2sin (example 3.3)

Percentage Accurate: 62.0% → 99.9%

Time: 12.8s

Alternatives: 14

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: 62.0% 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

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

FPCore

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

Derivation

1. Initial program 62.6%

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

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

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

   17. cos-neg-rev N/A

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

   18. lower-cos.f64 N/A

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

   19. distribute-neg-frac2 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. metadata-eval N/A

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

   7. lift-fma.f64 N/A

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

   8. count-2-rev N/A

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. lower-sin.f64 N/A

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

   12. +-commutative N/A

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

   13. associate-+r+ N/A

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

   14. count-2-rev N/A

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

   15. +-commutative N/A

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

   16. lift-fma.f64 N/A

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

   17. div-add-rev N/A

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

   18. lower-/.f64 N/A

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

9. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

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

   17. cos-neg-rev N/A

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

   18. lower-cos.f64 N/A

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

   19. distribute-neg-frac2 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x 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. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. distribute-neg-in N/A

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

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

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. mul-1-neg N/A

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

   16. remove-double-neg N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-sin.f64 N/A

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

   19. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 3: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

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

   17. cos-neg-rev N/A

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

   18. lower-cos.f64 N/A

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

   19. distribute-neg-frac2 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 4: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

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

   17. cos-neg-rev N/A

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

   18. lower-cos.f64 N/A

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

   19. distribute-neg-frac2 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 5: 99.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (* 0.5 eps) 2.0) (sin (/ (+ (PI) (fma x 2.0 eps)) 2.0))))

Derivation

1. Initial program 62.6%

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

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

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

   17. cos-neg-rev N/A

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

   18. lower-cos.f64 N/A

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

   19. distribute-neg-frac2 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. distribute-frac-neg N/A

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

   8. cos-neg-rev N/A

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

   9. sin-+PI/2 N/A

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

   10. lift-PI.f64 N/A

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

   11. div-add N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-fma.f64 99.3

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

9. Applied rewrites 99.3%

   \[\leadsto \left(\left(0.5 \cdot \varepsilon\right) \cdot 2\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right) + \mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right)} \]
10. Add Preprocessing

Alternative 6: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

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

   17. cos-neg-rev N/A

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

   18. lower-cos.f64 N/A

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

   19. distribute-neg-frac2 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around inf

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

   1. cos-neg-rev N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. distribute-neg-in N/A

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

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

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. mul-1-neg N/A

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

   12. remove-double-neg N/A

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

   13. +-commutative N/A

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

   14. lower-cos.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f64 99.3

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

10. Applied rewrites 99.3%

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

Alternative 7: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.6%

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

Alternative 8: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 62.6%

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

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

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

   17. cos-neg-rev N/A

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

   18. lower-cos.f64 N/A

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

   19. distribute-neg-frac2 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. metadata-eval N/A

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

   7. lift-fma.f64 N/A

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

   8. count-2-rev N/A

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. lower-sin.f64 N/A

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

   12. +-commutative N/A

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

   13. associate-+r+ N/A

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

   14. count-2-rev N/A

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

   15. +-commutative N/A

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

   16. lift-fma.f64 N/A

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

   17. div-add-rev N/A

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

   18. lower-/.f64 N/A

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-PI.f64 98.5

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

9. Applied rewrites 98.5%

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

Alternative 9: 99.1% 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 62.6%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 98.5

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

5. Applied rewrites 98.5%

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

Alternative 10: 98.5% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.4%

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

Alternative 11: 98.5% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.4%

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

Alternative 12: 98.5% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.3%

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

Alternative 13: 98.4% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.2%

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

Alternative 14: 98.0% 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 62.6%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.4%

   \[\leadsto 1 \cdot \varepsilon \]
9. 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 2024337 
(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)))