2cos (problem 3.3.5)

Percentage Accurate: 51.6% → 99.8%

Time: 18.6s

Alternatives: 17

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: 51.6% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \left(t\_0 \cdot \mathsf{fma}\left(t\_0, \cos x, \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right)\right) \cdot -2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))))
   (* (* t_0 (fma t_0 (cos x) (* (sin x) (cos (* 0.5 eps))))) -2.0)))

C

double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	return (t_0 * fma(t_0, cos(x), (sin(x) * cos((0.5 * eps))))) * -2.0;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	return Float64(Float64(t_0 * fma(t_0, cos(x), Float64(sin(x) * cos(Float64(0.5 * eps))))) * -2.0)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Initial program 54.9%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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 99.5%

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

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-rgt-in N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-fma.f64 99.6

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

7. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. sin-sum N/A

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

   3. lift-sin.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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 99.5%

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

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-rgt-in N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-fma.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 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 54.9%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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 99.5%

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

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-rgt-in N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-fma.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{-1}{645120}, \frac{1}{3840}\right)}, \frac{-1}{48}\right), \frac{1}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right) \cdot -2 \]

   14. unpow2 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{645120}, \frac{1}{3840}\right), \frac{-1}{48}\right), \frac{1}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right) \cdot -2 \]

   15. lower-*.f64 98.9

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

10. Applied rewrites 98.9%

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

11. Final simplification 98.9%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

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

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(eps, Float64(eps * fma(Float64(eps * 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[(eps * N[(eps * N[(N[(eps * eps), $MachinePrecision] * 0.00026041666666666666 + -0.020833333333333332), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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 99.5%

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

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-rgt-in N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-fma.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 98.7

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

10. Applied rewrites 98.7%

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

11. Final simplification 98.7%

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

Alternative 5: 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 54.9%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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 99.5%

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

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-rgt-in N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-fma.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.6

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

10. Applied rewrites 98.6%

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

11. Final simplification 98.6%

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

Alternative 6: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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 99.5%

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

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-rgt-in N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-fma.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 98.4

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

10. Applied rewrites 98.4%

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

11. Final simplification 98.4%

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

Alternative 7: 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 54.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 98.0

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

8. Applied rewrites 98.0%

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

Alternative 8: 98.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right) - \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (* eps (fma x (* x 0.25) -0.5))
   (fma
    (fma
     x
     (* x (fma x (* x -0.0001984126984126984) 0.008333333333333333))
     -0.16666666666666666)
    (* x (* x x))
    x))))

C

double code(double x, double eps) {
	return eps * ((eps * fma(x, (x * 0.25), -0.5)) - fma(fma(x, (x * fma(x, (x * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666), (x * (x * x)), x));
}

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * fma(x, Float64(x * 0.25), -0.5)) - fma(fma(x, Float64(x * fma(x, Float64(x * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666), Float64(x * Float64(x * x)), x)))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(x * N[(x * 0.25), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[(x * N[(x * N[(x * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} - \sin x\right) \]

   2. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{4}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) - \sin x\right) \]

   3. unpow2 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{4} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) - \sin x\right) \]

   4. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{4}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) - \sin x\right) \]

   5. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \frac{1}{4}\right) + \color{blue}{\frac{-1}{2}}\right) - \sin x\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right)} - \sin x\right) \]

   7. lower-*.f64 98.0

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \color{blue}{\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x\right)}\right) \]

    3. *-lft-identity N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + \color{blue}{x}\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \left(\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{2}\right)} \cdot x + x\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right)} + x\right)\right) \]

    6. pow-plus N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{x}^{\left(2 + 1\right)}} + x\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{\color{blue}{3}} + x\right)\right) \]

    8. cube-unmult N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + x\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) + x\right)\right) \]

    10. lower-fma.f64 N/A

        \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}, x \cdot {x}^{2}, x\right)}\right) \]

11. Applied rewrites 97.8%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}\right) \]
12. Add Preprocessing

Alternative 9: 98.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right) - \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (* eps (fma x (* x 0.25) -0.5))
   (fma
    (* x x)
    (* x (fma (* x x) 0.008333333333333333 -0.16666666666666666))
    x))))

C

double code(double x, double eps) {
	return eps * ((eps * fma(x, (x * 0.25), -0.5)) - fma((x * x), (x * fma((x * x), 0.008333333333333333, -0.16666666666666666)), x));
}

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * fma(x, Float64(x * 0.25), -0.5)) - fma(Float64(x * x), Float64(x * fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666)), x)))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(x * N[(x * 0.25), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} - \sin x\right) \]

   2. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{4}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) - \sin x\right) \]

   3. unpow2 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{4} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) - \sin x\right) \]

   4. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{4}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) - \sin x\right) \]

   5. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \frac{1}{4}\right) + \color{blue}{\frac{-1}{2}}\right) - \sin x\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right)} - \sin x\right) \]

   7. lower-*.f64 98.0

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right) \]
10. Step-by-step derivation

