2cos (problem 3.3.5)

Percentage Accurate: 38.1% → 99.1%

Time: 15.2s

Alternatives: 13

Speedup: 1.9×

Specification

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: 38.1% 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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.4 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\cos x \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4e-5) (not (<= eps 4.4e-5)))
   (fma (+ -1.0 (cos eps)) (cos x) (* (sin eps) (- (sin x))))
   (- (* -0.5 (* eps (* (cos x) eps))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4e-5) || !(eps <= 4.4e-5)) {
		tmp = fma((-1.0 + cos(eps)), cos(x), (sin(eps) * -sin(x)));
	} else {
		tmp = (-0.5 * (eps * (cos(x) * eps))) - (eps * sin(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4e-5) || !(eps <= 4.4e-5))
		tmp = fma(Float64(-1.0 + cos(eps)), cos(x), Float64(sin(eps) * Float64(-sin(x))));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(cos(x) * eps))) - Float64(eps * sin(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4e-5], N[Not[LessEqual[eps, 4.4e-5]], $MachinePrecision]], N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.00000000000000033e-5 or 4.3999999999999999e-5 < eps

   1. Initial program 54.3%

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

      1. sub-neg 54.3%

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

      2. cos-sum 99.2%

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

      3. associate-+l- 99.2%

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

      4. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. fma-neg 99.2%

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

      2. *-commutative 99.2%

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

      3. *-commutative 99.2%

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

      4. fma-neg 99.2%

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

      5. remove-double-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in eps around inf 99.2%

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

      1. associate--r+ 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. distribute-lft-out-- 99.3%

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

      4. sub-neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. +-commutative 99.3%

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

   8. Simplified 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. *-commutative 99.3%

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

      3. fma-def 99.4%

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

   10. Applied egg-rr 99.4%

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

   if -4.00000000000000033e-5 < eps < 4.3999999999999999e-5

   1. Initial program 19.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. unpow2 99.8%

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

      4. associate-*l* 99.8%

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

   4. Simplified 99.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.4 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\cos x \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 2: 99.0% 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 38.2%

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

   1. sub-neg 38.2%

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

   2. cos-sum 62.6%

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

   3. associate-+l- 62.6%

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

   4. fma-neg 62.6%

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

3. Applied egg-rr 62.6%

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

   1. fma-neg 62.6%

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

   2. *-commutative 62.6%

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

   3. *-commutative 62.6%

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

   4. fma-neg 62.6%

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

   5. remove-double-neg 62.6%

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

5. Simplified 62.6%

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

6. Taylor expanded in eps around inf 62.6%

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

   1. associate--r+ 90.2%

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

   2. *-rgt-identity 90.2%

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

   3. distribute-lft-out-- 90.3%

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

   4. sub-neg 90.3%

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

   5. metadata-eval 90.3%

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

   6. +-commutative 90.3%

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

8. Simplified 90.3%

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

   1. flip-+ 89.7%

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

   2. metadata-eval 89.7%

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

   3. 1-sub-cos 98.9%

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

   4. pow2 98.8%

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

10. Applied egg-rr 98.8%

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

11. Final simplification 98.8%

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

Alternative 3: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\cos x \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.6e-5) (not (<= eps 3.6e-5)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (- (* -0.5 (* eps (* (cos x) eps))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.6e-5) || !(eps <= 3.6e-5)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = (-0.5 * (eps * (cos(x) * eps))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4.6d-5)) .or. (.not. (eps <= 3.6d-5))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = ((-0.5d0) * (eps * (cos(x) * eps))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.6e-5) || !(eps <= 3.6e-5)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (-0.5 * (eps * (Math.cos(x) * eps))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.6e-5) or not (eps <= 3.6e-5):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (-0.5 * (eps * (math.cos(x) * eps))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.6e-5) || !(eps <= 3.6e-5))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(cos(x) * eps))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.6e-5) || ~((eps <= 3.6e-5)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = (-0.5 * (eps * (cos(x) * eps))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.6e-5], N[Not[LessEqual[eps, 3.6e-5]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.6e-5 or 3.60000000000000009e-5 < eps

   1. Initial program 54.3%

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

      1. sub-neg 54.3%

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

      2. cos-sum 99.2%

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

      3. associate-+l- 99.2%

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

      4. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. fma-neg 99.2%

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

      2. *-commutative 99.2%

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

      3. *-commutative 99.2%

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

      4. fma-neg 99.2%

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

      5. remove-double-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in eps around inf 99.2%

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

      1. associate--r+ 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. distribute-lft-out-- 99.3%

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

      4. sub-neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. +-commutative 99.3%

