2cos (problem 3.3.5)

Percentage Accurate: 52.8% → 99.8%

Time: 20.3s

Alternatives: 17

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: 52.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot 2\right)\\ \left(\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\varepsilon, 0.5, 0\right)\right), \cos t\_0, \cos \left(\mathsf{fma}\left(\varepsilon, 0.5, 0\right)\right) \cdot \sin t\_0\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot -2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x 2.0))))
   (*
    (*
     (fma
      (sin (fma eps 0.5 0.0))
      (cos t_0)
      (* (cos (fma eps 0.5 0.0)) (sin t_0)))
     (sin (* eps 0.5)))
    -2.0)))

C

double code(double x, double eps) {
	double t_0 = 0.5 * (x * 2.0);
	return (fma(sin(fma(eps, 0.5, 0.0)), cos(t_0), (cos(fma(eps, 0.5, 0.0)) * sin(t_0))) * sin((eps * 0.5))) * -2.0;
}

Julia

function code(x, eps)
	t_0 = Float64(0.5 * Float64(x * 2.0))
	return Float64(Float64(fma(sin(fma(eps, 0.5, 0.0)), cos(t_0), Float64(cos(fma(eps, 0.5, 0.0)) * sin(t_0))) * sin(Float64(eps * 0.5))) * -2.0)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(0.5 * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Sin[N[(eps * 0.5 + 0.0), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision] + N[(N[Cos[N[(eps * 0.5 + 0.0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Initial program 48.9%

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

   \[\leadsto \color{blue}{\left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(x, 2, \varepsilon\right) \cdot 0.5\right)\right) \cdot -2} \]
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 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

   \[\leadsto \color{blue}{\left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(x, 2, \varepsilon\right) \cdot 0.5\right)\right) \cdot -2} \]
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 99.8%

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

7. Taylor expanded in eps around inf

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

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

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

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

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

   3. *-commutative N/A

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

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

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 99.8

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

9. Simplified 99.8%

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

10. Taylor expanded in eps around 0

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

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

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

    2. +-commutative N/A

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

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

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

    4. unpow2 N/A

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

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

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

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

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

    9. unpow2 N/A

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

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

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

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

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

    14. unpow2 N/A

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

    15. *-lowering-*.f64 99.8

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  -2.0
  (*
   (*
    eps
    (fma
     (* eps eps)
     (fma
      eps
      (* eps (fma eps (* eps -1.5500992063492063e-6) 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 * fma(eps, (eps * -1.5500992063492063e-6), 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(Float64(eps * eps), fma(eps, Float64(eps * fma(eps, Float64(eps * -1.5500992063492063e-6), 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[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * N[(eps * N[(eps * -1.5500992063492063e-6), $MachinePrecision] + 0.00026041666666666666), $MachinePrecision]), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * N[(x * 2.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.9%

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

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

5. Taylor expanded in eps around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

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

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

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

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

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

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

   17. *-lowering-*.f64 99.6

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

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

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 99.5

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(sin(Float64(0.5 * fma(x, 2.0, eps))) * Float64(eps * fma(eps, Float64(eps * -0.020833333333333332), 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 * N[(eps * N[(eps * -0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.9%

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

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

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 99.3

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 6: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

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

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 98.7

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

7. Simplified 98.7%

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

8. Taylor expanded in eps around inf

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

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

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

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

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

   10. metadata-eval N/A

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

   11. distribute-lft-in N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. *-lft-identity N/A

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

   16. accelerator-lowering-fma.f64 98.7

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

10. Simplified 98.7%

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

Alternative 7: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in eps around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. +-rgt-identity N/A

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

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

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

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

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

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

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

   10. sin-lowering-sin.f64 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 97.7

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

8. Simplified 97.7%

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

Alternative 8: 98.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.16666666666666666, \varepsilon \cdot \varepsilon, -1\right)\\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.25, -0.16666666666666666 \cdot \left(\varepsilon \cdot \left(x \cdot t\_0\right)\right)\right), \varepsilon \cdot t\_0\right), \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma 0.16666666666666666 (* eps eps) -1.0)))
   (fma
    x
    (fma
     x
     (fma eps (* eps 0.25) (* -0.16666666666666666 (* eps (* x t_0))))
     (* eps t_0))
    (* (* eps eps) -0.5))))

