2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.8%

Time: 14.2s

Alternatives: 10

Speedup: 34.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.9%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 64.9

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

4. Applied rewrites 64.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-sin.f64 N/A

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

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

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

   6. lift-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-sin.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.9%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 64.9

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

4. Applied rewrites 64.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.9%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 64.9

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

4. Applied rewrites 64.9%

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

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. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 64.9

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

4. Applied rewrites 64.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

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 99.1%

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

11. Final simplification 99.1%

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

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (fma (pow (tan x) 2.0) eps eps))

C

double code(double x, double eps) {
	return fma(pow(tan(x), 2.0), eps, eps);
}

Julia

function code(x, eps)
	return fma((tan(x) ^ 2.0), eps, eps)
end

Wolfram

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

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.4

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 98.4%

   \[\leadsto \mathsf{fma}\left({\tan x}^{2}, \color{blue}{\varepsilon}, \varepsilon\right) \]
8. Add Preprocessing

Alternative 6: 98.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot \varepsilon\right), x, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma
    (fma (* eps eps) 1.3333333333333333 1.0)
    x
    (* (fma (* eps eps) 0.6666666666666666 1.0) eps))
   x
   (* 0.3333333333333333 (* eps eps)))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(fma((eps * eps), 1.3333333333333333, 1.0), x, (fma((eps * eps), 0.6666666666666666, 1.0) * eps)), x, (0.3333333333333333 * (eps * eps))), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(fma(Float64(eps * eps), 1.3333333333333333, 1.0), x, Float64(fma(Float64(eps * eps), 0.6666666666666666, 1.0) * eps)), x, Float64(0.3333333333333333 * Float64(eps * eps))), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * 1.3333333333333333 + 1.0), $MachinePrecision] * x + N[(N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666 + 1.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * x + N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \]

9. Final simplification 98.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot \varepsilon\right), x, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \varepsilon\right) \]
10. Add Preprocessing

Alternative 7: 98.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot \varepsilon\right), x, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma 1.0 x (* (fma (* eps eps) 0.6666666666666666 1.0) eps))
   x
   (* 0.3333333333333333 (* eps eps)))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(1.0, x, (fma((eps * eps), 0.6666666666666666, 1.0) * eps)), x, (0.3333333333333333 * (eps * eps))), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(1.0, x, Float64(fma(Float64(eps * eps), 0.6666666666666666, 1.0) * eps)), x, Float64(0.3333333333333333 * Float64(eps * eps))), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(1.0 * x + N[(N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666 + 1.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * x + N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \varepsilon, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 98.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.8888888888888888, 1.3333333333333333\right) \cdot x, \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \]

12. Final simplification 98.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot \varepsilon\right), x, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 8: 98.3% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.9%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 64.9

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

4. Applied rewrites 64.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{{\cos x}^{2}} + \frac{\varepsilon \cdot \sin x}{{\cos x}^{3}}\right) \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{{\cos x}^{2}} + \frac{\varepsilon \cdot \sin x}{{\cos x}^{3}}\right) \cdot \varepsilon} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{\varepsilon \cdot \sin x}{{\cos x}^{3}} + \frac{1}{{\cos x}^{2}}\right)} \cdot \varepsilon \]

   4. associate-/l* N/A

      \[\leadsto \left(\color{blue}{\varepsilon \cdot \frac{\sin x}{{\cos x}^{3}}} + \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\cos x}^{3}}, \frac{1}{{\cos x}^{2}}\right)} \cdot \varepsilon \]

   6. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{\sin x}{{\cos x}^{3}}}, \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   7. lower-sin.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\color{blue}{\sin x}}{{\cos x}^{3}}, \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\sin x}{\color{blue}{{\cos x}^{3}}}, \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   9. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\color{blue}{\cos x}}^{3}}, \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\cos x}^{3}}, \color{blue}{\frac{1}{{\cos x}^{2}}}\right) \cdot \varepsilon \]

   11. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\cos x}^{3}}, \frac{1}{\color{blue}{{\cos x}^{2}}}\right) \cdot \varepsilon \]

   12. lower-cos.f64 98.9

       \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\cos x}^{3}}, \frac{1}{{\color{blue}{\cos x}}^{2}}\right) \cdot \varepsilon \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\cos x}^{3}}, \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.0%

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

11. Final simplification 98.0%

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

Alternative 9: 98.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.0%

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

Alternative 10: 97.9% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return 1.0 * eps;
}

Python

def code(x, eps):
	return 1.0 * eps

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 64.9

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

4. Applied rewrites 64.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{{\cos x}^{2}} + \frac{\varepsilon \cdot \sin x}{{\cos x}^{3}}\right) \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{{\cos x}^{2}} + \frac{\varepsilon \cdot \sin x}{{\cos x}^{3}}\right) \cdot \varepsilon} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{\varepsilon \cdot \sin x}{{\cos x}^{3}} + \frac{1}{{\cos x}^{2}}\right)} \cdot \varepsilon \]

   4. associate-/l* N/A

      \[\leadsto \left(\color{blue}{\varepsilon \cdot \frac{\sin x}{{\cos x}^{3}}} + \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\cos x}^{3}}, \frac{1}{{\cos x}^{2}}\right)} \cdot \varepsilon \]

   6. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{\sin x}{{\cos x}^{3}}}, \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   7. lower-sin.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\color{blue}{\sin x}}{{\cos x}^{3}}, \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\sin x}{\color{blue}{{\cos x}^{3}}}, \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   9. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\color{blue}{\cos x}}^{3}}, \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\cos x}^{3}}, \color{blue}{\frac{1}{{\cos x}^{2}}}\right) \cdot \varepsilon \]

   11. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\cos x}^{3}}, \frac{1}{\color{blue}{{\cos x}^{2}}}\right) \cdot \varepsilon \]

   12. lower-cos.f64 98.9

       \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\cos x}^{3}}, \frac{1}{{\color{blue}{\cos x}}^{2}}\right) \cdot \varepsilon \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{\sin x}{{\cos x}^{3}}, \frac{1}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.7%

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

Developer Target 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024283 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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