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

   8. Simplified 99.3%

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

   if -4.6e-5 < eps < 3.60000000000000009e-5

   1. Initial program 19.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. unpow2 99.8%

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

      4. associate-*l* 99.8%

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

   4. Simplified 99.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\cos x \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 4: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0125 \lor \neg \left(\varepsilon \leq 0.0075\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\cos x \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0125) (not (<= eps 0.0075)))
   (- (cos eps) (cos x))
   (- (* -0.5 (* eps (* (cos x) eps))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0125) || !(eps <= 0.0075)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = (-0.5 * (eps * (cos(x) * eps))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0125d0)) .or. (.not. (eps <= 0.0075d0))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = ((-0.5d0) * (eps * (cos(x) * eps))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0125) || !(eps <= 0.0075)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = (-0.5 * (eps * (Math.cos(x) * eps))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0125) or not (eps <= 0.0075):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = (-0.5 * (eps * (math.cos(x) * eps))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0125) || !(eps <= 0.0075))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(cos(x) * eps))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0125) || ~((eps <= 0.0075)))
		tmp = cos(eps) - cos(x);
	else
		tmp = (-0.5 * (eps * (cos(x) * eps))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0125], N[Not[LessEqual[eps, 0.0075]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.012500000000000001 or 0.0074999999999999997 < eps

   1. Initial program 54.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 57.5%

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

   if -0.012500000000000001 < eps < 0.0074999999999999997

   1. Initial program 19.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. unpow2 99.8%

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

      4. associate-*l* 99.8%

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

   4. Simplified 99.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0125 \lor \neg \left(\varepsilon \leq 0.0075\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\cos x \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 5: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

   1. diff-cos 47.4%

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

   2. div-inv 47.4%

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

   3. metadata-eval 47.4%

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

   4. div-inv 47.4%

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

   5. +-commutative 47.4%

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

   6. metadata-eval 47.4%

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

3. Applied egg-rr 47.4%

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

   1. *-commutative 47.4%

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

   2. +-commutative 47.4%

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

   3. associate--l+ 76.1%

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

   4. +-inverses 76.1%

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

   5. distribute-lft-in 76.1%

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

   6. metadata-eval 76.1%

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

   7. *-commutative 76.1%

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

   8. +-commutative 76.1%

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

5. Simplified 76.1%

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

6. Taylor expanded in x around -inf 76.0%

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

7. Final simplification 76.0%

   \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

Alternative 6: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

   1. diff-cos 47.4%

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

   2. div-inv 47.4%

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

   3. metadata-eval 47.4%

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

   4. div-inv 47.4%

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

   5. +-commutative 47.4%

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

   6. metadata-eval 47.4%

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

3. Applied egg-rr 47.4%

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

   1. *-commutative 47.4%

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

   2. +-commutative 47.4%

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

   3. associate--l+ 76.1%

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

   4. +-inverses 76.1%

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

   5. distribute-lft-in 76.1%

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

   6. metadata-eval 76.1%

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

   7. *-commutative 76.1%

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

   8. +-commutative 76.1%

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

5. Simplified 76.1%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. log1p-expm1-u 76.1%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)\right)}\right) \]

   2. log1p-udef 59.6%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)\right)}\right) \]

   3. +-commutative 59.6%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \log \left(1 + \mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \color{blue}{\left(x + \varepsilon\right)}\right)\right)\right)\right)\right) \]

7. Applied egg-rr 59.6%

   \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 59.6%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \log \left(1 + \mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)\right)\right)\right)\right)} \]

   2. expm1-udef 39.0%

      \[\leadsto -2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \log \left(1 + \mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)\right)\right)\right)} - 1\right)} \]

   3. *-commutative 39.0%

      \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)}\right)} - 1\right) \]

   4. log1p-def 39.0%

      \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)\right)} - 1\right) \]

   5. log1p-expm1-u 39.0%

      \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)\right)} - 1\right) \]

   6. +-commutative 39.0%

      \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)\right)} - 1\right) \]

   7. +-rgt-identity 39.0%

      \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \varepsilon\right)}\right)} - 1\right) \]

   8. *-commutative 39.0%

      \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right) \cdot \sin \color{blue}{\left(\varepsilon \cdot 0.5\right)}\right)} - 1\right) \]

