2cos (problem 3.3.5)

Percentage Accurate: 53.7% → 99.8%

Time: 18.1s

Alternatives: 15

Speedup: 14.8×

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: 53.7% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.020833333333333332, 0.25\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  -2.0
  (+
   (* (sin (* eps 0.5)) (* (cos (* eps 0.5)) (sin x)))
   (* eps (* eps (* (cos x) (fma eps (* eps -0.020833333333333332) 0.25)))))))

C

double code(double x, double eps) {
	return -2.0 * ((sin((eps * 0.5)) * (cos((eps * 0.5)) * sin(x))) + (eps * (eps * (cos(x) * fma(eps, (eps * -0.020833333333333332), 0.25)))));
}

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(Float64(sin(Float64(eps * 0.5)) * Float64(cos(Float64(eps * 0.5)) * sin(x))) + Float64(eps * Float64(eps * Float64(cos(x) * fma(eps, Float64(eps * -0.020833333333333332), 0.25))))))
end

Wolfram

code[x_, eps_] := N[(-2.0 * N[(N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * N[(N[Cos[x], $MachinePrecision] * N[(eps * N[(eps * -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in eps around 0

    \[\leadsto \left({\varepsilon}^{2} \cdot \left(\frac{-1}{48} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{4} \cdot \cos x\right) + \color{blue}{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)} \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot -2 \]
11. Step-by-step derivation

12. Applied rewrites 99.5%

    \[\leadsto \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.020833333333333332, 0.25\right)\right)\right) + \color{blue}{\left(\cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x\right)} \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot -2 \]

13. Final simplification 99.5%

    \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.020833333333333332, 0.25\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 4: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 5: 99.6% accurate, 1.3× speedup

Math

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

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 (fma eps 0.5 x)))))

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(fma(eps, 0.5, x)));
}

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(fma(eps, 0.5, x))))
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[(eps * 0.5 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 99.3

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

10. Applied rewrites 99.3%

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

11. Final simplification 99.3%

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

Alternative 6: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.3%

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

11. Final simplification 99.3%

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

Alternative 7: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.2%

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

11. Final simplification 99.2%

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

Alternative 8: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 98.7%

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

11. Final simplification 98.7%

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

Alternative 9: 98.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 48.7

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

4. Applied rewrites 48.7%

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

5. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-sin.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.2%

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

Alternative 10: 98.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 48.7

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

4. Applied rewrites 48.7%

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

5. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-sin.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.1%

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

Alternative 11: 98.2% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 48.7

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

4. Applied rewrites 48.7%

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

5. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-sin.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 96.9%

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

Alternative 12: 97.7% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 48.7

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

4. Applied rewrites 48.7%

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

5. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-sin.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 96.3%

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

11. Final simplification 96.3%

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

Alternative 13: 97.7% 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 48.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 48.7

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

4. Applied rewrites 48.7%

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

5. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-sin.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 96.3%

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

Alternative 14: 53.6% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in x around 0

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

   1. 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. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-cos.f64 47.1

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

5. Applied rewrites 47.1%

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

6. Taylor expanded in eps around 0

   \[\leadsto \frac{-1}{2} \cdot \color{blue}{{\varepsilon}^{2}} \]
7. Step-by-step derivation

8. Applied rewrites 48.7%

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

Alternative 15: 52.3% accurate, 51.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in x around 0

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

   1. 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. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-cos.f64 47.1

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

5. Applied rewrites 47.1%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 46.9%

   \[\leadsto -1 + 1 \]
9. 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 2024228 
(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)))