2cos (problem 3.3.5)

Percentage Accurate: 52.1% → 99.8%

Time: 15.7s

Alternatives: 13

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: 52.1% 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: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.001388888888888889, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5\right) \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (sin x)
  (- (sin eps))
  (*
   (*
    (fma
     (fma (* eps eps) -0.001388888888888889 0.041666666666666664)
     (* eps eps)
     -0.5)
    (cos x))
   (* eps eps))))

C

double code(double x, double eps) {
	return fma(sin(x), -sin(eps), ((fma(fma((eps * eps), -0.001388888888888889, 0.041666666666666664), (eps * eps), -0.5) * cos(x)) * (eps * eps)));
}

Julia

function code(x, eps)
	return fma(sin(x), Float64(-sin(eps)), Float64(Float64(fma(fma(Float64(eps * eps), -0.001388888888888889, 0.041666666666666664), Float64(eps * eps), -0.5) * cos(x)) * Float64(eps * eps)))
end

Wolfram

code[x_, eps_] := N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. cos-sum N/A

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

   7. associate-+r- N/A

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

   8. lower--.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-sin.f64 80.4

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

4. Applied rewrites 80.4%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\left(\cos x + \left(-1 \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{24} \cdot \cos x\right)\right)\right)\right)} - \sin \varepsilon \cdot \sin x \]
6. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(\cos x + -1 \cdot \cos x\right) + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{24} \cdot \cos x\right)\right)\right)} - \sin \varepsilon \cdot \sin x \]

   2. distribute-rgt1-in N/A

      \[\leadsto \left(\color{blue}{\left(-1 + 1\right) \cdot \cos x} + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{24} \cdot \cos x\right)\right)\right) - \sin \varepsilon \cdot \sin x \]

   3. metadata-eval N/A

      \[\leadsto \left(\color{blue}{0} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{24} \cdot \cos x\right)\right)\right) - \sin \varepsilon \cdot \sin x \]

   4. mul0-lft N/A

      \[\leadsto \left(\color{blue}{0} + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{24} \cdot \cos x\right)\right)\right) - \sin \varepsilon \cdot \sin x \]

   5. +-lft-identity N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{24} \cdot \cos x\right)\right)} - \sin \varepsilon \cdot \sin x \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{24} \cdot \cos x\right)\right) \cdot {\varepsilon}^{2}} - \sin \varepsilon \cdot \sin x \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{24} \cdot \cos x\right)\right) \cdot {\varepsilon}^{2}} - \sin \varepsilon \cdot \sin x \]

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(-0.001388888888888889, \varepsilon \cdot \varepsilon, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5 \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} - \sin \varepsilon \cdot \sin x \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(\frac{-1}{720}, \varepsilon \cdot \varepsilon, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \sin \varepsilon \cdot \sin x} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(\frac{-1}{720}, \varepsilon \cdot \varepsilon, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\sin \varepsilon \cdot \sin x\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin \varepsilon \cdot \sin x\right)\right) + \mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(\frac{-1}{720}, \varepsilon \cdot \varepsilon, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sin \varepsilon \cdot \sin x}\right)\right) + \mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(\frac{-1}{720}, \varepsilon \cdot \varepsilon, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   5. lift-sin.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\sin \varepsilon \cdot \color{blue}{\sin x}\right)\right) + \mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(\frac{-1}{720}, \varepsilon \cdot \varepsilon, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sin \varepsilon}\right)\right) + \mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(\frac{-1}{720}, \varepsilon \cdot \varepsilon, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sin \varepsilon\right)\right)} + \mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(\frac{-1}{720}, \varepsilon \cdot \varepsilon, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, \mathsf{neg}\left(\sin \varepsilon\right), \mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(\frac{-1}{720}, \varepsilon \cdot \varepsilon, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   9. lift-sin.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin x}, \mathsf{neg}\left(\sin \varepsilon\right), \mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(\frac{-1}{720}, \varepsilon \cdot \varepsilon, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

