2cos (problem 3.3.5)

Percentage Accurate: 52.6% → 100.0%

Time: 15.7s

Alternatives: 11

Speedup: 25.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.6% 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: 100.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. *-commutative N/A

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

   7. sin-sum N/A

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

   8. +-lft-identity N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lower-fma.f64 N/A

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

6. Applied rewrites 99.8%

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sin \left(\varepsilon \cdot 0.5\right), \cos \left(1 \cdot x\right), \cos \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(1 \cdot x\right)\right)} \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x, \sin \left(\varepsilon \cdot 0.5\right), 0.25 \cdot \left(\left(\cos x \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot -2 \end{array} \]

FPCore

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

C

double code(double x, double eps) {
	return fma((cos((eps * 0.5)) * sin(x)), sin((eps * 0.5)), (0.25 * ((cos(x) * eps) * eps))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(fma(Float64(cos(Float64(eps * 0.5)) * sin(x)), sin(Float64(eps * 0.5)), Float64(0.25 * Float64(Float64(cos(x) * eps) * eps))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] + N[(0.25 * N[(N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. *-commutative N/A

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

   7. sin-sum N/A

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

   8. +-lft-identity N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lower-fma.f64 N/A

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

6. Applied rewrites 99.8%

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sin \left(\varepsilon \cdot 0.5\right), \cos \left(1 \cdot x\right), \cos \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(1 \cdot x\right)\right)} \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. *-commutative N/A

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

    4. unpow2 N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lower-cos.f64 99.8

        \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right), \left(\left(\color{blue}{\cos x} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot 0.25\right) \cdot -2 \]

11. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right), \color{blue}{\left(\left(\cos x \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot 0.25}\right) \cdot -2 \]

12. Final simplification 99.8%

    \[\leadsto \mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x, \sin \left(\varepsilon \cdot 0.5\right), 0.25 \cdot \left(\left(\cos x \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot -2 \]
13. Add Preprocessing

Alternative 3: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 5: 99.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 6: 79.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x}{\frac{-1}{\varepsilon}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (sin x) (/ -1.0 eps)))

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return Math.sin(x) / (-1.0 / eps);
}

Python

def code(x, eps):
	return math.sin(x) / (-1.0 / eps)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 80.1

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

5. Applied rewrites 80.1%

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

7. Applied rewrites 80.0%

   \[\leadsto \frac{\varepsilon \cdot \varepsilon}{-\varepsilon} \cdot \sin \color{blue}{x} \]
8. Step-by-step derivation

9. Applied rewrites 80.1%

   \[\leadsto \frac{\sin x}{\color{blue}{\frac{-1}{\varepsilon}}} \]
10. Add Preprocessing

Alternative 7: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 80.1

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

5. Applied rewrites 80.1%

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

Alternative 8: 78.7% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 80.1

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

5. Applied rewrites 80.1%

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

7. Applied rewrites 80.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 79.8%

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

11. Final simplification 79.8%

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

Alternative 9: 78.6% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 79.7%

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

9. Final simplification 79.7%

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

Alternative 10: 78.4% accurate, 25.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 79.6%

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

Alternative 11: 51.0% accurate, 51.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 48.9

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

5. Applied rewrites 48.9%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 48.9%

   \[\leadsto 1 - 1 \]
9. Add Preprocessing

Developer Target 1: 98.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024276 
(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)))