C

double code(double x, double eps) {
	double t_0 = fma(0.16666666666666666, (eps * eps), -1.0);
	return fma(x, fma(x, fma(eps, (eps * 0.25), (-0.16666666666666666 * (eps * (x * t_0)))), (eps * t_0)), ((eps * eps) * -0.5));
}

Julia

function code(x, eps)
	t_0 = fma(0.16666666666666666, Float64(eps * eps), -1.0)
	return fma(x, fma(x, fma(eps, Float64(eps * 0.25), Float64(-0.16666666666666666 * Float64(eps * Float64(x * t_0)))), Float64(eps * t_0)), Float64(Float64(eps * eps) * -0.5))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(eps * eps), $MachinePrecision] + -1.0), $MachinePrecision]}, N[(x * N[(x * N[(eps * N[(eps * 0.25), $MachinePrecision] + N[(-0.16666666666666666 * N[(eps * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in eps around 0

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

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

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

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

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

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

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

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

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

8. Simplified 97.0%

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

9. Final simplification 97.0%

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

Alternative 9: 98.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), 0 - \varepsilon\right), \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  x
  (fma x (* eps (fma eps 0.25 (* x 0.16666666666666666))) (- 0.0 eps))
  (* eps (* eps -0.5))))

C

double code(double x, double eps) {
	return fma(x, fma(x, (eps * fma(eps, 0.25, (x * 0.16666666666666666))), (0.0 - eps)), (eps * (eps * -0.5)));
}

Julia

function code(x, eps)
	return fma(x, fma(x, Float64(eps * fma(eps, 0.25, Float64(x * 0.16666666666666666))), Float64(0.0 - eps)), Float64(eps * Float64(eps * -0.5)))
end

Wolfram

code[x_, eps_] := N[(x * N[(x * N[(eps * N[(eps * 0.25 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 - eps), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in eps around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. +-rgt-identity N/A

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

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

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

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

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

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

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

   10. sin-lowering-sin.f64 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. +-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

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

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

   16. mul-1-neg N/A

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

   17. neg-sub0 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4}, x \cdot \frac{1}{6}\right), \color{blue}{0 - \varepsilon}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   18. --lowering--.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4}, x \cdot \frac{1}{6}\right), \color{blue}{0 - \varepsilon}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4}, x \cdot \frac{1}{6}\right), 0 - \varepsilon\right), \color{blue}{{\varepsilon}^{2} \cdot \frac{-1}{2}}\right) \]

8. Simplified 97.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), 0 - \varepsilon\right), \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right)} \]
9. Add Preprocessing

Alternative 10: 98.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, -0.5, \mathsf{fma}\left(x, 0.16666666666666666, \mathsf{fma}\left(\varepsilon, 0.25, 0\right)\right) \cdot \left(x \cdot x\right)\right) - x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (fma eps -0.5 (* (fma x 0.16666666666666666 (fma eps 0.25 0.0)) (* x x)))
   x)))

C

double code(double x, double eps) {
	return eps * (fma(eps, -0.5, (fma(x, 0.16666666666666666, fma(eps, 0.25, 0.0)) * (x * x))) - x);
}

Julia

function code(x, eps)
	return Float64(eps * Float64(fma(eps, -0.5, Float64(fma(x, 0.16666666666666666, fma(eps, 0.25, 0.0)) * Float64(x * x))) - x))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5 + N[(N[(x * 0.16666666666666666 + N[(eps * 0.25 + 0.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in eps around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. +-rgt-identity N/A

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

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

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

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

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

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

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

   10. sin-lowering-sin.f64 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   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 96.9

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

8. Simplified 96.9%

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

   1. distribute-lft-in N/A

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

   2. associate-+r+ N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

   10. +-commutative N/A

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

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

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

   12. *-commutative N/A

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

   13. +-rgt-identity N/A

       \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{4} \cdot \color{blue}{\left(\varepsilon + 0\right)}\right) \cdot \left(x \cdot x\right)\right) + -1 \cdot x\right) \]

   14. distribute-rgt-in N/A

       \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \mathsf{fma}\left(x, \frac{1}{6}, \color{blue}{\varepsilon \cdot \frac{1}{4} + 0 \cdot \frac{1}{4}}\right) \cdot \left(x \cdot x\right)\right) + -1 \cdot x\right) \]

   15. metadata-eval N/A

       \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \mathsf{fma}\left(x, \frac{1}{6}, \varepsilon \cdot \frac{1}{4} + \color{blue}{0}\right) \cdot \left(x \cdot x\right)\right) + -1 \cdot x\right) \]