   10. lower-neg.f64 100.0

       \[\leadsto \mathsf{fma}\left(\sin x, \color{blue}{-\sin \varepsilon}, \mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(-0.001388888888888889, \varepsilon \cdot \varepsilon, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5 \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

9. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.001388888888888889, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

10. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.001388888888888889, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5\right) \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]
11. Add Preprocessing

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.041666666666666664, -0.5\right) \cdot \left(\left(\cos x \cdot \varepsilon\right) \cdot \varepsilon\right) - \sin \varepsilon \cdot \sin x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (-
  (* (fma (* eps eps) 0.041666666666666664 -0.5) (* (* (cos x) eps) eps))
  (* (sin eps) (sin x))))

C

double code(double x, double eps) {
	return (fma((eps * eps), 0.041666666666666664, -0.5) * ((cos(x) * eps) * eps)) - (sin(eps) * sin(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(eps * eps), 0.041666666666666664, -0.5) * Float64(Float64(cos(x) * eps) * eps)) - Float64(sin(eps) * sin(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(eps * eps), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. cos-sum N/A

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

   7. associate-+r- N/A

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

   8. lower--.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-sin.f64 80.4

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

4. Applied rewrites 80.4%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\left(\cos x + \left(-1 \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{24} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right)\right)} - \sin \varepsilon \cdot \sin x \]
6. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(\cos x + -1 \cdot \cos x\right) + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{24} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right)} - \sin \varepsilon \cdot \sin x \]

   2. distribute-rgt1-in N/A

      \[\leadsto \left(\color{blue}{\left(-1 + 1\right) \cdot \cos x} + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{24} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right) - \sin \varepsilon \cdot \sin x \]

   3. metadata-eval N/A

      \[\leadsto \left(\color{blue}{0} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{24} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right) - \sin \varepsilon \cdot \sin x \]

   4. mul0-lft N/A

      \[\leadsto \left(\color{blue}{0} + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{24} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right) - \sin \varepsilon \cdot \sin x \]

   5. +-lft-identity N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{24} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} - \sin \varepsilon \cdot \sin x \]

   6. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \cos x\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right)} - \sin \varepsilon \cdot \sin x \]

   7. *-commutative N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \color{blue}{\left(\cos x \cdot \frac{-1}{2}\right)} + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right) - \sin \varepsilon \cdot \sin x \]

   8. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left({\varepsilon}^{2} \cdot \cos x\right) \cdot \frac{-1}{2}} + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right) - \sin \varepsilon \cdot \sin x \]

   9. *-commutative N/A

      \[\leadsto \left(\color{blue}{\frac{-1}{2} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)} + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right) - \sin \varepsilon \cdot \sin x \]

   10. associate-*r* N/A

       \[\leadsto \left(\frac{-1}{2} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \color{blue}{\left({\varepsilon}^{2} \cdot \frac{1}{24}\right) \cdot \left({\varepsilon}^{2} \cdot \cos x\right)}\right) - \sin \varepsilon \cdot \sin x \]

   11. *-commutative N/A

       \[\leadsto \left(\frac{-1}{2} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{2}\right)} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) - \sin \varepsilon \cdot \sin x \]

   12. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\left({\varepsilon}^{2} \cdot \cos x\right) \cdot \left(\frac{-1}{2} + \frac{1}{24} \cdot {\varepsilon}^{2}\right)} - \sin \varepsilon \cdot \sin x \]

   13. +-commutative N/A

       \[\leadsto \left({\varepsilon}^{2} \cdot \cos x\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{2} + \frac{-1}{2}\right)} - \sin \varepsilon \cdot \sin x \]

   14. metadata-eval N/A

       \[\leadsto \left({\varepsilon}^{2} \cdot \cos x\right) \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) - \sin \varepsilon \cdot \sin x \]

   15. sub-neg N/A

       \[\leadsto \left({\varepsilon}^{2} \cdot \cos x\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)} - \sin \varepsilon \cdot \sin x \]

