2cos (problem 3.3.5)

Percentage Accurate: 52.9% → 99.8%

Time: 21.6s

Alternatives: 7

Speedup: 51.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\cos x \cdot {\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (-
  (/ (* (cos x) (pow (sin eps) 2.0)) (- -1.0 (cos eps)))
  (* (sin eps) (sin x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. sub-neg 54.3%

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

   2. cos-sum 54.5%

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

   3. associate-+l- 54.4%

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

   4. add-sqr-sqrt 53.7%

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

   5. associate-*l* 53.7%

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

   6. fma-neg 53.7%

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

4. Applied egg-rr 53.7%

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

5. Taylor expanded in x around inf 54.4%

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

   1. associate--r+ 80.6%

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

   2. sub-neg 80.6%

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

   3. *-rgt-identity 80.6%

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

   4. cancel-sign-sub-inv 80.6%

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

   5. *-commutative 80.6%

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

   6. distribute-lft-out-- 80.6%

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

   7. sub-neg 80.6%

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

   8. metadata-eval 80.6%

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

   9. +-commutative 80.6%

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

   10. *-commutative 80.6%

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

7. Simplified 80.6%

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

   1. flip-+ 80.7%

      \[\leadsto \cos x \cdot \color{blue}{\frac{-1 \cdot -1 - \cos \varepsilon \cdot \cos \varepsilon}{-1 - \cos \varepsilon}} - \sin x \cdot \sin \varepsilon \]

   2. metadata-eval 80.7%

      \[\leadsto \cos x \cdot \frac{\color{blue}{1} - \cos \varepsilon \cdot \cos \varepsilon}{-1 - \cos \varepsilon} - \sin x \cdot \sin \varepsilon \]

   3. 1-sub-cos 99.7%

      \[\leadsto \cos x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-1 - \cos \varepsilon} - \sin x \cdot \sin \varepsilon \]

   4. pow2 99.7%

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

9. Applied egg-rr 99.7%

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

    1. associate-*r/ 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \frac{\cos x \cdot {\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \]
13. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (-
  (* (cos x) (/ (pow (sin eps) 2.0) (- -1.0 (cos eps))))
  (* (sin eps) (sin x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. sub-neg 54.3%

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

   2. cos-sum 54.5%

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

   3. associate-+l- 54.4%

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

   4. add-sqr-sqrt 53.7%

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

   5. associate-*l* 53.7%

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

   6. fma-neg 53.7%

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

4. Applied egg-rr 53.7%

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

5. Taylor expanded in x around inf 54.4%

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

   1. associate--r+ 80.6%

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

   2. sub-neg 80.6%

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

   3. *-rgt-identity 80.6%

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

   4. cancel-sign-sub-inv 80.6%

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

   5. *-commutative 80.6%

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

   6. distribute-lft-out-- 80.6%

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

   7. sub-neg 80.6%

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

   8. metadata-eval 80.6%

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

   9. +-commutative 80.6%

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

   10. *-commutative 80.6%

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

7. Simplified 80.6%

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

   1. flip-+ 80.7%

      \[\leadsto \cos x \cdot \color{blue}{\frac{-1 \cdot -1 - \cos \varepsilon \cdot \cos \varepsilon}{-1 - \cos \varepsilon}} - \sin x \cdot \sin \varepsilon \]

   2. metadata-eval 80.7%

      \[\leadsto \cos x \cdot \frac{\color{blue}{1} - \cos \varepsilon \cdot \cos \varepsilon}{-1 - \cos \varepsilon} - \sin x \cdot \sin \varepsilon \]

   3. 1-sub-cos 99.7%

      \[\leadsto \cos x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-1 - \cos \varepsilon} - \sin x \cdot \sin \varepsilon \]

   4. pow2 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

    \[\leadsto \cos x \cdot \frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \]
11. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (sin (* eps 0.5))
  (* -2.0 (log1p (expm1 (sin (* 0.5 (- eps (* x -2.0)))))))))

C

double code(double x, double eps) {
	return sin((eps * 0.5)) * (-2.0 * log1p(expm1(sin((0.5 * (eps - (x * -2.0)))))));
}

Java

public static double code(double x, double eps) {
	return Math.sin((eps * 0.5)) * (-2.0 * Math.log1p(Math.expm1(Math.sin((0.5 * (eps - (x * -2.0)))))));
}

Python

def code(x, eps):
	return math.sin((eps * 0.5)) * (-2.0 * math.log1p(math.expm1(math.sin((0.5 * (eps - (x * -2.0)))))))

Julia

function code(x, eps)
	return Float64(sin(Float64(eps * 0.5)) * Float64(-2.0 * log1p(expm1(sin(Float64(0.5 * Float64(eps - Float64(x * -2.0))))))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Log[1 + N[(Exp[N[Sin[N[(0.5 * N[(eps - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. diff-cos 81.2%

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

   2. div-inv 81.2%

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

   3. associate--l+ 81.2%

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

   4. metadata-eval 81.2%

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

   5. div-inv 81.2%

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

   6. +-commutative 81.2%

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

   7. associate-+l+ 81.2%

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

   8. metadata-eval 81.2%

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

4. Applied egg-rr 81.2%

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

   1. associate-*r* 81.2%

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

   2. *-commutative 81.2%

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

   3. associate-*l* 81.2%

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

   4. sub-neg 81.2%

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

   5. mul-1-neg 81.2%

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

   6. +-commutative 81.2%

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

   7. associate-+r+ 99.6%

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

   8. mul-1-neg 99.6%

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

   9. sub-neg 99.6%

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

   10. +-inverses 99.6%

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

   11. remove-double-neg 99.6%

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

   12. mul-1-neg 99.6%

       \[\leadsto \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   13. sub-neg 99.6%

       \[\leadsto \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   14. neg-sub0 99.6%

