2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.9%

Time: 13.3s

Alternatives: 13

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: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\left(\cos \left(x - \left(x + \varepsilon\right)\right) + \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot 0.5} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/ (sin eps) (* (+ (cos (- x (+ x eps))) (cos (fma 2.0 x eps))) 0.5)))

C

double code(double x, double eps) {
	return sin(eps) / ((cos((x - (x + eps))) + cos(fma(2.0, x, eps))) * 0.5);
}

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(Float64(cos(Float64(x - Float64(x + eps))) + cos(fma(2.0, x, eps))) * 0.5))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[(N[Cos[N[(x - N[(x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(2.0 * x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.2%

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

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 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

9. Applied rewrites 100.0%

   \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\left(\cos \left(x - \left(x + \varepsilon\right)\right) + \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot 0.5}} \]
10. 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 62.2%

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

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.2%

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

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 100.0

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

7. Applied rewrites 100.0%

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

Alternative 3: 99.7% 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 62.2%

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

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.2%

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

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 99.7%

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

Alternative 4: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.5, 1\right) + \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot 0.5} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/ (sin eps) (* (+ (fma (* eps eps) -0.5 1.0) (cos (fma 2.0 x eps))) 0.5)))

C

double code(double x, double eps) {
	return sin(eps) / ((fma((eps * eps), -0.5, 1.0) + cos(fma(2.0, x, eps))) * 0.5);
}

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(Float64(fma(Float64(eps * eps), -0.5, 1.0) + cos(fma(2.0, x, eps))) * 0.5))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[(N[(N[(eps * eps), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[Cos[N[(2.0 * x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.2%

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

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 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 99.7

       \[\leadsto \frac{\sin \varepsilon}{\left(\mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, -0.5, 1\right) + \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot 0.5} \]

12. Applied rewrites 99.7%

    \[\leadsto \frac{\sin \varepsilon}{\left(\color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.5, 1\right)} + \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot 0.5} \]
13. Add Preprocessing

Alternative 5: 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 62.2%

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

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.2%

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

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 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{-1 \cdot \left(\cos x \cdot \sin x\right) + \left(\varepsilon \cdot \left({\cos x}^{2} + {\sin x}^{2}\right) + \cos x \cdot \sin x\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
9. 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 99.5

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

10. Applied rewrites 99.5%

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

Alternative 6: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\cos \left(-2 \cdot x\right) + 1} \cdot 2 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (/ eps (+ (cos (* -2.0 x)) 1.0)) 2.0))

C

double code(double x, double eps) {
	return (eps / (cos((-2.0 * x)) + 1.0)) * 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps / (cos(((-2.0d0) * x)) + 1.0d0)) * 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (eps / (Math.cos((-2.0 * x)) + 1.0)) * 2.0;
}

Python

def code(x, eps):
	return (eps / (math.cos((-2.0 * x)) + 1.0)) * 2.0

Julia

function code(x, eps)
	return Float64(Float64(eps / Float64(cos(Float64(-2.0 * x)) + 1.0)) * 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps / (cos((-2.0 * x)) + 1.0)) * 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(eps / N[(N[Cos[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. inv-pow N/A

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

   8. clear-num N/A

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

   9. tan-quot N/A

      \[\leadsto {\color{blue}{\tan \left(x + \varepsilon\right)}}^{1} - \tan x \]

   10. lift-tan.f64 N/A

       \[\leadsto {\color{blue}{\tan \left(x + \varepsilon\right)}}^{1} - \tan x \]

   11. pow-to-exp N/A

       \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   12. lower-exp.f64 N/A

       \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   13. lower-*.f64 N/A

       \[\leadsto e^{\color{blue}{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   14. lower-log.f64 23.0

       \[\leadsto e^{\color{blue}{\log \tan \left(x + \varepsilon\right)} \cdot 1} - \tan x \]