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\left(\cos x \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.041666666666666664, -0.5\right)} - \sin \varepsilon \cdot \sin x \]

8. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.041666666666666664, -0.5\right) \cdot \left(\left(\cos x \cdot \varepsilon\right) \cdot \varepsilon\right) - \sin \varepsilon \cdot \sin x \]
9. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.5%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666, \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (*
    (fma
     (fma 0.00026041666666666666 (* eps eps) -0.020833333333333332)
     (* eps eps)
     0.5)
    eps)
   (sin (* (fma 2.0 x eps) 0.5)))
  -2.0))

C

double code(double x, double eps) {
	return ((fma(fma(0.00026041666666666666, (eps * eps), -0.020833333333333332), (eps * eps), 0.5) * eps) * sin((fma(2.0, x, eps) * 0.5))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(fma(0.00026041666666666666, Float64(eps * eps), -0.020833333333333332), Float64(eps * eps), 0.5) * eps) * sin(Float64(fma(2.0, x, eps) * 0.5))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.00026041666666666666 * N[(eps * eps), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.5), $MachinePrecision] * eps), $MachinePrecision] * N[Sin[N[(N[(2.0 * x + eps), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 50.5%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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

   \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)}\right) \cdot -2 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   3. +-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) + \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

   4. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) \cdot {\varepsilon}^{2}} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

   6. sub-neg N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{3840} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   7. metadata-eval N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \color{blue}{\frac{-1}{48}}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3840}, {\varepsilon}^{2}, \frac{-1}{48}\right)}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   9. unpow2 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3840}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{48}\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3840}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{48}\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   11. unpow2 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3840}, \varepsilon \cdot \varepsilon, \frac{-1}{48}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   12. lower-*.f64 99.7

       \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666, \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \color{blue}{\varepsilon \cdot \varepsilon}, 0.5\right) \cdot \varepsilon\right)\right) \cdot -2 \]

7. Applied rewrites 99.7%

   \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666, \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

8. Final simplification 99.7%

   \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666, \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right)\right) \cdot -2 \]
9. Add Preprocessing

Alternative 5: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 6: 99.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.5%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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

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

   1. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 7: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(-0.5 \cdot \varepsilon - \sin x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (- (* -0.5 eps) (sin x)) eps))

C

double code(double x, double eps) {
	return ((-0.5 * eps) - sin(x)) * eps;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((-0.5d0) * eps) - sin(x)) * eps
end function

Java

public static double code(double x, double eps) {
	return ((-0.5 * eps) - Math.sin(x)) * eps;
}

Python

def code(x, eps):
	return ((-0.5 * eps) - math.sin(x)) * eps

Julia

function code(x, eps)
	return Float64(Float64(Float64(-0.5 * eps) - sin(x)) * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = ((-0.5 * eps) - sin(x)) * eps;
end

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 99.4

       \[\leadsto \left(\left(\cos x \cdot -0.5\right) \cdot \varepsilon - \color{blue}{\sin x}\right) \cdot \varepsilon \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(\left(\cos x \cdot -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(\frac{-1}{2} \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

   \[\leadsto \left(-0.5 \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 8: 97.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5 \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.25 \cdot \varepsilon\right) \cdot \varepsilon, x, -\varepsilon\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (* -0.5 eps)
  eps
  (* (fma (* (fma 0.16666666666666666 x (* 0.25 eps)) eps) x (- eps)) x)))

C

double code(double x, double eps) {
	return fma((-0.5 * eps), eps, (fma((fma(0.16666666666666666, x, (0.25 * eps)) * eps), x, -eps) * x));
}

Julia

function code(x, eps)
	return fma(Float64(-0.5 * eps), eps, Float64(fma(Float64(fma(0.16666666666666666, x, Float64(0.25 * eps)) * eps), x, Float64(-eps)) * x))
end

