float64(eps * 0.6666666666666666)), 1.0), x), Float64(0.3333333333333333 * Float64(eps * eps))), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(x * N[(eps * N[(eps * N[(x * 1.3333333333333333 + N[(eps * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision] + N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right), \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

8. Simplified 97.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right), \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 1.3333333333333333, x\right)\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 97.5

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 1.3333333333333333, \color{blue}{\varepsilon \cdot 0.6666666666666666}\right), 1\right), x\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right) \]

11. Simplified 97.5%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 1.3333333333333333, \varepsilon \cdot 0.6666666666666666\right), 1\right), x\right)}, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right) \]
12. Add Preprocessing

Alternative 5: 98.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 1.3333333333333333, 0.3333333333333333\right), x\right), x \cdot x\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (fma
   eps
   (fma eps (fma x (* x 1.3333333333333333) 0.3333333333333333) x)
   (* x x))
  eps))

C

double code(double x, double eps) {
	return fma(eps, fma(eps, fma(eps, fma(x, (x * 1.3333333333333333), 0.3333333333333333), x), (x * x)), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(eps, fma(eps, fma(x, Float64(x * 1.3333333333333333), 0.3333333333333333), x), Float64(x * x)), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(eps * N[(x * N[(x * 1.3333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + x), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right), \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

8. Simplified 97.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right), \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 1.3333333333333333, x\right)\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. lower-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. *-commutative N/A

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

    9. lower-fma.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. unpow2 N/A

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

    13. lower-*.f64 97.5

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

11. Simplified 97.5%

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

Alternative 6: 98.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right), \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

8. Simplified 97.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right), \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 1.3333333333333333, x\right)\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. lower-+.f64 97.5

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

11. Simplified 97.5%

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

Alternative 7: 98.2% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right), \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

8. Simplified 97.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right), \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 1.3333333333333333, x\right)\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. unpow2 N/A

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

    2. distribute-rgt-out N/A

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

    3. lower-*.f64 N/A

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

    4. +-commutative N/A

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

    5. lower-+.f64 97.4

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

11. Simplified 97.4%

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

Alternative 8: 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 63.8%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.3

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

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 97.4

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

8. Simplified 97.4%

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

Alternative 9: 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 63.8%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.3

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

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 97.4

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

8. Simplified 97.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot x}, \varepsilon\right) \]
9. 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 97.3

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

10. Applied egg-rr 97.3%

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

11. Final simplification 97.3%

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

Alternative 10: 6.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.3

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

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 97.4

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

8. Simplified 97.4%

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

9. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 6.6

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

11. Simplified 6.6%

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

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 N/A

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

    4. lower-*.f64 6.6

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

13. Applied egg-rr 6.6%

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

14. Final simplification 6.6%

    \[\leadsto x \cdot \left(x \cdot \varepsilon\right) \]
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: 62.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024215 
(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)))