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

   15. mul-1-neg 99.6%

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

   16. remove-double-neg 99.6%

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

   17. *-commutative 99.6%

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

   18. +-commutative 99.6%

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

6. Simplified 99.6%

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

   1. log1p-expm1-u 99.7%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right)\right)}\right) \]

8. Applied egg-rr 99.7%

   \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right)\right)}\right) \]

9. Taylor expanded in x around -inf 99.7%

   \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)}\right)\right)\right) \]

10. Final simplification 99.7%

    \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. diff-cos 81.2%

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

   2. div-inv 81.2%

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

   3. associate--l+ 81.2%

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

   4. metadata-eval 81.2%

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

   5. div-inv 81.2%

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

   6. +-commutative 81.2%

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

   7. associate-+l+ 81.2%

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

   8. metadata-eval 81.2%

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

4. Applied egg-rr 81.2%

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

   1. associate-*r* 81.2%

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

   2. *-commutative 81.2%

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

   3. associate-*l* 81.2%

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

   4. sub-neg 81.2%

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

   5. mul-1-neg 81.2%

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

   6. +-commutative 81.2%

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

   7. associate-+r+ 99.6%

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

   8. mul-1-neg 99.6%

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

   9. sub-neg 99.6%

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

   10. +-inverses 99.6%

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

   11. remove-double-neg 99.6%

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

   12. mul-1-neg 99.6%

       \[\leadsto \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   13. sub-neg 99.6%

       \[\leadsto \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   14. neg-sub0 99.6%

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

   15. mul-1-neg 99.6%

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

   16. remove-double-neg 99.6%

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

   17. *-commutative 99.6%

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

   18. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around -inf 99.6%

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

8. Final simplification 99.6%

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

Alternative 5: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in eps around 0 98.6%

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)} \]
4. Step-by-step derivation

   1. +-commutative 98.6%

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

   2. mul-1-neg 98.6%

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

   3. unsub-neg 98.6%

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

   4. associate-*r* 98.6%

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

   5. *-commutative 98.6%

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

5. Simplified 98.6%

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

   1. add-cube-cbrt 98.6%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)} \cdot \sqrt[3]{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)}\right) \cdot \sqrt[3]{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)}} - \varepsilon \cdot \sin x \]

   2. pow3 98.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)}\right)}^{3}} - \varepsilon \cdot \sin x \]

7. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)}\right)}^{3}} - \varepsilon \cdot \sin x \]

8. Taylor expanded in x around 0 97.5%

   \[\leadsto {\color{blue}{\left({\left(1 \cdot {\varepsilon}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{-0.5}\right)}}^{3} - \varepsilon \cdot \sin x \]
9. Step-by-step derivation

   1. unpow1/3 97.7%

      \[\leadsto {\left(\color{blue}{\sqrt[3]{1 \cdot {\varepsilon}^{2}}} \cdot \sqrt[3]{-0.5}\right)}^{3} - \varepsilon \cdot \sin x \]

   2. *-lft-identity 97.7%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{{\varepsilon}^{2}}} \cdot \sqrt[3]{-0.5}\right)}^{3} - \varepsilon \cdot \sin x \]

10. Simplified 97.7%

    \[\leadsto {\color{blue}{\left(\sqrt[3]{{\varepsilon}^{2}} \cdot \sqrt[3]{-0.5}\right)}}^{3} - \varepsilon \cdot \sin x \]

11. Taylor expanded in x around inf 97.7%

    \[\leadsto \color{blue}{-0.5 \cdot \left({1}^{0.3333333333333333} \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x} \]
12. Step-by-step derivation

    1. *-commutative 97.7%

       \[\leadsto \color{blue}{\left({1}^{0.3333333333333333} \cdot {\varepsilon}^{2}\right) \cdot -0.5} - \varepsilon \cdot \sin x \]

    2. fma-neg 97.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left({1}^{0.3333333333333333} \cdot {\varepsilon}^{2}, -0.5, -\varepsilon \cdot \sin x\right)} \]

    3. pow-base-1 97.7%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot {\varepsilon}^{2}, -0.5, -\varepsilon \cdot \sin x\right) \]

    4. *-lft-identity 97.7%

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

    5. rem-square-sqrt 0.0%

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

    6. fma-define 0.0%

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

    7. unpow2 0.0%

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

    8. swap-sqr 0.0%

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

    9. associate-*l* 0.0%

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

    10. distribute-rgt-neg-in 0.0%

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

    11. distribute-lft-out 0.0%

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

    12. *-commutative 0.0%

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

    13. associate-*r* 0.0%

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

    14. rem-square-sqrt 97.8%

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

13. Simplified 97.8%

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

14. Final simplification 97.8%

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

Alternative 6: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in eps around 0 79.2%

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

   1. mul-1-neg 79.2%

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

   2. *-commutative 79.2%

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

   3. distribute-rgt-neg-in 79.2%

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

5. Simplified 79.2%

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

6. Final simplification 79.2%

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

Alternative 7: 79.0% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in eps around 0 79.2%

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

   1. mul-1-neg 79.2%

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

   2. *-commutative 79.2%

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

   3. distribute-rgt-neg-in 79.2%

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

5. Simplified 79.2%

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

6. Taylor expanded in x around 0 77.6%

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

   1. associate-*r* 77.6%

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

   2. neg-mul-1 77.6%

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

8. Simplified 77.6%

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

9. Final simplification 77.6%

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

Developer target: 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]

Reproduce

herbie shell --seed 2024040 
(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)))

  :herbie-target
  (* (* -2.0 (sin (+ x (/ eps 2.0)))) (sin (/ eps 2.0)))

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