2cos (problem 3.3.5)

Percentage Accurate: 52.5% → 99.7%

Time: 17.9s

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: 52.5% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -1.5500992063492063 \cdot 10^{-6}, 0.00026041666666666666\right), -0.020833333333333332\right), 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  -2.0
  (*
   (sin (fma 0.5 eps x))
   (*
    eps
    (fma
     (* eps eps)
     (fma
      (* eps eps)
      (fma (* eps eps) -1.5500992063492063e-6 0.00026041666666666666)
      -0.020833333333333332)
     0.5)))))

C

double code(double x, double eps) {
	return -2.0 * (sin(fma(0.5, eps, x)) * (eps * fma((eps * eps), fma((eps * eps), fma((eps * eps), -1.5500992063492063e-6, 0.00026041666666666666), -0.020833333333333332), 0.5)));
}

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(sin(fma(0.5, eps, x)) * Float64(eps * fma(Float64(eps * eps), fma(Float64(eps * eps), fma(Float64(eps * eps), -1.5500992063492063e-6, 0.00026041666666666666), -0.020833333333333332), 0.5))))
end

Wolfram

code[x_, eps_] := N[(-2.0 * N[(N[Sin[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision] * N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * -1.5500992063492063e-6 + 0.00026041666666666666), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

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

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

   \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)}\right) \cdot -2 \]
9. Step-by-step derivation

10. Applied rewrites 99.5%

    \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -1.5500992063492063 \cdot 10^{-6}, 0.00026041666666666666\right), -0.020833333333333332\right), 0.5\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)}\right) \cdot -2 \]

11. Final simplification 99.5%

    \[\leadsto -2 \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -1.5500992063492063 \cdot 10^{-6}, 0.00026041666666666666\right), -0.020833333333333332\right), 0.5\right)\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.5%

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

11. Final simplification 99.5%

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

Alternative 4: 99.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 5: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.2%

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

11. Final simplification 99.2%

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

Alternative 6: 98.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\cos \varepsilon + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + \color{blue}{-1} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   5. lower-cos.f64 49.7

      \[\leadsto -1 + \color{blue}{\cos \varepsilon} \]

5. Applied rewrites 49.7%

   \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

6. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 99.2

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.8%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 - \sin \color{blue}{x}\right) \]
12. Add Preprocessing

Alternative 7: 98.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\cos \varepsilon + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + \color{blue}{-1} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   5. lower-cos.f64 49.7

      \[\leadsto -1 + \color{blue}{\cos \varepsilon} \]

5. Applied rewrites 49.7%

   \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

6. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 99.2

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.2%

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

Alternative 8: 98.1% accurate, 6.1× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma eps -0.5 (* x (fma x (fma x 0.16666666666666666 (* eps 0.25)) -1.0)))))

C

double code(double x, double eps) {
	return eps * fma(eps, -0.5, (x * fma(x, fma(x, 0.16666666666666666, (eps * 0.25)), -1.0)));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, -0.5, Float64(x * fma(x, fma(x, 0.16666666666666666, Float64(eps * 0.25)), -1.0))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * -0.5 + N[(x * N[(x * N[(x * 0.16666666666666666 + N[(eps * 0.25), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\cos \varepsilon + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + \color{blue}{-1} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   5. lower-cos.f64 49.7

      \[\leadsto -1 + \color{blue}{\cos \varepsilon} \]

5. Applied rewrites 49.7%

   \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

6. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 99.2

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.2%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, \varepsilon \cdot 0.25\right), -1\right)}, \varepsilon \cdot -0.5\right) \]
12. Step-by-step derivation

13. Applied rewrites 98.2%

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

Alternative 9: 98.1% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\cos \varepsilon + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + \color{blue}{-1} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   5. lower-cos.f64 49.7

      \[\leadsto -1 + \color{blue}{\cos \varepsilon} \]

5. Applied rewrites 49.7%

   \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

6. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 99.2

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.2%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 98.0%

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

Alternative 10: 97.9% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\cos \varepsilon + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + \color{blue}{-1} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   5. lower-cos.f64 49.7

      \[\leadsto -1 + \color{blue}{\cos \varepsilon} \]

5. Applied rewrites 49.7%

   \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

6. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 99.2

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.2%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 97.6%

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

Alternative 11: 97.7% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. associate--l- N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 50.4

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

5. Applied rewrites 50.4%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 97.3%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot -0.5 - x\right)} \]
9. Add Preprocessing

Alternative 12: 78.9% accurate, 25.9× speedup

Math

\[\begin{array}{l} \\ -\varepsilon \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 78.5

      \[\leadsto -\varepsilon \cdot \color{blue}{\sin x} \]

5. Applied rewrites 78.5%

   \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.6%

   \[\leadsto -x \cdot \varepsilon \]

9. Final simplification 77.6%

   \[\leadsto -\varepsilon \cdot x \]
10. Add Preprocessing

Alternative 13: 51.1% accurate, 51.8× speedup

Math

\[\begin{array}{l} \\ -1 + 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ -1.0 1.0))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -1.0 + 1.0

Julia

function code(x, eps)
	return Float64(-1.0 + 1.0)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(-1.0 + 1.0), $MachinePrecision]

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\cos \varepsilon + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + \color{blue}{-1} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

   5. lower-cos.f64 49.7

      \[\leadsto -1 + \color{blue}{\cos \varepsilon} \]

5. Applied rewrites 49.7%

   \[\leadsto \color{blue}{-1 + \cos \varepsilon} \]

6. Taylor expanded in eps around 0

   \[\leadsto -1 + 1 \]
7. Step-by-step derivation

8. Applied rewrites 49.5%

   \[\leadsto -1 + 1 \]
9. 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 2024234 
(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)))