    1. distribute-rgt-in N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \color{blue}{\left(1 \cdot x + \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right)}\right) \]

    2. *-lft-identity N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \left(\color{blue}{x} + \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right)\right) \]

    3. +-commutative N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + x\right)}\right) \]

    4. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \left(\color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right)} + x\right)\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)}\right) \]

    6. unpow2 N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)\right) \]

    7. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x}, x\right)\right) \]

    9. sub-neg N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot x, x\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \mathsf{fma}\left(x \cdot x, \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \mathsf{fma}\left(x \cdot x, \left({x}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right) \cdot x, x\right)\right) \]

    12. lower-fma.f64 N/A

        \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120}, \frac{-1}{6}\right)} \cdot x, x\right)\right) \]

    13. unpow2 N/A

        \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right) - \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120}, \frac{-1}{6}\right) \cdot x, x\right)\right) \]

    14. lower-*.f64 97.7

        \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right) - \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.008333333333333333, -0.16666666666666666\right) \cdot x, x\right)\right) \]

11. Applied rewrites 97.7%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right) - \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot x, x\right)}\right) \]

12. Final simplification 97.7%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right) - \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x\right)\right) \]
13. Add Preprocessing

Alternative 10: 98.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) + \frac{-1}{2} \cdot {\varepsilon}^{2}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right)} \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right) + -1 \cdot \varepsilon}, \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right)}, \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\left(\varepsilon \cdot x\right) \cdot \frac{1}{6}} + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \left(x \cdot \frac{1}{6}\right)} + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \color{blue}{\left(\frac{1}{6} \cdot x\right)} + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{6} \cdot x, \frac{1}{4} \cdot {\varepsilon}^{2}\right)}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \frac{1}{6}}, \frac{1}{4} \cdot {\varepsilon}^{2}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \frac{1}{6}}, \frac{1}{4} \cdot {\varepsilon}^{2}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{4}}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{4}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\frac{1}{4} \cdot \varepsilon\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \color{blue}{\varepsilon \cdot \left(\frac{1}{4} \cdot \varepsilon\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   18. neg-mul-1 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \color{blue}{\mathsf{neg}\left(\varepsilon\right)}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   19. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \color{blue}{\mathsf{neg}\left(\varepsilon\right)}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \mathsf{neg}\left(\varepsilon\right)\right), \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}}\right) \]

   21. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   22. lower-*.f64 97.5

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

8. Applied rewrites 97.5%

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

9. Final simplification 97.5%

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

Alternative 11: 98.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} - \sin x\right) \]

   2. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{4}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) - \sin x\right) \]

   3. unpow2 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{4} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) - \sin x\right) \]

   4. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{4}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) - \sin x\right) \]

   5. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \frac{1}{4}\right) + \color{blue}{\frac{-1}{2}}\right) - \sin x\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right)} - \sin x\right) \]

   7. lower-*.f64 98.0

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in x around 0

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

    1. distribute-rgt-in N/A

       \[\leadsto \frac{-1}{2} \cdot {\varepsilon}^{2} + \color{blue}{\left(\left(-1 \cdot \varepsilon\right) \cdot x + \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)} \]

    2. associate-+r+ N/A

       \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {\varepsilon}^{2} + \left(-1 \cdot \varepsilon\right) \cdot x\right) + \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) \cdot x} \]

    3. +-commutative N/A

       \[\leadsto \color{blue}{\left(\left(-1 \cdot \varepsilon\right) \cdot x + \frac{-1}{2} \cdot {\varepsilon}^{2}\right)} + \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

    4. associate-*r* N/A

       \[\leadsto \left(\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)} + \frac{-1}{2} \cdot {\varepsilon}^{2}\right) + \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

    5. *-commutative N/A

       \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + \frac{-1}{2} \cdot {\varepsilon}^{2}\right) + \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

    6. associate-*r* N/A

       \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot \varepsilon} + \frac{-1}{2} \cdot {\varepsilon}^{2}\right) + \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

    7. unpow2 N/A

       \[\leadsto \left(\left(-1 \cdot x\right) \cdot \varepsilon + \frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) + \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

    8. associate-*r* N/A

       \[\leadsto \left(\left(-1 \cdot x\right) \cdot \varepsilon + \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right) \cdot \varepsilon}\right) + \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

    9. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\varepsilon \cdot \left(-1 \cdot x + \frac{-1}{2} \cdot \varepsilon\right)} + \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

    10. *-commutative N/A

        \[\leadsto \varepsilon \cdot \left(-1 \cdot x + \frac{-1}{2} \cdot \varepsilon\right) + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)} \]

    11. lower-fma.f64 N/A

        \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, -1 \cdot x + \frac{-1}{2} \cdot \varepsilon, x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)\right)} \]

    12. +-commutative N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{2} \cdot \varepsilon + -1 \cdot x}, x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)\right) \]

    13. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \varepsilon + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}, x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)\right) \]

