2cos (problem 3.3.5)

Percentage Accurate: 53.3% → 99.9%

Time: 21.0s

Alternatives: 13

Speedup: 41.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{-2}\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)}, -2\right) \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \varepsilon\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right)\right), -2\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), -2\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)\right), -2\right) \]

   11. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(x \cdot 2\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot \left(2 \cdot \frac{1}{2}\right) + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot 1 + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   15. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \frac{1}{2} \cdot \varepsilon\right)\right)\right), -2\right) \]

   17. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\right), -2\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   19. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

7. Simplified 99.8%

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

   1. sin-sum N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right), -2\right) \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\varepsilon \cdot \frac{1}{2}\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\sin x, \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\varepsilon \cdot \frac{1}{2}\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\varepsilon \cdot \frac{1}{2}\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), \sin \left(\varepsilon \cdot \frac{1}{2}\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), \sin \left(\varepsilon \cdot \frac{1}{2}\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(\left(\frac{\varepsilon}{2}\right)\right)\right), \sin \left(\varepsilon \cdot \frac{1}{2}\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \sin \left(\varepsilon \cdot \frac{1}{2}\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   10. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   12. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{sin.f64}\left(\left(\frac{\varepsilon}{2}\right)\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\varepsilon, 2\right)\right), \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\varepsilon, 2\right)\right), \mathsf{*.f64}\left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(x + \frac{1}{\frac{2}{\varepsilon}}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* -2.0 (* (sin (* eps 0.5)) (sin (+ x (/ 1.0 (/ 2.0 eps)))))))

C

double code(double x, double eps) {
	return -2.0 * (sin((eps * 0.5)) * sin((x + (1.0 / (2.0 / eps)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (-2.0d0) * (sin((eps * 0.5d0)) * sin((x + (1.0d0 / (2.0d0 / eps)))))
end function

Java

public static double code(double x, double eps) {
	return -2.0 * (Math.sin((eps * 0.5)) * Math.sin((x + (1.0 / (2.0 / eps)))));
}

Python

def code(x, eps):
	return -2.0 * (math.sin((eps * 0.5)) * math.sin((x + (1.0 / (2.0 / eps)))))

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(sin(Float64(eps * 0.5)) * sin(Float64(x + Float64(1.0 / Float64(2.0 / eps))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = -2.0 * (sin((eps * 0.5)) * sin((x + (1.0 / (2.0 / eps)))));
end

Wolfram

code[x_, eps_] := N[(-2.0 * N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(x + N[(1.0 / N[(2.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{-2}\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)}, -2\right) \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \varepsilon\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right)\right), -2\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), -2\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)\right), -2\right) \]

   11. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(x \cdot 2\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot \left(2 \cdot \frac{1}{2}\right) + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot 1 + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   15. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \frac{1}{2} \cdot \varepsilon\right)\right)\right), -2\right) \]

   17. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\right), -2\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   19. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

7. Simplified 99.8%

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

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   3. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{\varepsilon + 0}{2}\right)\right)\right)\right), -2\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{\frac{2}{\varepsilon + 0}}\right)\right)\right)\right), -2\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{2}{\varepsilon + 0}\right)\right)\right)\right)\right), -2\right) \]

   6. +-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{2}{\varepsilon}\right)\right)\right)\right)\right), -2\right) \]

   7. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \varepsilon\right)\right)\right)\right)\right), -2\right) \]

9. Applied egg-rr 99.8%

   \[\leadsto \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(x + \color{blue}{\frac{1}{\frac{2}{\varepsilon}}}\right)\right) \cdot -2 \]

10. Final simplification 99.8%

    \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(x + \frac{1}{\frac{2}{\varepsilon}}\right)\right) \]
11. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{-2}\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)}, -2\right) \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \varepsilon\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right)\right), -2\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), -2\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)\right), -2\right) \]

   11. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(x \cdot 2\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot \left(2 \cdot \frac{1}{2}\right) + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot 1 + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   15. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \frac{1}{2} \cdot \varepsilon\right)\right)\right), -2\right) \]

   17. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\right), -2\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   19. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (-2.0d0) * (sin((x + (eps * 0.5d0))) * (eps * (0.5d0 + (eps * (eps * ((-0.020833333333333332d0) + ((eps * eps) * 0.00026041666666666666d0)))))))
end function

