2tan (problem 3.3.2)

Percentage Accurate: 61.9% → 99.9%

Time: 13.4s

Alternatives: 13

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(1.0 / N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[eps], $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-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 62.5

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

4. Applied rewrites 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. tan-quot N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-sub N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. sin-diff N/A

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

   14. lift--.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. clear-num N/A

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

   18. associate-/r/ N/A

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

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{1}{\cos \left(\varepsilon + x\right) \cdot \cos x} \cdot \sin \left(0 + \varepsilon\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. +-inverses N/A

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

   7. associate--l+ N/A

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

   8. associate-/r/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. inv-pow N/A

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(1.0 / N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[eps], $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-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 62.5

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

4. Applied rewrites 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. tan-quot N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-sub N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. sin-diff N/A

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

   14. lift--.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. clear-num N/A

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

   18. associate-/r/ N/A

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

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{1}{\cos \left(\varepsilon + x\right) \cdot \cos x} \cdot \sin \left(0 + \varepsilon\right)} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-lft-identity 99.9

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[N[(x + 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. 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 62.5

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

   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 62.5

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

4. Applied rewrites 62.5%

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

5. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. remove-double-neg N/A

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

   4. mul-1-neg N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. mul-1-neg N/A

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

   13. remove-double-neg N/A

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

   14. +-commutative N/A

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

   15. lower-cos.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 4: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Cos[N[(x + eps), $MachinePrecision]], $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-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 62.5

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

4. Applied rewrites 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. tan-quot N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-sub N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. sin-diff N/A

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

   14. lift--.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. clear-num N/A

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

   18. associate-/r/ N/A

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

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{1}{\cos \left(\varepsilon + x\right) \cdot \cos x} \cdot \sin \left(0 + \varepsilon\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. +-inverses N/A

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

   7. associate--l+ N/A

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

   8. associate-/r/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. inv-pow N/A

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 99.7

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

11. Applied rewrites 99.7%

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

Alternative 5: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(1.0 / N[(N[(N[Cos[x], $MachinePrecision] / eps), $MachinePrecision] * N[Cos[N[(x + eps), $MachinePrecision]], $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. 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 62.5

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

   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 62.5

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

4. Applied rewrites 62.5%

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

5. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. +-commutative N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. sub-neg N/A

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

   7. lower-*.f64 N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. remove-double-neg N/A

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

   11. +-commutative N/A

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

   12. lower-cos.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-sin.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 6: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\cos \left(x + x\right) \cdot 0.5 + 0.5} \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (/ 1.0 (+ (* (cos (+ x x)) 0.5) 0.5)) eps))

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (1.0d0 / ((cos((x + x)) * 0.5d0) + 0.5d0)) * eps
end function

Java

public static double code(double x, double eps) {
	return (1.0 / ((Math.cos((x + x)) * 0.5) + 0.5)) * eps;
}

Python

def code(x, eps):
	return (1.0 / ((math.cos((x + x)) * 0.5) + 0.5)) * eps

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (1.0 / ((cos((x + x)) * 0.5) + 0.5)) * eps;
end

Wolfram

code[x_, eps_] := N[(N[(1.0 / N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * eps), $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 62.5

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

   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 62.5

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

4. Applied rewrites 62.5%

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

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 99.2

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

7. Applied rewrites 99.2%

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

9. Applied rewrites 99.3%

   \[\leadsto \frac{1}{0.5 + 0.5 \cdot \cos \left(x + x\right)} \cdot \color{blue}{\varepsilon} \]

10. Final simplification 99.3%

    \[\leadsto \frac{1}{\cos \left(x + x\right) \cdot 0.5 + 0.5} \cdot \varepsilon \]
11. Add Preprocessing

Alternative 7: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / ((cos((x + x)) * 0.5d0) + 0.5d0)
end function

Java

public static double code(double x, double eps) {
	return eps / ((Math.cos((x + x)) * 0.5) + 0.5);
}

Python

def code(x, eps):
	return eps / ((math.cos((x + x)) * 0.5) + 0.5)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps / ((cos((x + x)) * 0.5) + 0.5);
end

Wolfram

code[x_, eps_] := N[(eps / N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + 0.5), $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 62.5

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

   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 62.5

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

4. Applied rewrites 62.5%

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

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 99.2

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

7. Applied rewrites 99.2%

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

9. Applied rewrites 99.3%

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

10. Final simplification 99.3%

    \[\leadsto \frac{\varepsilon}{\cos \left(x + x\right) \cdot 0.5 + 0.5} \]
11. Add Preprocessing

Alternative 8: 98.5% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.37777777777777777 * eps), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(0.6666666666666666 * eps), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + eps), $MachinePrecision] * N[(x * x), $MachinePrecision] + eps), $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 62.5

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

   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 62.5

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

4. Applied rewrites 62.5%

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

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 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.5%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.7%

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

14. Final simplification 98.7%

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

Alternative 9: 98.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(eps / N[(N[(N[(x * x), $MachinePrecision] * 0.3333333333333333 + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $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 62.5

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

   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 62.5

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

4. Applied rewrites 62.5%

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

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 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.7%

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

Alternative 10: 98.5% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(N[(N[(0.6666666666666666 * eps), $MachinePrecision] * N[(x * x), $MachinePrecision] + eps), $MachinePrecision] * N[(x * x), $MachinePrecision] + eps), $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 62.5

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

   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 62.5

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

4. Applied rewrites 62.5%

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

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 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.6%

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

Alternative 11: 98.5% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.6%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.6%

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

12. Final simplification 98.6%

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

Alternative 12: 98.4% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(N[(x * eps), $MachinePrecision] * x + eps), $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 62.5

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

   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 62.5

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

4. Applied rewrites 62.5%

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

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 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.5%

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

Alternative 13: 6.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(x * eps), $MachinePrecision] * x), $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 62.5

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

   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 62.5

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

4. Applied rewrites 62.5%

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

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 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.5%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 6.3%

    \[\leadsto \left(x \cdot \varepsilon\right) \cdot x \]
14. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 62.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 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 2024235 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (/ (sin eps) (* (cos x) (cos (+ x eps)))))

  :alt
  (! :herbie-platform default (- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x)))

  :alt
  (! :herbie-platform default (+ eps (* eps (tan x) (tan x))))

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