2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 100.0%

Time: 12.9s

Alternatives: 9

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-cos.f64 63.0

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 63.0

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

4. Applied rewrites 63.0%

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2} \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. +-commutative N/A

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

   6. cos-sum N/A

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

   7. lower--.f64 N/A

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

   8. +-lft-identity N/A

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

   9. lift-+.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. count-2 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. +-lft-identity N/A

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

   17. +-lft-identity N/A

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

   18. lift-+.f64 N/A

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

   19. lift-sin.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lower-sin.f64 N/A

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

   22. count-2 N/A

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

   23. lower-*.f64 100.0

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

   24. lift-+.f64 N/A

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

   25. +-lft-identity 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-cos.f64 63.0

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 63.0

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

4. Applied rewrites 63.0%

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2} \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. +-commutative N/A

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

   6. cos-sum N/A

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

   7. lower--.f64 N/A

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

   8. +-lft-identity N/A

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

   9. lift-+.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. count-2 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. +-lft-identity N/A

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

   17. +-lft-identity N/A

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

   18. lift-+.f64 N/A

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

   19. lift-sin.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lower-sin.f64 N/A

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

   22. count-2 N/A

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

   23. lower-*.f64 100.0

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

   24. lift-+.f64 N/A

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

   25. +-lft-identity 100.0

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

8. Applied rewrites 100.0%

   \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\cos \left(0 + \varepsilon\right) + \color{blue}{\left(\cos \left(2 \cdot x\right) \cdot \cos \varepsilon - \sin \left(2 \cdot x\right) \cdot \sin \varepsilon\right)}} \cdot 2 \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. +-lft-identity N/A

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

   7. lift-+.f64 N/A

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

   8. distribute-rgt1-in N/A

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

   9. lower-fma.f64 N/A

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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)) ** (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 63.1

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

4. Applied rewrites 63.1%

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

5. Taylor expanded in x around 0

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

   1. lower-sin.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-cos.f64 63.0

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 63.0

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

4. Applied rewrites 63.0%

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-lft-identity N/A

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

   12. lift-+.f64 N/A

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

   13. +-rgt-identity N/A

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

   14. +-inverses N/A

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

   15. associate--l+ N/A

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

   16. lift-+.f64 N/A

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

   17. metadata-eval N/A

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

8. Applied rewrites 99.8%

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

Alternative 5: 99.1% 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 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-cos.f64 63.0

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 63.0

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

4. Applied rewrites 63.0%

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

6. Applied rewrites 99.8%

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

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 98.9

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

9. Applied rewrites 98.9%

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

Alternative 6: 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 63.0%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 63.1

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

4. Applied rewrites 63.1%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.9

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

7. Applied rewrites 98.9%

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

8. 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 98.1%

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

Alternative 7: 98.5% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 63.1

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

4. Applied rewrites 63.1%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.9

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.9%

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

Alternative 8: 6.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 63.1

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

4. Applied rewrites 63.1%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.9

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.9%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 6.5%

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

Alternative 9: 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 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-cos.f64 63.0

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 63.0

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in eps around 0

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft 5.3

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

7. Applied rewrites 5.3%

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

Developer Target 1: 99.1% 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 2024298 
(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)))