2cos (problem 3.3.5)

Percentage Accurate: 38.2% → 99.3%

Time: 17.6s

Alternatives: 18

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: 38.2% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0048:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0055)
   (- (* (cos eps) (cos x)) (fma (sin eps) (sin x) (cos x)))
   (if (<= eps 0.0048)
     (-
      (*
       (cos x)
       (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
      (* (sin eps) (sin x)))
     (fma (cos x) (+ -1.0 (cos eps)) (* (sin eps) (- (sin x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0055) {
		tmp = (cos(eps) * cos(x)) - fma(sin(eps), sin(x), cos(x));
	} else if (eps <= 0.0048) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) - (sin(eps) * sin(x));
	} else {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0055)
		tmp = Float64(Float64(cos(eps) * cos(x)) - fma(sin(eps), sin(x), cos(x)));
	elseif (eps <= 0.0048)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - Float64(sin(eps) * sin(x)));
	else
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0055], N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0048], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0054999999999999997

   1. Initial program 49.4%

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

      1. sub-neg 49.4%

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

      2. cos-sum 99.1%

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

      3. associate-+l- 99.1%

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

      4. fma-neg 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. fma-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. fma-neg 99.1%

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

      5. remove-double-neg 99.1%

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

   5. Simplified 99.1%

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

   if -0.0054999999999999997 < eps < 0.00479999999999999958

   1. Initial program 25.7%

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

      1. sub-neg 25.7%

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

      2. cos-sum 27.0%

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

      3. associate-+l- 27.0%

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

      4. fma-neg 27.0%

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

   3. Applied egg-rr 27.0%

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

      1. fma-neg 27.0%

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

      2. *-commutative 27.0%

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

      3. *-commutative 27.0%

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

      4. fma-neg 27.0%

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

      5. remove-double-neg 27.0%

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

   5. Simplified 27.0%

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

   6. Taylor expanded in eps around inf 27.0%

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

      1. associate--r+ 79.6%

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

      2. *-commutative 79.6%

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

      3. *-rgt-identity 79.6%

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

      4. distribute-lft-out-- 79.5%

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

      5. sub-neg 79.5%

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

      6. metadata-eval 79.5%

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

      7. +-commutative 79.5%

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

      8. *-commutative 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in eps around 0 99.9%

      \[\leadsto \cos x \cdot \color{blue}{\left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} - \sin x \cdot \sin \varepsilon \]

   if 0.00479999999999999958 < eps

   1. Initial program 46.5%

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

      1. sub-neg 46.5%

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

      2. cos-sum 98.7%

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

      3. associate-+l- 98.7%

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

      4. fma-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. fma-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-neg 98.8%

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

      5. remove-double-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around inf 98.7%

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

      1. associate--r+ 98.8%

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

      2. *-commutative 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. distribute-lft-out-- 98.9%

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

      5. sub-neg 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

      8. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around inf 98.9%

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

       1. fma-neg 98.9%

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

       2. sub-neg 98.9%

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

       3. metadata-eval 98.9%

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

       4. distribute-rgt-neg-in 98.9%

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

   11. Simplified 98.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0048:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.9%

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

   1. sub-neg 37.9%

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

   2. cos-sum 66.3%

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

   3. associate-+l- 66.3%

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

   4. fma-neg 66.3%

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

3. Applied egg-rr 66.3%

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

   1. fma-neg 66.3%

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

   2. *-commutative 66.3%

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

   3. *-commutative 66.3%

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

   4. fma-neg 66.3%

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

   5. remove-double-neg 66.3%

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

5. Simplified 66.3%

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

6. Taylor expanded in eps around inf 66.3%

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

   1. associate--r+ 90.2%

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

   2. *-commutative 90.2%

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

   3. *-rgt-identity 90.2%

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

   4. distribute-lft-out-- 90.2%

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

   5. sub-neg 90.2%

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

   6. metadata-eval 90.2%

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

   7. +-commutative 90.2%

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

   8. *-commutative 90.2%

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

8. Simplified 90.2%

   \[\leadsto \color{blue}{\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon} \]
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. associate-*r/ 89.7%