    14. unsub-neg N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{2} \cdot \varepsilon - x}, x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)\right) \]

    15. lower--.f64 N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{2} \cdot \varepsilon - x}, x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \frac{-1}{2}} - x, x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)\right) \]

    17. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \frac{-1}{2}} - x, x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)\right) \]

11. Applied rewrites 97.3%

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

Alternative 12: 98.0% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right) + \frac{-1}{2} \cdot \varepsilon\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1, \frac{-1}{2} \cdot \varepsilon\right)} \]

   3. sub-neg N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) + \left(\mathsf{neg}\left(1\right)\right)}, \frac{-1}{2} \cdot \varepsilon\right) \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) + \color{blue}{-1}, \frac{-1}{2} \cdot \varepsilon\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon, -1\right)}, \frac{-1}{2} \cdot \varepsilon\right) \]

   6. +-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{4} \cdot \varepsilon + \frac{1}{6} \cdot x}, -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   7. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \frac{1}{4}} + \frac{1}{6} \cdot x, -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{4}, \frac{1}{6} \cdot x\right)}, -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   9. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \frac{1}{4}, \color{blue}{x \cdot \frac{1}{6}}\right), -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \frac{1}{4}, \color{blue}{x \cdot \frac{1}{6}}\right), -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   11. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \frac{1}{4}, x \cdot \frac{1}{6}\right), -1\right), \color{blue}{\varepsilon \cdot \frac{-1}{2}}\right) \]

   12. lower-*.f64 97.3

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

8. Applied rewrites 97.3%

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

Alternative 13: 97.7% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \frac{-1}{2}} + x \cdot \left(\frac{1}{4} \cdot \left(\varepsilon \cdot x\right) - 1\right)\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\frac{1}{4} \cdot \left(\varepsilon \cdot x\right) - 1\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \color{blue}{x \cdot \left(\frac{1}{4} \cdot \left(\varepsilon \cdot x\right) - 1\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(\varepsilon \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot \varepsilon\right) \cdot x} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\left(\frac{1}{4} \cdot \varepsilon\right) \cdot x + \color{blue}{-1}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)} \cdot x + -1\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\varepsilon \cdot \left(\frac{1}{4} \cdot x\right)} + -1\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)}\right) \]

   10. lower-*.f64 96.8

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

8. Applied rewrites 96.8%

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

   1. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} + x \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{4} \cdot x\right)} + -1\right)\right) \]

   2. lift-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} + x \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} + \color{blue}{x \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)}\right) \]

   4. distribute-lft-in N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2}\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)\right) \cdot \varepsilon} + \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right), \varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2}\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{1}{4} \cdot x}, -1\right), \varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \frac{1}{4}}, -1\right), \varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2}\right)\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \frac{1}{4}}, -1\right), \varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{4}, -1\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \varepsilon}\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{4}, -1\right), \varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)} \cdot \varepsilon\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{4}, -1\right), \varepsilon, \color{blue}{\frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{4}, -1\right), \varepsilon, \frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   15. lower-*.f64 97.0

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

10. Applied rewrites 97.0%

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

11. Final simplification 97.0%

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

Alternative 14: 97.5% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \frac{-1}{2}} + x \cdot \left(\frac{1}{4} \cdot \left(\varepsilon \cdot x\right) - 1\right)\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\frac{1}{4} \cdot \left(\varepsilon \cdot x\right) - 1\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \color{blue}{x \cdot \left(\frac{1}{4} \cdot \left(\varepsilon \cdot x\right) - 1\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(\varepsilon \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot \varepsilon\right) \cdot x} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\left(\frac{1}{4} \cdot \varepsilon\right) \cdot x + \color{blue}{-1}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)} \cdot x + -1\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\varepsilon \cdot \left(\frac{1}{4} \cdot x\right)} + -1\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)}\right) \]

   10. lower-*.f64 96.8

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

8. Applied rewrites 96.8%

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

   1. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} + x \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{4} \cdot x\right)} + -1\right)\right) \]

   2. lift-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} + x \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} + \color{blue}{x \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)}\right) \]

   4. lift-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot x, -1\right)\right) \cdot \varepsilon} \]

   6. lower-*.f64 96.8

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

10. Applied rewrites 96.8%

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

11. Final simplification 96.8%

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

Alternative 15: 97.5% 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 54.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 96.7

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

8. Applied rewrites 96.7%

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

Alternative 16: 78.5% 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 54.9%

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 79.3

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

8. Applied rewrites 79.3%

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

9. Final simplification 79.3%

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

Alternative 17: 50.1% 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 54.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-+.f64 N/A

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

   4. lower-cos.f64 52.8

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{1} + -1 \]
7. Step-by-step derivation

8. Applied rewrites 52.7%

   \[\leadsto \color{blue}{1} + -1 \]
9. Step-by-step derivation

   1. metadata-eval 52.7

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

10. Applied rewrites 52.7%

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

Developer Target 1: 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 2024216 
(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)))