2cos (problem 3.3.5)

Percentage Accurate: 51.1% → 99.6%

Time: 17.9s

Alternatives: 11

Speedup: 23.0×

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: 51.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -1.5500992063492063 \cdot 10^{-6}, 0.00026041666666666666\right), -0.020833333333333332\right), 0.5\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (*
    eps
    (fma
     (* eps eps)
     (fma
      (* eps eps)
      (fma (* eps eps) -1.5500992063492063e-6 0.00026041666666666666)
      -0.020833333333333332)
     0.5))
   (fma (sin (fma 0.5 eps 0.0)) (cos x) (* (sin x) (cos (fma 0.5 eps 0.0)))))
  -2.0))

C

double code(double x, double eps) {
	return ((eps * fma((eps * eps), fma((eps * eps), fma((eps * eps), -1.5500992063492063e-6, 0.00026041666666666666), -0.020833333333333332), 0.5)) * fma(sin(fma(0.5, eps, 0.0)), cos(x), (sin(x) * cos(fma(0.5, eps, 0.0))))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(eps * fma(Float64(eps * eps), fma(Float64(eps * eps), fma(Float64(eps * eps), -1.5500992063492063e-6, 0.00026041666666666666), -0.020833333333333332), 0.5)) * fma(sin(fma(0.5, eps, 0.0)), cos(x), Float64(sin(x) * cos(fma(0.5, eps, 0.0))))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(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] * N[(N[Sin[N[(0.5 * eps + 0.0), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(0.5 * eps + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.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 egg-rr 99.7%

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. +-rgt-identity N/A

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

   5. sin-sum N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. sin-lowering-sin.f64 N/A

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

   8. +-rgt-identity N/A

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

   9. *-commutative N/A

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

   10. +-rgt-identity N/A

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

   11. distribute-rgt-in N/A

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

   12. metadata-eval N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. cos-lowering-cos.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. *-lowering-*.f64 N/A

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

   17. *-lowering-*.f64 N/A

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

6. Applied egg-rr 99.7%

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

   1. +-rgt-identity N/A

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

   2. +-rgt-identity N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. +-rgt-identity N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. +-rgt-identity N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. *-rgt-identity N/A

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

   13. *-commutative N/A

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

   14. +-rgt-identity N/A

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

   15. *-lowering-*.f64 N/A

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in eps around 0

   \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right)} \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right)} \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    2. +-commutative N/A

       \[\leadsto \left(\left(\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    3. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, {\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}, \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    4. unpow2 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, {\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, {\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    6. sub-neg N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    7. metadata-eval N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, {\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) + \color{blue}{\frac{-1}{48}}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    8. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}, \frac{-1}{48}\right)}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    9. unpow2 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}, \frac{-1}{48}\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}, \frac{-1}{48}\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    11. +-commutative N/A

        \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{-1}{645120} \cdot {\varepsilon}^{2} + \frac{1}{3840}}, \frac{-1}{48}\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    12. *-commutative N/A

        \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{-1}{645120}} + \frac{1}{3840}, \frac{-1}{48}\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    13. accelerator-lowering-fma.f64 N/A

        \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{-1}{645120}, \frac{1}{3840}\right)}, \frac{-1}{48}\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    14. unpow2 N/A

        \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{645120}, \frac{1}{3840}\right), \frac{-1}{48}\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    15. *-lowering-*.f64 99.7

        \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, -1.5500992063492063 \cdot 10^{-6}, 0.00026041666666666666\right), -0.020833333333333332\right), 0.5\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

11. Simplified 99.7%

    \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -1.5500992063492063 \cdot 10^{-6}, 0.00026041666666666666\right), -0.020833333333333332\right), 0.5\right)\right)} \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]
12. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.00026041666666666666, -0.020833333333333332\right), 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  -2.0
  (*
   (fma (sin (fma 0.5 eps 0.0)) (cos x) (* (sin x) (cos (fma 0.5 eps 0.0))))
   (*
    eps
    (fma
     (* eps eps)
     (fma eps (* eps 0.00026041666666666666) -0.020833333333333332)
     0.5)))))

