2sin (example 3.3)

Percentage Accurate: 62.5% → 99.9%

Time: 12.4s

Alternatives: 10

Speedup: 34.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. +-inverses N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. +-inverses N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

10. Applied rewrites 99.9%

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

11. Taylor expanded in eps around 0

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

13. Applied rewrites 99.5%

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

14. Final simplification 99.5%

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

Alternative 4: 99.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 5: 98.4% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(x \cdot \left(\varepsilon + x\right), -0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, -0.16666666666666666\right) \cdot \varepsilon, \varepsilon, 1\right)\right) \cdot \color{blue}{\varepsilon} \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.9%

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

12. Final simplification 98.9%

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

Alternative 6: 98.4% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

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

Alternative 7: 98.3% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 98.8%

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

10. Applied rewrites 98.8%

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

12. Applied rewrites 98.8%

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

Alternative 8: 98.3% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 98.8%

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

Alternative 9: 97.9% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(x \cdot \left(\varepsilon + x\right), -0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, -0.16666666666666666\right) \cdot \varepsilon, \varepsilon, 1\right)\right) \cdot \color{blue}{\varepsilon} \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.5%

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

Alternative 10: 97.9% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 98.8%

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

9. Taylor expanded in x around 0

   \[\leadsto 1 \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 98.5%

    \[\leadsto 1 \cdot \varepsilon \]
12. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024248 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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