9. Applied egg-rr 39.0%

   \[\leadsto -2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 76.0%

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

    2. expm1-log1p 76.1%

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

    3. *-commutative 76.1%

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

11. Simplified 76.1%

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

12. Final simplification 76.1%

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

Alternative 7: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -7 \cdot 10^{-39} \lor \neg \left(x \leq 1.55 \cdot 10^{-30}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -7e-39) (not (<= x 1.55e-30)))
     (* -2.0 (* (sin x) t_0))
     (* -2.0 (pow t_0 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -7e-39) || !(x <= 1.55e-30)) {
		tmp = -2.0 * (sin(x) * t_0);
	} else {
		tmp = -2.0 * pow(t_0, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((x <= (-7d-39)) .or. (.not. (x <= 1.55d-30))) then
        tmp = (-2.0d0) * (sin(x) * t_0)
    else
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((x <= -7e-39) || !(x <= 1.55e-30)) {
		tmp = -2.0 * (Math.sin(x) * t_0);
	} else {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -7e-39) or not (x <= 1.55e-30):
		tmp = -2.0 * (math.sin(x) * t_0)
	else:
		tmp = -2.0 * math.pow(t_0, 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -7e-39) || !(x <= 1.55e-30))
		tmp = Float64(-2.0 * Float64(sin(x) * t_0));
	else
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((x <= -7e-39) || ~((x <= 1.55e-30)))
		tmp = -2.0 * (sin(x) * t_0);
	else
		tmp = -2.0 * (t_0 ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -7e-39], N[Not[LessEqual[x, 1.55e-30]], $MachinePrecision]], N[(-2.0 * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.99999999999999999e-39 or 1.54999999999999995e-30 < x

   1. Initial program 10.8%

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

      1. diff-cos 11.0%

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

      2. div-inv 11.0%

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

      3. metadata-eval 11.0%

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

      4. div-inv 11.0%

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

      5. +-commutative 11.0%

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

      6. metadata-eval 11.0%

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

   3. Applied egg-rr 11.0%

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

      1. *-commutative 11.0%

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

      2. +-commutative 11.0%

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

      3. associate--l+ 57.3%

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

      4. +-inverses 57.3%

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

      5. distribute-lft-in 57.3%

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

      6. metadata-eval 57.3%

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

      7. *-commutative 57.3%

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

      8. +-commutative 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in eps around 0 53.9%

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

   if -6.99999999999999999e-39 < x < 1.54999999999999995e-30

   1. Initial program 72.2%

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

      1. diff-cos 92.8%

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

      2. div-inv 92.8%

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

      3. metadata-eval 92.8%

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

      4. div-inv 92.8%

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

      5. +-commutative 92.8%

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

      6. metadata-eval 92.8%

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

   3. Applied egg-rr 92.8%

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

      1. *-commutative 92.8%

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

      2. +-commutative 92.8%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 91.9%

      \[\leadsto -2 \cdot \color{blue}{{\sin \left(0.5 \cdot \varepsilon\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-39} \lor \neg \left(x \leq 1.55 \cdot 10^{-30}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 8: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-38} \lor \neg \left(x \leq 2.15 \cdot 10^{-30}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.65e-38) (not (<= x 2.15e-30)))
   (* eps (- (sin x)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.65e-38) || !(x <= 2.15e-30)) {
		tmp = eps * -sin(x);
	} else {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-1.65d-38)) .or. (.not. (x <= 2.15d-30))) then
        tmp = eps * -sin(x)
    else
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.65e-38) || !(x <= 2.15e-30)) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.65e-38) or not (x <= 2.15e-30):
		tmp = eps * -math.sin(x)
	else:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.65e-38) || !(x <= 2.15e-30))
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.65e-38) || ~((x <= 2.15e-30)))
		tmp = eps * -sin(x);
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.65e-38], N[Not[LessEqual[x, 2.15e-30]], $MachinePrecision]], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6500000000000001e-38 or 2.14999999999999983e-30 < x

   1. Initial program 10.8%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 50.5%

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

      1. associate-*r* 50.5%

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

      2. mul-1-neg 50.5%

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

   4. Simplified 50.5%

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

   if -1.6500000000000001e-38 < x < 2.14999999999999983e-30

   1. Initial program 72.2%

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

      1. diff-cos 92.8%

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

      2. div-inv 92.8%

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

      3. metadata-eval 92.8%

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

      4. div-inv 92.8%

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

      5. +-commutative 92.8%

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

      6. metadata-eval 92.8%

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

   3. Applied egg-rr 92.8%

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

      1. *-commutative 92.8%

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

      2. +-commutative 92.8%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 91.9%

      \[\leadsto -2 \cdot \color{blue}{{\sin \left(0.5 \cdot \varepsilon\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-38} \lor \neg \left(x \leq 2.15 \cdot 10^{-30}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 9: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.8e-7) (not (<= eps 5.2e-7)))
   (- (cos eps) (cos x))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.8e-7) || !(eps <= 5.2e-7)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4.8d-7)) .or. (.not. (eps <= 5.2d-7))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.8e-7) || !(eps <= 5.2e-7)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.8e-7) or not (eps <= 5.2e-7):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.8e-7) || !(eps <= 5.2e-7))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.8e-7) || ~((eps <= 5.2e-7)))
		tmp = cos(eps) - cos(x);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.8e-7], N[Not[LessEqual[eps, 5.2e-7]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.79999999999999957e-7 or 5.19999999999999998e-7 < eps

