2cos (problem 3.3.5)

Percentage Accurate: 52.4% → 99.7%

Time: 16.4s

Alternatives: 11

Speedup: 25.9×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\left(-\varepsilon\right) \cdot \varepsilon}{-1}, \frac{0.5}{\varepsilon}, x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  -2.0
  (* (sin (* 0.5 eps)) (sin (fma (/ (* (- eps) eps) -1.0) (/ 0.5 eps) x)))))

C

double code(double x, double eps) {
	return -2.0 * (sin((0.5 * eps)) * sin(fma(((-eps * eps) / -1.0), (0.5 / eps), x)));
}

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(sin(Float64(0.5 * eps)) * sin(fma(Float64(Float64(Float64(-eps) * eps) / -1.0), Float64(0.5 / eps), x))))
end

Wolfram

code[x_, eps_] := N[(-2.0 * N[(N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(N[((-eps) * eps), $MachinePrecision] / -1.0), $MachinePrecision] * N[(0.5 / eps), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. +-lft-identity N/A

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

   6. flip-+ N/A

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

   7. neg-sub0 N/A

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

   8. lift-neg.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lift-neg.f64 N/A

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

   11. neg-mul-1 N/A

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

   12. metadata-eval N/A

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

   13. times-frac N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub0-neg N/A

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

   18. lower-neg.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. metadata-eval N/A

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

   21. lower-/.f64 N/A

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

   22. *-commutative N/A

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

   23. associate-*r* N/A

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

   24. metadata-eval N/A

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

   25. lower-*.f64 99.8

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{-\varepsilon \cdot \varepsilon}{-1}, \frac{0.5}{\varepsilon}, \color{blue}{1 \cdot x}\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

6. Applied rewrites 99.8%

   \[\leadsto \left(\sin \color{blue}{\left(\mathsf{fma}\left(\frac{-\varepsilon \cdot \varepsilon}{-1}, \frac{0.5}{\varepsilon}, 1 \cdot x\right)\right)} \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

7. Final simplification 99.8%

   \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\left(-\varepsilon\right) \cdot \varepsilon}{-1}, \frac{0.5}{\varepsilon}, x\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(-0.020833333333333332 \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(\frac{0.5}{\varepsilon}, \varepsilon \cdot \varepsilon, x\right)\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (* (fma (* -0.020833333333333332 eps) eps 0.5) eps)
   (sin (fma (/ 0.5 eps) (* eps eps) x)))
  -2.0))

C

double code(double x, double eps) {
	return ((fma((-0.020833333333333332 * eps), eps, 0.5) * eps) * sin(fma((0.5 / eps), (eps * eps), x))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(-0.020833333333333332 * eps), eps, 0.5) * eps) * sin(fma(Float64(0.5 / eps), Float64(eps * eps), x))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(-0.020833333333333332 * eps), $MachinePrecision] * eps + 0.5), $MachinePrecision] * eps), $MachinePrecision] * N[Sin[N[(N[(0.5 / eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]

5. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}\right) \cdot -2 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   3. +-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{48} \cdot {\varepsilon}^{2} + \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

   4. unpow2 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\left(\frac{-1}{48} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   5. associate-*r* N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{-1}{48} \cdot \varepsilon\right) \cdot \varepsilon} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

   7. lower-*.f64 99.5

      \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{-0.020833333333333332 \cdot \varepsilon}, \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot -2 \]

7. Applied rewrites 99.5%

   \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.020833333333333332 \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot -2 \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   2. lift-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   3. +-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\varepsilon + 2 \cdot x\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   4. distribute-lft-in N/A

      \[\leadsto \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + \frac{1}{2} \cdot \left(2 \cdot x\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   5. *-commutative N/A

      \[\leadsto \left(\sin \left(\color{blue}{\varepsilon \cdot \frac{1}{2}} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   6. +-lft-identity N/A

      \[\leadsto \left(\sin \left(\color{blue}{\left(0 + \varepsilon\right)} \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   7. flip-+ N/A