Wolfram

code[x_, eps_] := N[(N[(-0.5 * eps), $MachinePrecision] * eps + N[(N[(N[(N[(0.16666666666666666 * x + N[(0.25 * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x + (-eps)), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 99.4

       \[\leadsto \left(\left(\cos x \cdot -0.5\right) \cdot \varepsilon - \color{blue}{\sin x}\right) \cdot \varepsilon \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(\left(\cos x \cdot -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{2} \cdot {\varepsilon}^{2} + \color{blue}{x \cdot \left(-1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 98.6%

   \[\leadsto \mathsf{fma}\left(-0.5 \cdot \varepsilon, \color{blue}{\varepsilon}, \mathsf{fma}\left(\varepsilon \cdot \mathsf{fma}\left(0.16666666666666666, x, 0.25 \cdot \varepsilon\right), x, -\varepsilon\right) \cdot x\right) \]

9. Final simplification 98.6%

   \[\leadsto \mathsf{fma}\left(-0.5 \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.25 \cdot \varepsilon\right) \cdot \varepsilon, x, -\varepsilon\right) \cdot x\right) \]
10. Add Preprocessing

Alternative 9: 97.9% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, x, -1\right), x, -0.5 \cdot \varepsilon\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (fma (fma (* 0.16666666666666666 x) x -1.0) x (* -0.5 eps)) eps))

C

double code(double x, double eps) {
	return fma(fma((0.16666666666666666 * x), x, -1.0), x, (-0.5 * eps)) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(Float64(0.16666666666666666 * x), x, -1.0), x, Float64(-0.5 * eps)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + N[(-0.5 * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 99.4

       \[\leadsto \left(\left(\cos x \cdot -0.5\right) \cdot \varepsilon - \color{blue}{\sin x}\right) \cdot \varepsilon \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(\left(\cos x \cdot -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.25 \cdot \varepsilon\right), x, -1\right), x, -0.5 \cdot \varepsilon\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot x, x, -1\right), x, \frac{-1}{2} \cdot \varepsilon\right) \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, x, -1\right), x, -0.5 \cdot \varepsilon\right) \cdot \varepsilon \]
12. Add Preprocessing

Alternative 10: 97.6% accurate, 10.9× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (fma (- x) eps (* -0.5 (* eps eps))))

C

double code(double x, double eps) {
	return fma(-x, eps, (-0.5 * (eps * eps)));
}

Julia

function code(x, eps)
	return fma(Float64(-x), eps, Float64(-0.5 * Float64(eps * eps)))
end

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 99.4

       \[\leadsto \left(\left(\cos x \cdot -0.5\right) \cdot \varepsilon - \color{blue}{\sin x}\right) \cdot \varepsilon \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(\left(\cos x \cdot -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.25 \cdot \varepsilon\right), x, -1\right), x, -0.5 \cdot \varepsilon\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.0%

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

12. Final simplification 98.0%

    \[\leadsto \mathsf{fma}\left(-x, \varepsilon, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]
13. Add Preprocessing

Alternative 11: 97.4% accurate, 14.8× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* (fma -0.5 eps (- x)) eps))

C

double code(double x, double eps) {
	return fma(-0.5, eps, -x) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(-0.5, eps, Float64(-x)) * eps)
end

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 99.4

       \[\leadsto \left(\left(\cos x \cdot -0.5\right) \cdot \varepsilon - \color{blue}{\sin x}\right) \cdot \varepsilon \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(\left(\cos x \cdot -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.9%

   \[\leadsto \mathsf{fma}\left(-0.5, \varepsilon, -x\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 12: 78.2% accurate, 25.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 80.0

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

5. Applied rewrites 80.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 79.0%

   \[\leadsto \left(-x\right) \cdot \color{blue}{\varepsilon} \]
9. Add Preprocessing

Alternative 13: 50.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 50.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. cos-sum N/A

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

   7. associate-+r- N/A

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

   8. lower--.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-sin.f64 80.4

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

4. Applied rewrites 80.4%

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

5. Taylor expanded in eps around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft 49.7

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

7. Applied rewrites 49.7%

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

Developer Target 1: 98.8% 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 2024275 
(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 (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

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