Java

public static double code(double x, double eps) {
	return -2.0 * (Math.sin((x + (eps * 0.5))) * (eps * (0.5 + (eps * (eps * (-0.020833333333333332 + ((eps * eps) * 0.00026041666666666666)))))));
}

Python

def code(x, eps):
	return -2.0 * (math.sin((x + (eps * 0.5))) * (eps * (0.5 + (eps * (eps * (-0.020833333333333332 + ((eps * eps) * 0.00026041666666666666)))))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = -2.0 * (sin((x + (eps * 0.5))) * (eps * (0.5 + (eps * (eps * (-0.020833333333333332 + ((eps * eps) * 0.00026041666666666666)))))));
end

Wolfram

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

Derivation

1. Initial program 49.5%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{-2}\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)}, -2\right) \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \varepsilon\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right)\right), -2\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), -2\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)\right), -2\right) \]

   11. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(x \cdot 2\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot \left(2 \cdot \frac{1}{2}\right) + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot 1 + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   15. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \frac{1}{2} \cdot \varepsilon\right)\right)\right), -2\right) \]

   17. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\right), -2\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   19. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

7. Simplified 99.8%

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

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{48} + \frac{1}{3840} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{48}, \left(\frac{1}{3840} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{48}, \left({\varepsilon}^{2} \cdot \frac{1}{3840}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \frac{1}{3840}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{3840}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   14. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{3840}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

10. Simplified 99.8%

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

11. Final simplification 99.8%

    \[\leadsto -2 \cdot \left(\sin \left(x + \varepsilon \cdot 0.5\right) \cdot \left(\varepsilon \cdot \left(0.5 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.020833333333333332 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.00026041666666666666\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 5: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (-2.0d0) * (sin((x + (eps * 0.5d0))) * (eps * (0.5d0 + ((-0.020833333333333332d0) * (eps * eps)))))
end function

Java

public static double code(double x, double eps) {
	return -2.0 * (Math.sin((x + (eps * 0.5))) * (eps * (0.5 + (-0.020833333333333332 * (eps * eps)))));
}

Python

def code(x, eps):
	return -2.0 * (math.sin((x + (eps * 0.5))) * (eps * (0.5 + (-0.020833333333333332 * (eps * eps)))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = -2.0 * (sin((x + (eps * 0.5))) * (eps * (0.5 + (-0.020833333333333332 * (eps * eps)))));
end

Wolfram

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

Derivation

1. Initial program 49.5%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{-2}\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)}, -2\right) \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \varepsilon\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right)\right), -2\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), -2\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)\right), -2\right) \]

   11. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(x \cdot 2\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot \left(2 \cdot \frac{1}{2}\right) + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot 1 + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   15. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \frac{1}{2} \cdot \varepsilon\right)\right)\right), -2\right) \]

   17. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\right), -2\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   19. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

7. Simplified 99.8%

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

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{48}, \left({\varepsilon}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{48}, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   5. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

10. Simplified 99.7%

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

11. Final simplification 99.7%

    \[\leadsto -2 \cdot \left(\sin \left(x + \varepsilon \cdot 0.5\right) \cdot \left(\varepsilon \cdot \left(0.5 + -0.020833333333333332 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 6: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(x + \frac{1}{\frac{2}{\varepsilon}}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* -2.0 (* (* eps 0.5) (sin (+ x (/ 1.0 (/ 2.0 eps)))))))

C

double code(double x, double eps) {
	return -2.0 * ((eps * 0.5) * sin((x + (1.0 / (2.0 / eps)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (-2.0d0) * ((eps * 0.5d0) * sin((x + (1.0d0 / (2.0d0 / eps)))))
end function

Java

public static double code(double x, double eps) {
	return -2.0 * ((eps * 0.5) * Math.sin((x + (1.0 / (2.0 / eps)))));
}

Python

def code(x, eps):
	return -2.0 * ((eps * 0.5) * math.sin((x + (1.0 / (2.0 / eps)))))

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(Float64(eps * 0.5) * sin(Float64(x + Float64(1.0 / Float64(2.0 / eps))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = -2.0 * ((eps * 0.5) * sin((x + (1.0 / (2.0 / eps)))));
end

Wolfram

code[x_, eps_] := N[(-2.0 * N[(N[(eps * 0.5), $MachinePrecision] * N[Sin[N[(x + N[(1.0 / N[(2.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{-2}\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)}, -2\right) \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \varepsilon\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right)\right), -2\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), -2\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)\right), -2\right) \]

