2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 99.7%

Time: 13.1s

Alternatives: 14

Speedup: 34.5×

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: 62.7% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. frac-add N/A

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

   10. *-commutative N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 62.5%

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

5. Taylor expanded in x around 0

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

    10. associate-*r* N/A

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

    11. distribute-rgt-out N/A

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

    12. lower-*.f64 N/A

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

    13. lower-sin.f64 N/A

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

    14. unpow2 N/A

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

    15. associate-*r* N/A

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

    16. lower-fma.f64 N/A

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

    17. lower-*.f64 100.0

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

12. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. frac-add N/A

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

   10. *-commutative N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 62.5%

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

5. Taylor expanded in x around 0

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, \varepsilon, 0\right), \varepsilon, 1\right), \varepsilon, 0\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (fma (fma (fma -0.16666666666666666 eps 0.0) eps 1.0) eps 0.0)
  (* (cos (+ eps x)) (cos x))))

C

double code(double x, double eps) {
	return fma(fma(fma(-0.16666666666666666, eps, 0.0), eps, 1.0), eps, 0.0) / (cos((eps + x)) * cos(x));
}

Julia

function code(x, eps)
	return Float64(fma(fma(fma(-0.16666666666666666, eps, 0.0), eps, 1.0), eps, 0.0) / Float64(cos(Float64(eps + x)) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(-0.16666666666666666 * eps + 0.0), $MachinePrecision] * eps + 1.0), $MachinePrecision] * eps + 0.0), $MachinePrecision] / N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. frac-add N/A

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

   10. *-commutative N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 62.5%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt1-in N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\cos x \cdot \sin x\right) + \left(\frac{1}{2} \cdot \left(\cos x \cdot \sin x\right) + \varepsilon \cdot \left(\frac{-1}{6} \cdot {\cos x}^{2} + \frac{-1}{6} \cdot {\sin x}^{2}\right)\right)\right) + \left({\cos x}^{2} + {\sin x}^{2}\right)\right) \cdot \varepsilon + \color{blue}{0} \cdot \left(\cos x \cdot \sin x\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   6. mul0-lft N/A

      \[\leadsto \frac{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\cos x \cdot \sin x\right) + \left(\frac{1}{2} \cdot \left(\cos x \cdot \sin x\right) + \varepsilon \cdot \left(\frac{-1}{6} \cdot {\cos x}^{2} + \frac{-1}{6} \cdot {\sin x}^{2}\right)\right)\right) + \left({\cos x}^{2} + {\sin x}^{2}\right)\right) \cdot \varepsilon + \color{blue}{0}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, \varepsilon, 0\right), \varepsilon, 1\right), \varepsilon, 0\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Add Preprocessing

Alternative 4: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. frac-add N/A

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

   10. *-commutative N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 62.5%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} \cdot \left(\cos x \cdot \sin x\right) + \varepsilon \cdot \left({\cos x}^{2} + {\sin x}^{2}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   5. mul0-lft N/A

      \[\leadsto \frac{\color{blue}{0} + \varepsilon \cdot \left({\cos x}^{2} + {\sin x}^{2}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   6. unpow2 N/A

      \[\leadsto \frac{0 + \varepsilon \cdot \left(\color{blue}{\cos x \cdot \cos x} + {\sin x}^{2}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   7. unpow2 N/A

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

   8. cos-sin-sum N/A

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

   9. *-rgt-identity N/A

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

   10. remove-double-neg N/A

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

   11. mul-1-neg N/A

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

   12. sub-neg N/A

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

   13. neg-sub0 N/A

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

   14. mul-1-neg N/A

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

   15. remove-double-neg 99.2

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

7. Applied rewrites 99.2%

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

Alternative 5: 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 62.5%

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

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 62.3

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 62.3

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

4. Applied rewrites 62.3%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

Alternative 6: 98.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} - \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot -0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\left(\varepsilon + \frac{-1}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

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

9. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right) + \color{blue}{x \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(\left(\varepsilon + \frac{-1}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right) + \varepsilon \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2}\right)} \]

11. Applied rewrites 98.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 1.3333333333333333, 1\right) \cdot \varepsilon, x, \varepsilon \cdot \varepsilon\right), \color{blue}{x}, \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333, \varepsilon\right)\right) \]
12. Step-by-step derivation

13. Applied rewrites 98.3%

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

Alternative 7: 98.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} - \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot -0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\left(\varepsilon + \frac{-1}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\left(\varepsilon + \frac{-1}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 98.3%

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

Alternative 8: 98.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} - \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot -0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\left(\varepsilon + \frac{-1}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.3%

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

Alternative 9: 98.2% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} - \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot -0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\left(\varepsilon + \frac{-1}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.3%

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

12. Taylor expanded in eps around 0

    \[\leadsto \left(1 + \left(\varepsilon \cdot \left(x + \varepsilon \cdot \left(\frac{1}{3} + \frac{4}{3} \cdot {x}^{2}\right)\right) + {x}^{2}\right)\right) \cdot \varepsilon \]
13. Step-by-step derivation

14. Applied rewrites 98.3%

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

Alternative 10: 98.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} - \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot -0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\left(\varepsilon + \frac{-1}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.3%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 98.3%

    \[\leadsto \mathsf{fma}\left(\varepsilon + x, x, \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, 1\right)\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 11: 98.1% accurate, 13.8× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* (fma x (+ eps x) 1.0) eps))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} - \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot -0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\left(\varepsilon + \frac{-1}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.3%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 98.1%

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

Alternative 12: 98.1% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} - \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot -0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\left(\varepsilon + \frac{-1}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.3%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 98.1%

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

Alternative 13: 97.7% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* 1.0 eps))

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return 1.0 * eps;
}

Python

def code(x, eps):
	return 1.0 * eps

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} - \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot -0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in eps around 0

   \[\leadsto 1 \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 98.0%

    \[\leadsto 1 \cdot \varepsilon \]
12. Add Preprocessing

Alternative 14: 5.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-cos.f64 62.5

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 62.5

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

4. Applied rewrites 62.5%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}} \]
6. Step-by-step derivation

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot \frac{\sin x}{\cos x} \]

   3. mul0-lft 5.3

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 5.3%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 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 2024320 
(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 (+ eps (* eps (tan x) (tan x))))

  (- (tan (+ x eps)) (tan x)))