C

double code(double x, double eps) {
	return -2.0 * (fma(sin(fma(0.5, eps, 0.0)), cos(x), (sin(x) * cos(fma(0.5, eps, 0.0)))) * (eps * fma((eps * eps), fma(eps, (eps * 0.00026041666666666666), -0.020833333333333332), 0.5)));
}

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(fma(sin(fma(0.5, eps, 0.0)), cos(x), Float64(sin(x) * cos(fma(0.5, eps, 0.0)))) * Float64(eps * fma(Float64(eps * eps), fma(eps, Float64(eps * 0.00026041666666666666), -0.020833333333333332), 0.5))))
end

Wolfram

code[x_, eps_] := N[(-2.0 * N[(N[(N[Sin[N[(0.5 * eps + 0.0), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(0.5 * eps + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * 0.00026041666666666666), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.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 egg-rr 99.7%

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. +-rgt-identity N/A

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

   5. sin-sum N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. sin-lowering-sin.f64 N/A

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

   8. +-rgt-identity N/A

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

   9. *-commutative N/A

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

   10. +-rgt-identity N/A

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

   11. distribute-rgt-in N/A

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

   12. metadata-eval N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. cos-lowering-cos.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. *-lowering-*.f64 N/A

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

   17. *-lowering-*.f64 N/A

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

6. Applied egg-rr 99.7%

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

   1. +-rgt-identity N/A

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

   2. +-rgt-identity N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. +-rgt-identity N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. +-rgt-identity N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. *-rgt-identity N/A

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

   13. *-commutative N/A

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

   14. +-rgt-identity N/A

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

   15. *-lowering-*.f64 N/A

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in eps around 0

   \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)} \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)} \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    2. +-commutative N/A

       \[\leadsto \left(\left(\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    3. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    4. unpow2 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    6. sub-neg N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{1}{3840} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    7. *-commutative N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{3840}} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    8. unpow2 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{3840} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    9. associate-*l* N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{3840}\right)} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    10. metadata-eval N/A

        \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{3840}\right) + \color{blue}{\frac{-1}{48}}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    11. accelerator-lowering-fma.f64 N/A

        \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{3840}, \frac{-1}{48}\right)}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

    12. *-lowering-*.f64 99.7

        \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot 0.00026041666666666666}, -0.020833333333333332\right), 0.5\right)\right) \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

11. Simplified 99.7%

    \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.00026041666666666666, -0.020833333333333332\right), 0.5\right)\right)} \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right)\right)\right) \cdot -2 \]

12. Final simplification 99.7%

    \[\leadsto -2 \cdot \left(\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right), \cos x, \sin x \cdot \cos \left(\mathsf{fma}\left(0.5, \varepsilon, 0\right)\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.00026041666666666666, -0.020833333333333332\right), 0.5\right)\right)\right) \]
13. Add Preprocessing

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.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 egg-rr 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 99.7

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot -2\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), 0.5\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (* eps -2.0)
  (* (fma -0.020833333333333332 (fma eps eps 0.0) 0.5) (sin (fma 0.5 eps x)))))

C

double code(double x, double eps) {
	return (eps * -2.0) * (fma(-0.020833333333333332, fma(eps, eps, 0.0), 0.5) * sin(fma(0.5, eps, x)));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * -2.0) * Float64(fma(-0.020833333333333332, fma(eps, eps, 0.0), 0.5) * sin(fma(0.5, eps, x))))
end

Wolfram

code[x_, eps_] := N[(N[(eps * -2.0), $MachinePrecision] * N[(N[(-0.020833333333333332 * N[(eps * eps + 0.0), $MachinePrecision] + 0.5), $MachinePrecision] * N[Sin[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.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 egg-rr 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 99.6

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

7. Simplified 99.6%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. +-rgt-identity N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. sin-lowering-sin.f64 N/A

