2sin (example 3.3)

Percentage Accurate: 62.6% → 100.0%

Time: 15.9s

Alternatives: 13

Speedup: 207.0×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := 0.5 \cdot \left(x \cdot 2\right)\\ 2 \cdot \left(t\_0 \cdot \left(\cos t\_1 \cdot \cos \left(\varepsilon \cdot 0.5\right) - \sin t\_1 \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))) (t_1 (* 0.5 (* x 2.0))))
   (* 2.0 (* t_0 (- (* (cos t_1) (cos (* eps 0.5))) (* (sin t_1) t_0))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double t_1 = 0.5 * (x * 2.0);
	return 2.0 * (t_0 * ((cos(t_1) * cos((eps * 0.5))) - (sin(t_1) * t_0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin((eps * 0.5d0))
    t_1 = 0.5d0 * (x * 2.0d0)
    code = 2.0d0 * (t_0 * ((cos(t_1) * cos((eps * 0.5d0))) - (sin(t_1) * t_0)))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double t_1 = 0.5 * (x * 2.0);
	return 2.0 * (t_0 * ((Math.cos(t_1) * Math.cos((eps * 0.5))) - (Math.sin(t_1) * t_0)));
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	t_1 = 0.5 * (x * 2.0)
	return 2.0 * (t_0 * ((math.cos(t_1) * math.cos((eps * 0.5))) - (math.sin(t_1) * t_0)))

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	t_1 = Float64(0.5 * Float64(x * 2.0))
	return Float64(2.0 * Float64(t_0 * Float64(Float64(cos(t_1) * cos(Float64(eps * 0.5))) - Float64(sin(t_1) * t_0))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin((eps * 0.5));
	t_1 = 0.5 * (x * 2.0);
	tmp = 2.0 * (t_0 * ((cos(t_1) * cos((eps * 0.5))) - (sin(t_1) * t_0)));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(t$95$0 * N[(N[(N[Cos[t$95$1], $MachinePrecision] * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.1%

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

   1. lift-+.f64 N/A

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

   2. diff-sin N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

4. Applied egg-rr 99.9%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. +-rgt-identity N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. cos-sum N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. +-rgt-identity N/A

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

   17. lift-sin.f64 N/A

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos \left(0.5 \cdot \left(x \cdot 2\right)\right) \cdot \cos \left(\varepsilon \cdot 0.5\right) - \sin \left(0.5 \cdot \left(x \cdot 2\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. lift-+.f64 N/A

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

   2. diff-sin N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.9

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 99.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. lift-+.f64 N/A

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

   2. diff-sin N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 99.9

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

7. Simplified 99.9%

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

9. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.020833333333333332, 0.5\right) \cdot \left(\varepsilon \cdot \left(2 \cdot \cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)\right)\right)} \]
10. Add Preprocessing

Alternative 4: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. lift-+.f64 N/A

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

   2. diff-sin N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 99.9

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

7. Simplified 99.9%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 99.7

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

10. Simplified 99.7%

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

11. Taylor expanded in eps around inf

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

    1. +-commutative N/A

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

    2. distribute-rgt-in N/A

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

    3. *-commutative N/A

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

    4. associate-*l* N/A

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

    5. metadata-eval N/A

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

    6. *-rgt-identity N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. metadata-eval N/A

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

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

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

    11. lower-cos.f64 N/A

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

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

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

    13. metadata-eval N/A

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

    14. +-commutative N/A

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

    15. lower-fma.f64 99.7

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

13. Simplified 99.7%

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

Alternative 5: 98.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

5. Simplified 99.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
6. Add Preprocessing

Alternative 6: 98.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon + \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + -0.5 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (+
   eps
   (*
    (* eps (fma (* x x) -0.001388888888888889 0.041666666666666664))
    (* (* x x) (* x x))))
  (* -0.5 (* eps (* x x)))))

C

double code(double x, double eps) {
	return (eps + ((eps * fma((x * x), -0.001388888888888889, 0.041666666666666664)) * ((x * x) * (x * x)))) + (-0.5 * (eps * (x * x)));
}

Julia

function code(x, eps)
	return Float64(Float64(eps + Float64(Float64(eps * fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664)) * Float64(Float64(x * x) * Float64(x * x)))) + Float64(-0.5 * Float64(eps * Float64(x * x))))
end

Wolfram

code[x_, eps_] := N[(N[(eps + N[(N[(eps * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right)\right) + \varepsilon} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right), \varepsilon\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right), \varepsilon\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right), \varepsilon\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right) + \frac{-1}{2} \cdot \varepsilon}, \varepsilon\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon, \frac{-1}{2} \cdot \varepsilon\right)}, \varepsilon\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot \varepsilon + \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \varepsilon + \frac{-1}{720} \cdot \color{blue}{\left({x}^{2} \cdot \varepsilon\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \varepsilon + \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right) \cdot \varepsilon}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   12. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right), \color{blue}{\varepsilon \cdot \frac{-1}{2}}\right), \varepsilon\right) \]