   11. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(x \cdot 2\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot \left(2 \cdot \frac{1}{2}\right) + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot 1 + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   15. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \frac{1}{2} \cdot \varepsilon\right)\right)\right), -2\right) \]

   17. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\right), -2\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   19. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

7. Simplified 99.8%

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

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   3. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{\varepsilon + 0}{2}\right)\right)\right)\right), -2\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{\frac{2}{\varepsilon + 0}}\right)\right)\right)\right), -2\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{2}{\varepsilon + 0}\right)\right)\right)\right)\right), -2\right) \]

   6. +-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{2}{\varepsilon}\right)\right)\right)\right)\right), -2\right) \]

   7. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \varepsilon\right)\right)\right)\right)\right), -2\right) \]

9. Applied egg-rr 99.8%

   \[\leadsto \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(x + \color{blue}{\frac{1}{\frac{2}{\varepsilon}}}\right)\right) \cdot -2 \]

10. Taylor expanded in eps around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \varepsilon\right)\right)\right)\right)\right), -2\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f64 99.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \varepsilon\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \varepsilon\right)\right)\right)\right)\right), -2\right) \]

12. Simplified 99.5%

    \[\leadsto \left(\color{blue}{\left(0.5 \cdot \varepsilon\right)} \cdot \sin \left(x + \frac{1}{\frac{2}{\varepsilon}}\right)\right) \cdot -2 \]

13. Final simplification 99.5%

    \[\leadsto -2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(x + \frac{1}{\frac{2}{\varepsilon}}\right)\right) \]
14. Add Preprocessing

Alternative 7: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{-2}\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)}, -2\right) \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \varepsilon\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), -2\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right)\right), -2\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), -2\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)\right), -2\right) \]

   11. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\left(x \cdot 2\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot \left(2 \cdot \frac{1}{2}\right) + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x \cdot 1 + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   15. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \varepsilon \cdot \frac{1}{2}\right)\right)\right), -2\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(x + \frac{1}{2} \cdot \varepsilon\right)\right)\right), -2\right) \]

   17. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\right), -2\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   19. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

7. Simplified 99.8%

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

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

   2. *-lowering-*.f64 99.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), -2\right) \]

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 8: 79.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0 - \sin x \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 0.0 (* (sin x) eps)))

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return 0.0 - (Math.sin(x) * eps);
}

Python

def code(x, eps):
	return 0.0 - (math.sin(x) * eps)

Julia

function code(x, eps)
	return Float64(0.0 - Float64(sin(x) * eps))
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(0.0 - N[(N[Sin[x], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(-1 \cdot \varepsilon\right)}\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{-1} \cdot \varepsilon\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\mathsf{neg}\left(\varepsilon\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(0 - \color{blue}{\varepsilon}\right)\right) \]

   7. --lowering--.f64 79.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{\_.f64}\left(0, \color{blue}{\varepsilon}\right)\right) \]

5. Simplified 79.0%

   \[\leadsto \color{blue}{\sin x \cdot \left(0 - \varepsilon\right)} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\mathsf{neg}\left(\varepsilon\right)\right)\right) \]

   2. neg-lowering-neg.f64 79.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{neg.f64}\left(\varepsilon\right)\right) \]

7. Applied egg-rr 79.0%

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

8. Final simplification 79.0%

   \[\leadsto 0 - \sin x \cdot \varepsilon \]
9. Add Preprocessing

Alternative 9: 79.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(x \cdot \left(-1 - \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (*
   x
   (-
    -1.0
    (*
     (* x x)
     (+
      -0.16666666666666666
      (*
       (* x x)
       (+ 0.008333333333333333 (* (* x x) -0.0001984126984126984)))))))))

C

double code(double x, double eps) {
	return eps * (x * (-1.0 - ((x * x) * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (x * ((-1.0d0) - ((x * x) * ((-0.16666666666666666d0) + ((x * x) * (0.008333333333333333d0 + ((x * x) * (-0.0001984126984126984d0))))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (x * (-1.0 - ((x * x) * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))));
}

Python

def code(x, eps):
	return eps * (x * (-1.0 - ((x * x) * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(x * Float64(-1.0 - Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.0001984126984126984))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (x * (-1.0 - ((x * x) * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(x * N[(-1.0 - N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(-1 \cdot \varepsilon\right)}\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{-1} \cdot \varepsilon\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\mathsf{neg}\left(\varepsilon\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(0 - \color{blue}{\varepsilon}\right)\right) \]

