2cos (problem 3.3.5)

Percentage Accurate: 52.4% → 99.7%

Time: 17.2s

Alternatives: 12

Speedup: 25.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.4% 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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 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. *-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

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

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

   7. metadata-eval N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 99.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

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

Alternative 4: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 53.3

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

5. Applied rewrites 53.3%

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

6. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-sin.f64 99.4

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

8. Applied rewrites 99.4%

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

10. Applied rewrites 99.5%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 99.0%

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

Alternative 5: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 53.3

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

5. Applied rewrites 53.3%

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

6. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-sin.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.9%

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

Alternative 6: 98.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 53.3

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

5. Applied rewrites 53.3%

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

6. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-sin.f64 99.4

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

8. Applied rewrites 99.4%

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

10. Applied rewrites 99.5%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.3%

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

14. Final simplification 98.3%

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

Alternative 7: 98.1% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 53.3

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

5. Applied rewrites 53.3%

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

6. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-sin.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.2%

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

Alternative 8: 97.8% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.9%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 97.9%

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

12. Final simplification 97.9%

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

Alternative 9: 97.8% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.9%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 97.9%

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

12. Final simplification 97.9%

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

Alternative 10: 97.7% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 53.3

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

5. Applied rewrites 53.3%

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

6. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-sin.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 97.8%

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

Alternative 11: 78.7% accurate, 25.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 80.6

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

5. Applied rewrites 80.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 79.7%

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

Alternative 12: 51.0% accurate, 51.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 53.3

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

5. Applied rewrites 53.3%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 53.1%

   \[\leadsto 1 - 1 \]
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 2024240 
(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)))