   20. lower-*.f64 98.6

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), \color{blue}{\varepsilon \cdot -0.5}\right), \varepsilon\right) \]

8. Simplified 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), \varepsilon \cdot -0.5\right), \varepsilon\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{720} + \frac{1}{24}\right)\right) + \varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \]

   2. lift-*.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{720} + \frac{1}{24}\right)\right) + \varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \]

   3. lift-*.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right)\right) + \varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \]

   4. lift-fma.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)}\right) + \varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \]

   5. lift-*.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right)} + \varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \]

   6. lift-*.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right) + \color{blue}{\varepsilon \cdot \frac{-1}{2}}\right) + \varepsilon \]

   7. lift-fma.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right), \varepsilon \cdot \frac{-1}{2}\right)} + \varepsilon \]

   8. +-commutative N/A

      \[\leadsto \color{blue}{\varepsilon + \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right), \varepsilon \cdot \frac{-1}{2}\right)} \]

   9. lift-fma.f64 N/A

      \[\leadsto \varepsilon + \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right) + \varepsilon \cdot \frac{-1}{2}\right)} \]

   10. distribute-lft-in N/A

       \[\leadsto \varepsilon + \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right)\right) + \left(x \cdot x\right) \cdot \left(\varepsilon \cdot \frac{-1}{2}\right)\right)} \]

   11. associate-+r+ N/A

       \[\leadsto \color{blue}{\left(\varepsilon + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right)\right)\right) + \left(x \cdot x\right) \cdot \left(\varepsilon \cdot \frac{-1}{2}\right)} \]

10. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{\left(\varepsilon + \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + -0.5 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)} \]
11. Add Preprocessing

Alternative 7: 98.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right)\right)\right), x, \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot -0.5, \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (*
   x
   (*
    (* x x)
    (* eps (fma (* x x) -0.001388888888888889 0.041666666666666664))))
  x
  (fma eps (* (* x x) -0.5) eps)))

C

double code(double x, double eps) {
	return fma((x * ((x * x) * (eps * fma((x * x), -0.001388888888888889, 0.041666666666666664)))), x, fma(eps, ((x * x) * -0.5), eps));
}

Julia

function code(x, eps)
	return fma(Float64(x * Float64(Float64(x * x) * Float64(eps * fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664)))), x, fma(eps, Float64(Float64(x * x) * -0.5), eps))
end

Wolfram

code[x_, eps_] := N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(eps * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(eps * N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right)\right) + \varepsilon} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right), \varepsilon\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right), \varepsilon\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right), \varepsilon\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right) + \frac{-1}{2} \cdot \varepsilon}, \varepsilon\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon, \frac{-1}{2} \cdot \varepsilon\right)}, \varepsilon\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot \varepsilon + \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \varepsilon + \frac{-1}{720} \cdot \color{blue}{\left({x}^{2} \cdot \varepsilon\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \varepsilon + \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right) \cdot \varepsilon}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   12. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right), \color{blue}{\varepsilon \cdot \frac{-1}{2}}\right), \varepsilon\right) \]

   20. lower-*.f64 98.6

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), \color{blue}{\varepsilon \cdot -0.5}\right), \varepsilon\right) \]

8. Simplified 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), \varepsilon \cdot -0.5\right), \varepsilon\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{720} + \frac{1}{24}\right)\right) + \varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \]

   2. lift-*.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{720} + \frac{1}{24}\right)\right) + \varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \]

   3. lift-*.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right)\right) + \varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \]

   4. lift-fma.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)}\right) + \varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \]

   5. lift-*.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right)} + \varepsilon \cdot \frac{-1}{2}\right) + \varepsilon \]

   6. lift-*.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right) + \color{blue}{\varepsilon \cdot \frac{-1}{2}}\right) + \varepsilon \]

   7. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot x\right) + \left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \left(x \cdot x\right)\right)} + \varepsilon \]

   8. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot x\right) + \left(\left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \left(x \cdot x\right) + \varepsilon\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left(\left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \left(x \cdot x\right) + \varepsilon\right) \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right)\right) \cdot x\right) \cdot x} + \left(\left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \left(x \cdot x\right) + \varepsilon\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right)\right)\right) \cdot x, x, \left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \left(x \cdot x\right) + \varepsilon\right)} \]

10. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right)\right)\right) \cdot x, x, \mathsf{fma}\left(\varepsilon, -0.5 \cdot \left(x \cdot x\right), \varepsilon\right)\right)} \]

11. Final simplification 98.6%

    \[\leadsto \mathsf{fma}\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right)\right)\right), x, \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot -0.5, \varepsilon\right)\right) \]
12. Add Preprocessing

Alternative 8: 98.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (* eps (* x x))
  (fma x (* x (fma (* x x) -0.001388888888888889 0.041666666666666664)) -0.5)
  eps))

