2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.9%

Time: 15.4s

Alternatives: 8

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

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

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 64.0%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. 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

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

   2. +-commutative N/A

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

   3. remove-double-neg N/A

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

   4. mul-1-neg N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. sub-neg N/A

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

   10. mul-1-neg N/A

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

   11. remove-double-neg N/A

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

   12. +-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. 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. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (* eps eps)
   (* eps (fma eps (* eps 0.008333333333333333) -0.16666666666666666))
   eps)
  (/ 1.0 (* (cos x) (cos (+ x eps))))))

C

double code(double x, double eps) {
	return fma((eps * eps), (eps * fma(eps, (eps * 0.008333333333333333), -0.16666666666666666)), eps) * (1.0 / (cos(x) * cos((x + eps))));
}

Julia

function code(x, eps)
	return Float64(fma(Float64(eps * eps), Float64(eps * fma(eps, Float64(eps * 0.008333333333333333), -0.16666666666666666)), eps) * Float64(1.0 / Float64(cos(x) * cos(Float64(x + eps)))))
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * N[(eps * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * N[(1.0 / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(x + eps), $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

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. 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. associate-*l* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	return Float64(Float64(1.0 / Float64(cos(x) * cos(Float64(x + eps)))) * 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[(x + eps), $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

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. 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%

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

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. 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: 98.4% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

8. Applied rewrites 98.0%

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

10. Applied rewrites 98.0%

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

11. Taylor expanded in x around 0

    \[\leadsto x \cdot \left(x \cdot \left(\varepsilon + x \cdot \left(\frac{2}{3} \cdot \left(\varepsilon \cdot x\right) + \varepsilon \cdot \left(\frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right)\right)\right) + {\varepsilon}^{2}\right) + \varepsilon \]
12. Step-by-step derivation

13. Applied rewrites 98.6%

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

14. Final simplification 98.6%

    \[\leadsto \varepsilon + x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 1.3333333333333333, \varepsilon \cdot \left(x \cdot 0.6666666666666666\right)\right), \varepsilon\right), \varepsilon \cdot \varepsilon\right) \]
15. Add Preprocessing

Alternative 6: 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

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

8. Applied rewrites 98.6%

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

Alternative 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 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

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

8. Applied rewrites 98.4%

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

9. Final simplification 98.4%

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

Alternative 8: 97.7% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

8. Applied rewrites 98.0%

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

9. Final simplification 98.0%

   \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \varepsilon, \varepsilon\right) \]
10. 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)))