   15. lift-+.f64 N/A

       \[\leadsto e^{\log \tan \color{blue}{\left(x + \varepsilon\right)} \cdot 1} - \tan x \]

   16. +-commutative N/A

       \[\leadsto e^{\log \tan \color{blue}{\left(\varepsilon + x\right)} \cdot 1} - \tan x \]

   17. lower-+.f64 23.0

       \[\leadsto e^{\log \tan \color{blue}{\left(\varepsilon + x\right)} \cdot 1} - \tan x \]

4. Applied rewrites 23.0%

   \[\leadsto \color{blue}{e^{\log \tan \left(\varepsilon + x\right) \cdot 1}} - \tan x \]

6. Applied rewrites 100.0%

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

7. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-neg-in N/A

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

   6. cos-neg N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{\varepsilon}{\cos \left(-2 \cdot x\right) + 1}} \cdot 2 \]
10. Add Preprocessing

Alternative 7: 98.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \varepsilon \cdot -0.18888888888888888, 0.3333333333333333 \cdot \varepsilon\right), x \cdot x, 0.5 \cdot \varepsilon\right), x \cdot x, 0.5 \cdot \varepsilon\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (fma
    (fma (* (- x) x) (* eps -0.18888888888888888) (* 0.3333333333333333 eps))
    (* x x)
    (* 0.5 eps))
   (* x x)
   (* 0.5 eps))
  2.0))

C

double code(double x, double eps) {
	return fma(fma(fma((-x * x), (eps * -0.18888888888888888), (0.3333333333333333 * eps)), (x * x), (0.5 * eps)), (x * x), (0.5 * eps)) * 2.0;
}

Julia

function code(x, eps)
	return Float64(fma(fma(fma(Float64(Float64(-x) * x), Float64(eps * -0.18888888888888888), Float64(0.3333333333333333 * eps)), Float64(x * x), Float64(0.5 * eps)), Float64(x * x), Float64(0.5 * eps)) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[((-x) * x), $MachinePrecision] * N[(eps * -0.18888888888888888), $MachinePrecision] + N[(0.3333333333333333 * eps), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(0.5 * eps), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(0.5 * eps), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. inv-pow N/A

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

   8. clear-num N/A

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

   9. tan-quot N/A

      \[\leadsto {\color{blue}{\tan \left(x + \varepsilon\right)}}^{1} - \tan x \]

   10. lift-tan.f64 N/A

       \[\leadsto {\color{blue}{\tan \left(x + \varepsilon\right)}}^{1} - \tan x \]

   11. pow-to-exp N/A

       \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   12. lower-exp.f64 N/A

       \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   13. lower-*.f64 N/A

       \[\leadsto e^{\color{blue}{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   14. lower-log.f64 23.0

       \[\leadsto e^{\color{blue}{\log \tan \left(x + \varepsilon\right)} \cdot 1} - \tan x \]

   15. lift-+.f64 N/A

       \[\leadsto e^{\log \tan \color{blue}{\left(x + \varepsilon\right)} \cdot 1} - \tan x \]

   16. +-commutative N/A

       \[\leadsto e^{\log \tan \color{blue}{\left(\varepsilon + x\right)} \cdot 1} - \tan x \]

   17. lower-+.f64 23.0

       \[\leadsto e^{\log \tan \color{blue}{\left(\varepsilon + x\right)} \cdot 1} - \tan x \]

4. Applied rewrites 23.0%

   \[\leadsto \color{blue}{e^{\log \tan \left(\varepsilon + x\right) \cdot 1}} - \tan x \]

6. Applied rewrites 100.0%

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

7. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-neg-in N/A

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

   6. cos-neg N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Taylor expanded in x around 0

    \[\leadsto \left(\frac{1}{2} \cdot \varepsilon + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + \left(\frac{-1}{45} \cdot \varepsilon + \frac{1}{3} \cdot \varepsilon\right)\right)\right) - \left(\frac{-1}{2} \cdot \varepsilon + \frac{1}{6} \cdot \varepsilon\right)\right) - \frac{-1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]
11. Step-by-step derivation