      \[\leadsto \left(\sin \left(\color{blue}{\frac{0 \cdot 0 - \varepsilon \cdot \varepsilon}{0 - \varepsilon}} \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   8. metadata-eval N/A

      \[\leadsto \left(\sin \left(\frac{\color{blue}{0} - \varepsilon \cdot \varepsilon}{0 - \varepsilon} \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{0 - \color{blue}{\varepsilon \cdot \varepsilon}}{0 - \varepsilon} \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   10. neg-sub0 N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)}}{0 - \varepsilon} \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   11. lift-neg.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)}}{0 - \varepsilon} \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   12. neg-sub0 N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)}{\color{blue}{\mathsf{neg}\left(\varepsilon\right)}} \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   13. lift-neg.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)}{\color{blue}{\mathsf{neg}\left(\varepsilon\right)}} \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   14. associate-*l/ N/A

       \[\leadsto \left(\sin \left(\color{blue}{\frac{\left(\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{1}{2}}{\mathsf{neg}\left(\varepsilon\right)}} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   15. lift-neg.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\left(\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{1}{2}}{\color{blue}{\mathsf{neg}\left(\varepsilon\right)}} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   16. neg-mul-1 N/A

       \[\leadsto \left(\sin \left(\frac{\left(\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{1}{2}}{\color{blue}{-1 \cdot \varepsilon}} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   17. frac-times N/A

       \[\leadsto \left(\sin \left(\color{blue}{\frac{\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)}{-1} \cdot \frac{\frac{1}{2}}{\varepsilon}} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   18. lift-/.f64 N/A

       \[\leadsto \left(\sin \left(\color{blue}{\frac{\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)}{-1}} \cdot \frac{\frac{1}{2}}{\varepsilon} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   19. lift-/.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)}{-1} \cdot \color{blue}{\frac{\frac{1}{2}}{\varepsilon}} + \frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   20. associate-*r* N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)}{-1} \cdot \frac{\frac{1}{2}}{\varepsilon} + \color{blue}{\left(\frac{1}{2} \cdot 2\right) \cdot x}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   21. metadata-eval N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)}{-1} \cdot \frac{\frac{1}{2}}{\varepsilon} + \color{blue}{1} \cdot x\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   22. lift-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)}{-1} \cdot \frac{\frac{1}{2}}{\varepsilon} + \color{blue}{1 \cdot x}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   23. lift-fma.f64 99.5

       \[\leadsto \left(\sin \color{blue}{\left(\mathsf{fma}\left(\frac{-\varepsilon \cdot \varepsilon}{-1}, \frac{0.5}{\varepsilon}, 1 \cdot x\right)\right)} \cdot \left(\mathsf{fma}\left(-0.020833333333333332 \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot -2 \]

9. Applied rewrites 99.5%

   \[\leadsto \left(\sin \color{blue}{\left(\mathsf{fma}\left(\frac{0.5}{\varepsilon}, \varepsilon \cdot \varepsilon, x\right)\right)} \cdot \left(\mathsf{fma}\left(-0.020833333333333332 \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot -2 \]

10. Final simplification 99.5%

    \[\leadsto \left(\left(\mathsf{fma}\left(-0.020833333333333332 \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(\frac{0.5}{\varepsilon}, \varepsilon \cdot \varepsilon, x\right)\right)\right) \cdot -2 \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (pow (cbrt (* (* -2.0 (sin (* 0.5 (fma 2.0 x eps)))) (sin (* 0.5 eps)))) 3.0))

C

double code(double x, double eps) {
	return pow(cbrt(((-2.0 * sin((0.5 * fma(2.0, x, eps)))) * sin((0.5 * eps)))), 3.0);
}

Julia

function code(x, eps)
	return cbrt(Float64(Float64(-2.0 * sin(Float64(0.5 * fma(2.0, x, eps)))) * sin(Float64(0.5 * eps)))) ^ 3.0
end

Wolfram

code[x_, eps_] := N[Power[N[Power[N[(N[(-2.0 * N[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Reproduce

herbie shell --seed 2024229 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (* -2 (sin (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform default (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

  (- (cos (+ x eps)) (cos x)))