2cos (problem 3.3.5)

Percentage Accurate: 51.9% → 99.5%

Time: 18.2s

Alternatives: 13

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: 51.9% 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: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. *-commutative N/A

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

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

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

   10. distribute-lft-neg-out N/A

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

   11. lift-neg.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 99.9

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

7. Applied egg-rr 99.9%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

Alternative 3: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0

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

8. Simplified 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 99.4

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0

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

8. Simplified 99.3%

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

Alternative 5: 98.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 - \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (* eps -0.5)
   (fma
    (fma
     x
     (* x (fma (* x x) -0.0001984126984126984 0.008333333333333333))
     -0.16666666666666666)
    (* x (* x x))
    x))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) - fma(fma(x, (x * fma((x * x), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (x * (x * x)), x));
}

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) - fma(fma(x, Float64(x * fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(x * Float64(x * x)), x)))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] - N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0

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

8. Simplified 99.3%

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

9. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} - \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} - x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} - \color{blue}{\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x\right)}\right) \]

    3. *-commutative N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} - \left(\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{2}\right)} \cdot x + 1 \cdot x\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} - \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right)} + 1 \cdot x\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} - \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + 1 \cdot x\right)\right) \]

    6. unpow3 N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} - \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{x}^{3}} + 1 \cdot x\right)\right) \]

    7. *-lft-identity N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} - \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{3} + \color{blue}{x}\right)\right) \]

    8. lower-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2} - \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}, {x}^{3}, x\right)}\right) \]

11. Simplified 98.5%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}\right) \]
12. Add Preprocessing

Alternative 6: 98.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

8. Simplified 98.3%

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

9. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

    4. *-commutative N/A

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

    5. unpow2 N/A

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 98.3

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

11. Simplified 98.3%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. +-commutative N/A

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

    7. lift-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. associate-*r* N/A

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

    11. lower-fma.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-*.f64 98.5

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

13. Applied egg-rr 98.5%

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

Alternative 7: 98.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

8. Simplified 98.3%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

    4. distribute-lft-neg-in N/A

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

    5. distribute-neg-in N/A

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

    6. +-commutative N/A

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

    7. mul-1-neg N/A

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 N/A

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

    11. mul-1-neg N/A

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

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

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

    13. +-commutative N/A

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

    14. distribute-neg-in N/A

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

    15. distribute-lft-neg-in N/A

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

    16. metadata-eval N/A

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

    17. sub-neg N/A

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

    18. lower-*.f64 N/A

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

11. Simplified 98.3%

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

Alternative 8: 98.2% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0

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

8. Simplified 99.3%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. *-rgt-identity N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 98.3

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

11. Simplified 98.3%

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

12. Final simplification 98.3%

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

Alternative 9: 98.2% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0

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

8. Simplified 99.3%

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

9. Taylor expanded in x around 0

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

    1. unpow2 N/A

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

    2. associate-*r* N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. distribute-rgt-in N/A

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

    7. metadata-eval N/A

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

    8. sub-neg N/A

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

    9. *-commutative N/A

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

    10. associate-*l* N/A

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

    11. distribute-rgt-out N/A

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

    12. lower-*.f64 N/A

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

    13. +-commutative N/A

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

    14. lower-fma.f64 N/A

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

11. Simplified 98.3%

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

12. Final simplification 98.3%

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

Alternative 10: 97.8% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

8. Simplified 98.3%

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

9. Taylor expanded in x around 0

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 97.9

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

11. Simplified 97.9%

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

Alternative 11: 97.8% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 97.8

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

8. Simplified 97.8%

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

Alternative 12: 78.6% accurate, 25.9× speedup

Math

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

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(x * Float64(-eps))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 80.9

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

5. Simplified 80.9%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 79.7

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

8. Simplified 79.7%

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

9. Final simplification 79.7%

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

Alternative 13: 50.5% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 52.0%

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

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

   4. lower-cos.f64 51.3

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

5. Simplified 51.3%

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

6. Taylor expanded in eps around 0

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

8. Simplified 51.3%

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

   1. metadata-eval 51.3

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

10. Applied egg-rr 51.3%

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

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024208 
(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)))