2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.7%

Time: 15.7s

Alternatives: 15

Speedup: 17.3×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f64 N/A

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

11. Applied rewrites 99.9%

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

Alternative 2: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in eps around 0

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

12. Applied rewrites 99.8%

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

13. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in eps around 0

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

    1. lower-/.f64 N/A

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

    2. lower-cos.f64 99.7

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

12. Applied rewrites 99.7%

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

Alternative 5: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in eps around 0

    \[\leadsto \frac{1 \cdot \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
11. Step-by-step derivation

12. Applied rewrites 99.7%

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

13. Final simplification 99.7%

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

Alternative 6: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\varepsilon \cdot \sin x}{{\cos x}^{3}} + \varepsilon \cdot \frac{1}{{\cos x}^{2}}} \]

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. associate-*r/ N/A

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

   11. *-rgt-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.1%

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

11. Final simplification 99.1%

    \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \varepsilon, \frac{\varepsilon}{{\cos x}^{2}}\right) \]
12. Add Preprocessing

Alternative 7: 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 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.1

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

7. Applied rewrites 99.1%

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

Alternative 8: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.1

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

7. Applied rewrites 99.1%

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

9. Applied rewrites 99.1%

   \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 + 0.5 \cdot \cos \left(x + x\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 99.1%

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

Alternative 9: 98.3% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (* x x)
  (fma
   (* x x)
   (fma (* x x) (* eps 0.37777777777777777) (* eps 0.6666666666666666))
   eps)
  eps))

C

double code(double x, double eps) {
	return fma((x * x), fma((x * x), fma((x * x), (eps * 0.37777777777777777), (eps * 0.6666666666666666)), eps), eps);
}

Julia

function code(x, eps)
	return fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), Float64(eps * 0.37777777777777777), Float64(eps * 0.6666666666666666)), eps), eps)
end

Wolfram

code[x_, eps_] := N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(eps * 0.37777777777777777), $MachinePrecision] + N[(eps * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.1

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

7. Applied rewrites 99.1%

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

9. Applied rewrites 99.1%

   \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 + 0.5 \cdot \cos \left(x + x\right)}} \]

10. Taylor expanded in x around 0

    \[\leadsto \varepsilon + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \varepsilon + \left(\frac{-2}{45} \cdot \varepsilon + \frac{2}{3} \cdot \varepsilon\right)\right)\right) - \left(-1 \cdot \varepsilon + \frac{1}{3} \cdot \varepsilon\right)\right) - -1 \cdot \varepsilon\right)} \]
11. Step-by-step derivation

12. Applied rewrites 98.4%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot 0.37777777777777777, \varepsilon \cdot 0.6666666666666666\right), \varepsilon\right)}, \varepsilon\right) \]
13. Add Preprocessing

Alternative 10: 98.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.1

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.4%

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

Alternative 11: 98.3% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.1

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.4%

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

Alternative 12: 98.2% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\varepsilon \cdot \sin x}{{\cos x}^{3}} + \varepsilon \cdot \frac{1}{{\cos x}^{2}}} \]

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. associate-*r/ N/A

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

   11. *-rgt-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.2%

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

11. Final simplification 98.2%

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

Alternative 13: 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 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.1

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.2%

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

Alternative 14: 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 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\varepsilon \cdot \sin x}{{\cos x}^{3}} + \varepsilon \cdot \frac{1}{{\cos x}^{2}}} \]

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. associate-*r/ N/A

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

   11. *-rgt-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.8%

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

11. Final simplification 97.8%

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

Alternative 15: 5.7% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\varepsilon \cdot \sin x}{{\cos x}^{3}} + \varepsilon \cdot \frac{1}{{\cos x}^{2}}} \]

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. associate-*r/ N/A

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

   11. *-rgt-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.8%

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

11. Taylor expanded in eps around inf

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

13. Applied rewrites 5.7%

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

14. Final simplification 5.7%

    \[\leadsto \varepsilon \cdot \left(x \cdot \varepsilon\right) \]
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: 62.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024221 
(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)))