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

   13. *-commutative N/A

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

   14. distribute-rgt-in N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. accelerator-lowering-fma.f64 N/A

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

   18. associate-*l* N/A

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

   19. metadata-eval N/A

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

   20. *-rgt-identity 99.6

       \[\leadsto \left(-2 \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), 0.5\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \varepsilon, \color{blue}{x}\right)\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(-2 \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), 0.5\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)\right)} \]

10. Final simplification 99.6%

    \[\leadsto \left(\varepsilon \cdot -2\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), 0.5\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)\right) \]
11. Add Preprocessing

Alternative 5: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.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 egg-rr 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 99.6

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

7. Simplified 99.6%

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

   1. *-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. +-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity 99.6

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

9. Applied egg-rr 99.6%

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

10. Final simplification 99.6%

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

Alternative 6: 99.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.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 egg-rr 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 99.3

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 7: 98.0% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. cos-sum N/A

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

   2. associate--l- N/A

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

   3. sub-neg N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. sin-lowering-sin.f64 N/A

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

   10. sin-lowering-sin.f64 N/A

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

   11. cos-lowering-cos.f64 54.9

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

4. Applied egg-rr 54.9%

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

5. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. --lowering--.f64 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cos-lowering-cos.f64 N/A

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

   14. sin-lowering-sin.f64 99.4

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

7. Simplified 99.4%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 98.5

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

10. Simplified 98.5%

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

Alternative 8: 97.5% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. cos-sum N/A

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

   2. associate--l- N/A

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

   3. sub-neg N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. sin-lowering-sin.f64 N/A

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

   10. sin-lowering-sin.f64 N/A

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

   11. cos-lowering-cos.f64 54.9

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

4. Applied egg-rr 54.9%

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

5. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. --lowering--.f64 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cos-lowering-cos.f64 N/A

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

   14. sin-lowering-sin.f64 99.4

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

7. Simplified 99.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{4} \cdot \left(\varepsilon \cdot x\right) - 1\right)\right)} \]
9. 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. accelerator-lowering-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. *-lowering-*.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. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-lowering-*.f64 98.3

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

10. Simplified 98.3%

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

Alternative 9: 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 54.8%

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

   1. cos-sum N/A

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

   2. associate--l- N/A

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

   3. sub-neg N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. sin-lowering-sin.f64 N/A

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

   10. sin-lowering-sin.f64 N/A

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

   11. cos-lowering-cos.f64 54.9

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

4. Applied egg-rr 54.9%

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

5. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. --lowering--.f64 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cos-lowering-cos.f64 N/A

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

   14. sin-lowering-sin.f64 99.4

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

7. Simplified 99.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-1 \cdot x + \frac{-1}{2} \cdot \varepsilon\right)} \]
9. 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. --lowering--.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. *-lowering-*.f64 98.3

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

10. Simplified 98.3%

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

Alternative 10: 77.8% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0 - \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* x (- 0.0 eps)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return x * (0.0 - eps)

Julia

function code(x, eps)
	return Float64(x * Float64(0.0 - eps))
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(x * N[(0.0 - eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\sin x, \color{blue}{\mathsf{neg}\left(\varepsilon\right)}, 0\right) \]

   8. neg-sub0 N/A

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

   9. --lowering--.f64 80.2

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

5. Simplified 80.2%

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

   1. +-rgt-identity N/A

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

   2. sub0-neg N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. neg-lowering-neg.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 80.2

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

7. Applied egg-rr 80.2%

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

8. Taylor expanded in x around 0

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

10. Simplified 79.6%

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

11. Final simplification 79.6%

    \[\leadsto x \cdot \left(0 - \varepsilon\right) \]
12. Add Preprocessing

Alternative 11: 49.6% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 54.8%

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

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

   4. cos-lowering-cos.f64 53.9

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

5. Simplified 53.9%

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

6. Taylor expanded in eps around 0

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

8. Simplified 53.8%

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

   1. metadata-eval 53.8

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

10. Applied egg-rr 53.8%

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

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024197 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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

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