12. Applied rewrites 98.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \varepsilon \cdot -0.18888888888888888, 0.3333333333333333 \cdot \varepsilon\right), x \cdot x, 0.5 \cdot \varepsilon\right), \color{blue}{x \cdot x}, 0.5 \cdot \varepsilon\right) \cdot 2 \]
13. Add Preprocessing

Alternative 8: 98.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right), x \cdot x, 1\right) + 1} \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (/ eps (+ (fma (fma 0.6666666666666666 (* x x) -2.0) (* x x) 1.0) 1.0))
  2.0))

C

double code(double x, double eps) {
	return (eps / (fma(fma(0.6666666666666666, (x * x), -2.0), (x * x), 1.0) + 1.0)) * 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(eps / Float64(fma(fma(0.6666666666666666, Float64(x * x), -2.0), Float64(x * x), 1.0) + 1.0)) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(eps / N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. inv-pow N/A

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

   8. clear-num N/A

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

   9. tan-quot N/A

      \[\leadsto {\color{blue}{\tan \left(x + \varepsilon\right)}}^{1} - \tan x \]

   10. lift-tan.f64 N/A

       \[\leadsto {\color{blue}{\tan \left(x + \varepsilon\right)}}^{1} - \tan x \]

   11. pow-to-exp N/A

       \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   12. lower-exp.f64 N/A

       \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   13. lower-*.f64 N/A

       \[\leadsto e^{\color{blue}{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   14. lower-log.f64 23.0

       \[\leadsto e^{\color{blue}{\log \tan \left(x + \varepsilon\right)} \cdot 1} - \tan x \]

   15. lift-+.f64 N/A

       \[\leadsto e^{\log \tan \color{blue}{\left(x + \varepsilon\right)} \cdot 1} - \tan x \]

   16. +-commutative N/A

       \[\leadsto e^{\log \tan \color{blue}{\left(\varepsilon + x\right)} \cdot 1} - \tan x \]

   17. lower-+.f64 23.0

       \[\leadsto e^{\log \tan \color{blue}{\left(\varepsilon + x\right)} \cdot 1} - \tan x \]

4. Applied rewrites 23.0%

   \[\leadsto \color{blue}{e^{\log \tan \left(\varepsilon + x\right) \cdot 1}} - \tan x \]

6. Applied rewrites 100.0%

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

7. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-neg-in N/A

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

   6. cos-neg N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 98.8%

    \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right), x \cdot x, 1\right) + 1} \cdot 2 \]
13. Add Preprocessing

Alternative 9: 98.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right), x \cdot x, 2\right)} \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (/ eps (fma (fma 0.6666666666666666 (* x x) -2.0) (* x x) 2.0)) 2.0))

C

double code(double x, double eps) {
	return (eps / fma(fma(0.6666666666666666, (x * x), -2.0), (x * x), 2.0)) * 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(eps / fma(fma(0.6666666666666666, Float64(x * x), -2.0), Float64(x * x), 2.0)) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(eps / N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. inv-pow N/A

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

   8. clear-num N/A

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

   9. tan-quot N/A

      \[\leadsto {\color{blue}{\tan \left(x + \varepsilon\right)}}^{1} - \tan x \]

   10. lift-tan.f64 N/A

       \[\leadsto {\color{blue}{\tan \left(x + \varepsilon\right)}}^{1} - \tan x \]

   11. pow-to-exp N/A

       \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   12. lower-exp.f64 N/A

       \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   13. lower-*.f64 N/A

       \[\leadsto e^{\color{blue}{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   14. lower-log.f64 23.0

       \[\leadsto e^{\color{blue}{\log \tan \left(x + \varepsilon\right)} \cdot 1} - \tan x \]

   15. lift-+.f64 N/A

       \[\leadsto e^{\log \tan \color{blue}{\left(x + \varepsilon\right)} \cdot 1} - \tan x \]

