2tan (problem 3.3.2)

Percentage Accurate: 62.9% → 100.0%

Time: 13.3s

Alternatives: 14

Speedup: 12.2×

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: 62.9% 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: 100.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. tan-quot N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 61.4

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

4. Applied rewrites 61.4%

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

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2} \]
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. lower-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-lft-identity N/A

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

   16. +-lft-identity N/A

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

   17. lift-+.f64 N/A

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

   18. lift-sin.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. lower-sin.f64 N/A

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

   21. lower-+.f64 100.0

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

   22. lift-+.f64 N/A

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

   23. +-lft-identity 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. tan-quot N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 61.4

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

4. Applied rewrites 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. tan-quot N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. remove-double-div N/A

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

   11. lift-tan.f64 N/A

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

   12. tan-quot N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. frac-sub N/A

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

6. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. cos-mult N/A

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

   9. associate-/r/ N/A

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

8. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. cos-sum N/A

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

   6. count-2 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. count-2 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. sub-neg N/A

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

   17. associate-+r+ N/A

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. tan-quot N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 61.4

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

4. Applied rewrites 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. tan-quot N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. remove-double-div N/A

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

   11. lift-tan.f64 N/A

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

   12. tan-quot N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. frac-sub N/A

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 0.6× speedup

Math

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

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(Float64(sin(eps) / 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[(N[Sin[eps], $MachinePrecision] / N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. tan-quot N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 61.4

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

4. Applied rewrites 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. tan-quot N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. remove-double-div N/A

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

   11. lift-tan.f64 N/A

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

   12. tan-quot N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. frac-sub N/A

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 5: 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 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around inf

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Final simplification 99.9%

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

Alternative 6: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.6

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 99.6

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

9. Applied rewrites 99.6%

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

Alternative 7: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 8: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.4%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. div-inv N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. lift-+.f64 N/A

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

    8. +-commutative N/A

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

    9. lift-+.f64 N/A

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

    10. lower-/.f64 99.4

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

12. Applied rewrites 99.4%

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

Alternative 9: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 62.1

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

4. Applied rewrites 62.1%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 10: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. tan-quot N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 61.4

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

4. Applied rewrites 61.4%

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

6. Applied rewrites 99.9%

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

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 99.0

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

9. Applied rewrites 99.0%

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

Alternative 11: 98.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.6%

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

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 98.4%

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

9. Final simplification 98.4%

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

Alternative 12: 98.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.4%

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

12. Final simplification 98.4%

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

Alternative 13: 97.9% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.1%

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

Alternative 14: 6.2% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (0.3333333333333333d0 * eps) * (eps * eps)
end function

Java

public static double code(double x, double eps) {
	return (0.3333333333333333 * eps) * (eps * eps);
}

Python

def code(x, eps):
	return (0.3333333333333333 * eps) * (eps * eps)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (0.3333333333333333 * eps) * (eps * eps);
end

Wolfram

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

Derivation

1. Initial program 62.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.1%

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

9. Taylor expanded in eps around inf

   \[\leadsto \frac{1}{3} \cdot {\varepsilon}^{\color{blue}{3}} \]
10. Step-by-step derivation

11. Applied rewrites 6.2%

    \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot 0.3333333333333333 \]
12. Step-by-step derivation

13. Applied rewrites 6.2%

    \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.3333333333333333 \cdot \varepsilon\right) \]

14. Final simplification 6.2%

    \[\leadsto \left(0.3333333333333333 \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]
15. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 63.0% 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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024240 
(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)))