C

double code(double x, double eps) {
	return fma((eps * (x * x)), fma(x, (x * fma((x * x), -0.001388888888888889, 0.041666666666666664)), -0.5), eps);
}

Julia

function code(x, eps)
	return fma(Float64(eps * Float64(x * x)), fma(x, Float64(x * fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664)), -0.5), eps)
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right)\right) + \varepsilon} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right), \varepsilon\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right), \varepsilon\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right), \varepsilon\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right) + \frac{-1}{2} \cdot \varepsilon}, \varepsilon\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon, \frac{-1}{2} \cdot \varepsilon\right)}, \varepsilon\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot \varepsilon + \frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \varepsilon + \frac{-1}{720} \cdot \color{blue}{\left({x}^{2} \cdot \varepsilon\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \varepsilon + \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right) \cdot \varepsilon}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   12. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right), \color{blue}{\varepsilon \cdot \frac{-1}{2}}\right), \varepsilon\right) \]

   20. lower-*.f64 98.6

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), \color{blue}{\varepsilon \cdot -0.5}\right), \varepsilon\right) \]

8. Simplified 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), \varepsilon \cdot -0.5\right), \varepsilon\right)} \]

9. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \]

    2. distribute-lft-in N/A

       \[\leadsto \color{blue}{\varepsilon \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) + \varepsilon \cdot 1} \]

    3. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\varepsilon \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} + \varepsilon \cdot 1 \]

    4. *-rgt-identity N/A

       \[\leadsto \left(\varepsilon \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + \color{blue}{\varepsilon} \]

    5. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot {x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, \varepsilon\right)} \]

11. Simplified 98.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), \varepsilon\right)} \]
12. Add Preprocessing

Alternative 9: 98.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   (* x x)
   (fma (* x x) (fma (* x x) -0.001388888888888889 0.041666666666666664) -0.5)
   1.0)))

C

double code(double x, double eps) {
	return eps * fma((x * x), fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
}

Julia

function code(x, eps)
	return Float64(eps * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \]

   3. unpow2 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \]

   5. sub-neg N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

   6. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) \]

   8. unpow2 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \]

   10. +-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \]

   11. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \]

   13. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

   14. lower-*.f64 98.6

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \]

8. Simplified 98.6%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \]
9. Add Preprocessing

Alternative 10: 98.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (fma x (* x (fma 0.041666666666666664 (* x x) -0.5)) 1.0)))

C

double code(double x, double eps) {
	return eps * fma(x, (x * fma(0.041666666666666664, (x * x), -0.5)), 1.0);
}

Julia

function code(x, eps)
	return Float64(eps * fma(x, Float64(x * fma(0.041666666666666664, Float64(x * x), -0.5)), 1.0))
end

Wolfram

code[x_, eps_] := N[(eps * N[(x * N[(x * N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} \]

   2. unpow2 N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right) \]

   3. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} + 1\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right), 1\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}, 1\right) \]

   6. sub-neg N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \]

   7. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}\right), 1\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right)}, 1\right) \]

   9. unpow2 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right), 1\right) \]

   10. lower-*.f64 98.4

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.041666666666666664, \color{blue}{x \cdot x}, -0.5\right), 1\right) \]

8. Simplified 98.4%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), 1\right)} \]
9. Add Preprocessing

Alternative 11: 98.2% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. distribute-lft-out N/A

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

   10. +-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 98.4

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

8. Simplified 98.4%

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

Alternative 12: 98.2% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 98.3

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

8. Simplified 98.3%

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

Alternative 13: 97.7% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0

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

8. Simplified 97.5%

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

   1. *-rgt-identity 97.5

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

10. Applied egg-rr 97.5%

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

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (2.0 * cos((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 * cos((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \left(\cos \varepsilon - 1\right) + \cos x \cdot \sin \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ (* (sin x) (- (cos eps) 1.0)) (* (cos x) (sin eps))))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return (math.sin(x) * (math.cos(eps) - 1.0)) + (math.cos(x) * math.sin(eps))

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (cos((0.5d0 * (eps - ((-2.0d0) * x)))) * sin((0.5d0 * eps))) * 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (Math.cos((0.5 * (eps - (-2.0 * x)))) * Math.sin((0.5 * eps))) * 2.0;
}

Python

def code(x, eps):
	return (math.cos((0.5 * (eps - (-2.0 * x)))) * math.sin((0.5 * eps))) * 2.0

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (cos((0.5 * (eps - (-2.0 * x)))) * sin((0.5 * eps))) * 2.0;
end

Wolfram

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

Reproduce

herbie shell --seed 2024210 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :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 (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform default (+ (* (sin x) (- (cos eps) 1)) (* (cos x) (sin eps))))

  :alt
  (! :herbie-platform default (* (cos (* 1/2 (- eps (* -2 x)))) (sin (* 1/2 eps)) 2))

  (- (sin (+ x eps)) (sin x)))