   7. --lowering--.f64 79.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{\_.f64}\left(0, \color{blue}{\varepsilon}\right)\right) \]

5. Simplified 79.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \varepsilon\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   17. *-lowering-*.f64 78.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

8. Simplified 78.5%

   \[\leadsto \color{blue}{\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)} \cdot \left(0 - \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. mul-1-neg N/A

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

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

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)}\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)}\right)\right) \]

    6. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

    8. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(-1 - \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

    9. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

    13. sub-neg N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

    15. +-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

11. Simplified 78.5%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot \left(-1 - \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)} \]
12. Add Preprocessing

Alternative 10: 79.1% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (x * ((-1.0d0) - (x * (x * (-0.16666666666666666d0)))))
end function

Java

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

Python

def code(x, eps):
	return eps * (x * (-1.0 - (x * (x * -0.16666666666666666))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * (x * (-1.0 - (x * (x * -0.16666666666666666))));
end

Wolfram

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

Derivation

1. Initial program 49.5%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(-1 \cdot \varepsilon\right)}\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{-1} \cdot \varepsilon\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\mathsf{neg}\left(\varepsilon\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(0 - \color{blue}{\varepsilon}\right)\right) \]

   7. --lowering--.f64 79.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{\_.f64}\left(0, \color{blue}{\varepsilon}\right)\right) \]

5. Simplified 79.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \varepsilon\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   17. *-lowering-*.f64 78.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

8. Simplified 78.5%

   \[\leadsto \color{blue}{\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)} \cdot \left(0 - \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. mul-1-neg N/A

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

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

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)}\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)}\right)\right) \]

    6. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

    8. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(-1 - \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

    9. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

    13. sub-neg N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

    15. +-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

11. Simplified 78.5%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot \left(-1 - \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)} \]

12. Taylor expanded in x around 0

    \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right) \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(x \cdot \color{blue}{\left(x \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(x \cdot \left(\frac{-1}{6} \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f64 78.5%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

14. Simplified 78.5%

    \[\leadsto \varepsilon \cdot \left(x \cdot \left(-1 - \color{blue}{x \cdot \left(x \cdot -0.16666666666666666\right)}\right)\right) \]
15. Add Preprocessing

Alternative 11: 78.8% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ 0 - x \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 0.0 (* x eps)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0 - (x * eps)

Julia

function code(x, eps)
	return Float64(0.0 - Float64(x * eps))
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(0.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(-1 \cdot \varepsilon\right)}\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{-1} \cdot \varepsilon\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\mathsf{neg}\left(\varepsilon\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(0 - \color{blue}{\varepsilon}\right)\right) \]

   7. --lowering--.f64 79.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{\_.f64}\left(0, \color{blue}{\varepsilon}\right)\right) \]

5. Simplified 79.0%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\varepsilon \cdot x} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\varepsilon \cdot x\right)}\right) \]

   4. *-lowering-*.f64 78.3%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\varepsilon, \color{blue}{x}\right)\right) \]

8. Simplified 78.3%

   \[\leadsto \color{blue}{0 - \varepsilon \cdot x} \]

9. Final simplification 78.3%

   \[\leadsto 0 - x \cdot \varepsilon \]
10. Add Preprocessing

Alternative 12: 53.1% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + -1 \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\cos \varepsilon, \color{blue}{-1}\right) \]

   4. cos-lowering-cos.f64 48.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\varepsilon\right), -1\right) \]

5. Simplified 48.3%

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

6. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{\frac{-1}{2}} \]

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

   7. *-lowering-*.f64 49.8%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\frac{-1}{2}}\right)\right) \]

8. Simplified 49.8%

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

Alternative 13: 51.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 49.5%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + -1 \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\cos \varepsilon, \color{blue}{-1}\right) \]

   4. cos-lowering-cos.f64 48.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\varepsilon\right), -1\right) \]

5. Simplified 48.3%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, -1\right) \]
7. Step-by-step derivation

8. Simplified 48.3%

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

   1. metadata-eval 48.3%

      \[\leadsto 0 \]

10. Applied egg-rr 48.3%

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

Developer Target 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

  :alt
  (! :herbie-platform default (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

  (- (cos (+ x eps)) (cos x)))