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

       \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \mathsf{fma}\left(x, \frac{1}{6}, \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{4}, 0\right)}\right) \cdot \left(x \cdot x\right)\right) + -1 \cdot x\right) \]

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

       \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \mathsf{fma}\left(x, \frac{1}{6}, \mathsf{fma}\left(\varepsilon, \frac{1}{4}, 0\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) + -1 \cdot x\right) \]

   18. mul-1-neg N/A

       \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \mathsf{fma}\left(x, \frac{1}{6}, \mathsf{fma}\left(\varepsilon, \frac{1}{4}, 0\right)\right) \cdot \left(x \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

   19. neg-lowering-neg.f64 96.9

       \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, -0.5, \mathsf{fma}\left(x, 0.16666666666666666, \mathsf{fma}\left(\varepsilon, 0.25, 0\right)\right) \cdot \left(x \cdot x\right)\right) + \color{blue}{\left(-x\right)}\right) \]

10. Applied egg-rr 96.9%

    \[\leadsto \varepsilon \cdot \color{blue}{\left(\mathsf{fma}\left(\varepsilon, -0.5, \mathsf{fma}\left(x, 0.16666666666666666, \mathsf{fma}\left(\varepsilon, 0.25, 0\right)\right) \cdot \left(x \cdot x\right)\right) + \left(-x\right)\right)} \]

11. Final simplification 96.9%

    \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, -0.5, \mathsf{fma}\left(x, 0.16666666666666666, \mathsf{fma}\left(\varepsilon, 0.25, 0\right)\right) \cdot \left(x \cdot x\right)\right) - x\right) \]
12. Add Preprocessing

Alternative 11: 98.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in eps around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. +-rgt-identity N/A

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

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

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

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

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

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

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

   10. sin-lowering-sin.f64 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   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 96.9

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

8. Simplified 96.9%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

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

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

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

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

    6. *-commutative N/A

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

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

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

    8. *-commutative N/A

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

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

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

    10. *-commutative N/A

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

    11. *-lowering-*.f64 96.9

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

11. Simplified 96.9%

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

Alternative 12: 98.3% 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 48.9%

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

3. Taylor expanded in eps around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. +-rgt-identity N/A

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

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

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

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

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

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

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

   10. sin-lowering-sin.f64 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   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 96.9

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

8. Simplified 96.9%

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

Alternative 13: 98.2% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in eps around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. +-rgt-identity N/A

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

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

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

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

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

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

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

   10. sin-lowering-sin.f64 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   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 96.9

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

8. Simplified 96.9%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f64 96.7

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

11. Simplified 96.7%

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

Alternative 14: 97.9% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in eps around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. +-rgt-identity N/A

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

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

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

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

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

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

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

   10. sin-lowering-sin.f64 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   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 96.9

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

8. Simplified 96.9%

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

9. Taylor expanded in x around 0

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

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

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

    4. mul-1-neg N/A

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

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

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

    6. mul-1-neg N/A

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

    7. neg-sub0 N/A

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

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

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

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

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

    10. unpow2 N/A

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

    11. *-lowering-*.f64 96.3

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

11. Simplified 96.3%

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

12. Final simplification 96.3%

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

Alternative 15: 97.8% 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 48.9%

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

3. Taylor expanded in eps around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. +-rgt-identity N/A

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

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

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

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

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

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

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

   10. sin-lowering-sin.f64 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. --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 96.2

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

8. Simplified 96.2%

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

Alternative 16: 78.9% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ 0 - \varepsilon \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in eps around 0

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

   1. +-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 78.5

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

5. Simplified 78.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, 0 - \varepsilon, 0\right)} \]
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 78.5

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

7. Applied egg-rr 78.5%

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

8. Taylor expanded in x around 0

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

10. Simplified 77.4%

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

11. Final simplification 77.4%

    \[\leadsto 0 - \varepsilon \cdot x \]
12. Add Preprocessing

Alternative 17: 51.6% 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 48.9%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

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

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

   4. cos-lowering-cos.f64 46.2

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

5. Simplified 46.2%

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

6. Taylor expanded in eps around 0

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

8. Simplified 45.9%

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

   1. metadata-eval 45.9

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

10. Applied egg-rr 45.9%

    \[\leadsto \color{blue}{0} \]
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: 98.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024194 
(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)))