   16. +-commutative N/A

       \[\leadsto e^{\log \tan \color{blue}{\left(\varepsilon + x\right)} \cdot 1} - \tan x \]

   17. lower-+.f64 23.0

       \[\leadsto e^{\log \tan \color{blue}{\left(\varepsilon + x\right)} \cdot 1} - \tan x \]

4. Applied rewrites 23.0%

   \[\leadsto \color{blue}{e^{\log \tan \left(\varepsilon + x\right) \cdot 1}} - \tan x \]

6. Applied rewrites 100.0%

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

7. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-neg-in N/A

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

   6. cos-neg N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 98.8%

    \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right), \color{blue}{x \cdot x}, 2\right)} \cdot 2 \]
13. Add Preprocessing

Alternative 10: 98.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(x \cdot x, \varepsilon, \varepsilon\right) \cdot 0.5\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* (fma (* x x) eps eps) 0.5) 2.0))

C

double code(double x, double eps) {
	return (fma((x * x), eps, eps) * 0.5) * 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(x * x), eps, eps) * 0.5) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(x * x), $MachinePrecision] * eps + eps), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. inv-pow N/A

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

   8. clear-num N/A

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

   9. tan-quot N/A

      \[\leadsto {\color{blue}{\tan \left(x + \varepsilon\right)}}^{1} - \tan x \]

   10. lift-tan.f64 N/A

       \[\leadsto {\color{blue}{\tan \left(x + \varepsilon\right)}}^{1} - \tan x \]

   11. pow-to-exp N/A

       \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   12. lower-exp.f64 N/A

       \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   13. lower-*.f64 N/A

       \[\leadsto e^{\color{blue}{\log \tan \left(x + \varepsilon\right) \cdot 1}} - \tan x \]

   14. lower-log.f64 23.0

       \[\leadsto e^{\color{blue}{\log \tan \left(x + \varepsilon\right)} \cdot 1} - \tan x \]

   15. lift-+.f64 N/A

       \[\leadsto e^{\log \tan \color{blue}{\left(x + \varepsilon\right)} \cdot 1} - \tan x \]

   16. +-commutative N/A

       \[\leadsto e^{\log \tan \color{blue}{\left(\varepsilon + x\right)} \cdot 1} - \tan x \]

   17. lower-+.f64 23.0

       \[\leadsto e^{\log \tan \color{blue}{\left(\varepsilon + x\right)} \cdot 1} - \tan x \]

4. Applied rewrites 23.0%

   \[\leadsto \color{blue}{e^{\log \tan \left(\varepsilon + x\right) \cdot 1}} - \tan x \]

6. Applied rewrites 100.0%

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

7. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-neg-in N/A

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

   6. cos-neg N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 98.5%

    \[\leadsto \left(\mathsf{fma}\left(x \cdot x, \varepsilon, \varepsilon\right) \cdot \color{blue}{0.5}\right) \cdot 2 \]
13. Add Preprocessing

Alternative 11: 97.8% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

   2. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \varepsilon}}{\cos \varepsilon} \]

   3. lower-cos.f64 97.9

      \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\cos \varepsilon}} \]

5. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 97.9%

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

Alternative 12: 97.8% 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 62.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

   2. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \varepsilon}}{\cos \varepsilon} \]

   3. lower-cos.f64 97.9

      \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\cos \varepsilon}} \]

5. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 97.9%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 97.9%

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

Alternative 13: 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 62.2%

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

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.2%

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

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 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in eps around 0

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

    1. associate-*r/ N/A

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

    2. distribute-lft1-in N/A

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

    3. metadata-eval N/A

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

    4. mul0-lft N/A

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

    5. metadata-eval N/A

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

    6. mul0-rgt N/A

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

    7. associate-*l/ N/A

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

    8. mul0-rgt 5.4

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

12. Applied rewrites 5.4%

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

Developer Target 1: 99.0% 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 2024313 
(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)))