2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.7%

Time: 15.8s

Alternatives: 11

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: 62.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right)\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \left(-x\right) \cdot \sin \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (fma
   eps
   (* eps (* eps (fma (* eps eps) 0.008333333333333333 -0.16666666666666666)))
   eps)
  (* (cos x) (fma (cos x) (cos eps) (* (sin (- x)) (sin eps))))))

C

double code(double x, double eps) {
	return fma(eps, (eps * (eps * fma((eps * eps), 0.008333333333333333, -0.16666666666666666))), eps) / (cos(x) * fma(cos(x), cos(eps), (sin(-x) * sin(eps))));
}

Julia

function code(x, eps)
	return Float64(fma(eps, Float64(eps * Float64(eps * fma(Float64(eps * eps), 0.008333333333333333, -0.16666666666666666))), eps) / Float64(cos(x) * fma(cos(x), cos(eps), Float64(sin(Float64(-x)) * sin(eps)))))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(eps * N[(eps * N[(N[(eps * eps), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[(-x)], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.1%

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

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) + 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) + \varepsilon \cdot 1}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\varepsilon}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right), \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{120} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \left({\varepsilon}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right)\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   14. lower-*.f64 99.9

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, 0.008333333333333333, -0.16666666666666666\right)\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

7. Applied rewrites 99.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right)\right), \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \cos \color{blue}{\left(x + \varepsilon\right)}} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \color{blue}{\cos \left(x + \varepsilon\right)}} \]

   3. cos-sum N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}} \]

   4. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)}} \]

   5. lift-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \left(\color{blue}{\cos x} \cdot \cos \varepsilon + \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)}} \]

   7. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\cos x, \color{blue}{\cos \varepsilon}, \left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \sin \varepsilon}\right)} \]

   9. sin-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \color{blue}{\sin \left(\mathsf{neg}\left(x\right)\right)} \cdot \sin \varepsilon\right)} \]

   10. lower-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \color{blue}{\sin \left(\mathsf{neg}\left(x\right)\right)} \cdot \sin \varepsilon\right)} \]

   11. lower-neg.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \sin \varepsilon\right)} \]

   12. lower-sin.f64 100.0

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right)\right), \varepsilon\right)}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \left(-x\right) \cdot \color{blue}{\sin \varepsilon}\right)} \]

9. Applied rewrites 100.0%

   \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right)\right), \varepsilon\right)}{\cos x \cdot \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \left(-x\right) \cdot \sin \varepsilon\right)}} \]
10. Add Preprocessing

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   eps
   (fma
    (* eps eps)
    (fma eps (* eps 0.008333333333333333) -0.16666666666666666)
    1.0))
  (* (cos x) (cos (+ x eps)))))

C

double code(double x, double eps) {
	return (eps * fma((eps * eps), fma(eps, (eps * 0.008333333333333333), -0.16666666666666666), 1.0)) / (cos(x) * cos((x + eps)));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * fma(Float64(eps * eps), fma(eps, Float64(eps * 0.008333333333333333), -0.16666666666666666), 1.0)) / Float64(cos(x) * cos(Float64(x + eps))))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.6%

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

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 62.6

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

4. Applied rewrites 62.6%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) + 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) + \varepsilon \cdot 1}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\varepsilon}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right), \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}, \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{120} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \left({\varepsilon}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right)\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   14. lower-*.f64 99.7

       \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, 0.008333333333333333, -0.16666666666666666\right)\right), \varepsilon\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right)\right), \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.7%

   \[\leadsto \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \color{blue}{\varepsilon}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

10. Final simplification 99.7%

    \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 62.8% 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 2024223 
(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)))