   1. Initial program 54.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 57.5%

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

   if -4.79999999999999957e-7 < eps < 5.19999999999999998e-7

   1. Initial program 19.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 79.8%

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

      1. associate-*r* 79.8%

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

      2. mul-1-neg 79.8%

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

   4. Simplified 79.8%

      \[\leadsto \color{blue}{\left(-\varepsilon\right) \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 10: 67.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0001 \lor \neg \left(\varepsilon \leq 10^{-6}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0001) (not (<= eps 1e-6)))
   (+ -1.0 (cos eps))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0001) || !(eps <= 1e-6)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0001d0)) .or. (.not. (eps <= 1d-6))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0001) || !(eps <= 1e-6)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0001) or not (eps <= 1e-6):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0001) || !(eps <= 1e-6))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0001) || ~((eps <= 1e-6)))
		tmp = -1.0 + cos(eps);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0001], N[Not[LessEqual[eps, 1e-6]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.00000000000000005e-4 or 9.99999999999999955e-7 < eps

   1. Initial program 54.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 56.0%

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

   if -1.00000000000000005e-4 < eps < 9.99999999999999955e-7

   1. Initial program 19.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 79.8%

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

      1. associate-*r* 79.8%

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

      2. mul-1-neg 79.8%

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

   4. Simplified 79.8%

      \[\leadsto \color{blue}{\left(-\varepsilon\right) \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0001 \lor \neg \left(\varepsilon \leq 10^{-6}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 11: 46.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.95 \cdot 10^{-20} \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.95e-20) (not (<= eps 0.00016)))
   (+ -1.0 (cos eps))
   (* eps (* eps -0.5))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.95e-20) || !(eps <= 0.00016)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = eps * (eps * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.95d-20)) .or. (.not. (eps <= 0.00016d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = eps * (eps * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.95e-20) || !(eps <= 0.00016)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = eps * (eps * -0.5);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.95e-20) or not (eps <= 0.00016):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = eps * (eps * -0.5)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.95e-20) || !(eps <= 0.00016))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(eps * Float64(eps * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.95e-20) || ~((eps <= 0.00016)))
		tmp = -1.0 + cos(eps);
	else
		tmp = eps * (eps * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.95e-20], N[Not[LessEqual[eps, 0.00016]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.95000000000000004e-20 or 1.60000000000000013e-4 < eps

   1. Initial program 53.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 54.9%

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

   if -1.95000000000000004e-20 < eps < 1.60000000000000013e-4

   1. Initial program 20.0%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 20.0%

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

   3. Taylor expanded in eps around 0 39.7%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 39.7%

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

      2. associate-*r* 39.7%

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

      3. *-commutative 39.7%

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

   5. Simplified 39.7%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot -0.5\right) \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.95 \cdot 10^{-20} \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 12: 21.0% 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 38.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]

2. Taylor expanded in x around 0 39.0%

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

3. Taylor expanded in eps around 0 20.0%

   \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
4. Step-by-step derivation

   1. unpow2 20.0%

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

   2. associate-*r* 20.0%

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

   3. *-commutative 20.0%

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

5. Simplified 20.0%

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

6. Final simplification 20.0%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5\right) \]

Alternative 13: 12.2% 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 38.2%

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

   1. add-cube-cbrt 38.0%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cos \left(x + \varepsilon\right)} \cdot \sqrt[3]{\cos \left(x + \varepsilon\right)}\right) \cdot \sqrt[3]{\cos \left(x + \varepsilon\right)}} - \cos x \]

   2. pow3 38.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right)}\right)}^{3}} - \cos x \]

3. Applied egg-rr 38.1%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right)}\right)}^{3}} - \cos x \]

4. Taylor expanded in eps around 0 10.8%

   \[\leadsto \color{blue}{\cos x \cdot {1}^{0.3333333333333333} - \cos x} \]
5. Step-by-step derivation

   1. pow-base-1 10.8%

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

   2. *-rgt-identity 10.8%

      \[\leadsto \color{blue}{\cos x} - \cos x \]

   3. +-inverses 10.8%

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

6. Simplified 10.8%

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

7. Final simplification 10.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023228 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))