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

   3. metadata-eval 89.7%

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

   4. 1-sub-cos 99.0%

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

   5. pow2 99.0%

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

10. Applied egg-rr 99.0%

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

    1. *-commutative 99.0%

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

    2. associate-/l* 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

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

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0048:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (<= eps -0.0055)
     (- (- (* (cos eps) (cos x)) t_0) (cos x))
     (if (<= eps 0.0048)
       (-
        (*
         (cos x)
         (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
        t_0)
       (fma (cos x) (+ -1.0 (cos eps)) (* (sin eps) (- (sin x))))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0055) {
		tmp = ((cos(eps) * cos(x)) - t_0) - cos(x);
	} else if (eps <= 0.0048) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) - t_0;
	} else {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0055)
		tmp = Float64(Float64(Float64(cos(eps) * cos(x)) - t_0) - cos(x));
	elseif (eps <= 0.0048)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - t_0);
	else
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0055], N[(N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0048], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0054999999999999997

   1. Initial program 49.4%

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

      1. cos-sum 99.1%

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

   3. Applied egg-rr 99.1%

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

   if -0.0054999999999999997 < eps < 0.00479999999999999958

   1. Initial program 25.7%

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

      1. sub-neg 25.7%

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

      2. cos-sum 27.0%

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

      3. associate-+l- 27.0%

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

      4. fma-neg 27.0%

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

   3. Applied egg-rr 27.0%

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

      1. fma-neg 27.0%

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

      2. *-commutative 27.0%

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

      3. *-commutative 27.0%

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

      4. fma-neg 27.0%

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

      5. remove-double-neg 27.0%

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

   5. Simplified 27.0%

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

   6. Taylor expanded in eps around inf 27.0%

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

      1. associate--r+ 79.6%

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

      2. *-commutative 79.6%

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

      3. *-rgt-identity 79.6%

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

      4. distribute-lft-out-- 79.5%

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

      5. sub-neg 79.5%

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

      6. metadata-eval 79.5%

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

      7. +-commutative 79.5%

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

      8. *-commutative 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in eps around 0 99.9%

      \[\leadsto \cos x \cdot \color{blue}{\left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} - \sin x \cdot \sin \varepsilon \]

   if 0.00479999999999999958 < eps

   1. Initial program 46.5%

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

      1. sub-neg 46.5%

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

      2. cos-sum 98.7%

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

      3. associate-+l- 98.7%

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

      4. fma-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. fma-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-neg 98.8%

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

      5. remove-double-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around inf 98.7%

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

      1. associate--r+ 98.8%

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

      2. *-commutative 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. distribute-lft-out-- 98.9%

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

      5. sub-neg 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

      8. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around inf 98.9%

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

       1. fma-neg 98.9%

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

       2. sub-neg 98.9%

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

       3. metadata-eval 98.9%

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

       4. distribute-rgt-neg-in 98.9%

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

   11. Simplified 98.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0048:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016 \lor \neg \left(\varepsilon \leq 0.00018\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00016) (not (<= eps 0.00018)))
   (fma (cos x) (+ -1.0 (cos eps)) (* (sin eps) (- (sin x))))
   (- (* (cos x) (* -0.5 (pow eps 2.0))) (* (sin eps) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00016) || !(eps <= 0.00018)) {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(eps) * -sin(x)));
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (sin(eps) * sin(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00016) || !(eps <= 0.00018))
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(eps) * Float64(-sin(x))));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(sin(eps) * sin(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00016], N[Not[LessEqual[eps, 0.00018]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.60000000000000013e-4 or 1.80000000000000011e-4 < eps

   1. Initial program 47.6%

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

      1. sub-neg 47.6%

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

      2. cos-sum 98.8%

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

      3. associate-+l- 98.8%

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

      4. fma-neg 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. fma-neg 98.8%

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

      2. *-commutative 98.8%

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

      3. *-commutative 98.8%

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

      4. fma-neg 98.8%

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

      5. remove-double-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around inf 98.8%

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

      1. associate--r+ 98.8%

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

      2. *-commutative 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. distribute-lft-out-- 98.9%

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

      5. sub-neg 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

      8. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around inf 98.9%

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

       1. fma-neg 98.9%

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

       2. sub-neg 98.9%

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

       3. metadata-eval 98.9%

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

       4. distribute-rgt-neg-in 98.9%

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

   11. Simplified 98.9%

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

   if -1.60000000000000013e-4 < eps < 1.80000000000000011e-4

   1. Initial program 25.9%

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

      1. sub-neg 25.9%

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

      2. cos-sum 26.5%

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

      3. associate-+l- 26.5%

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

      4. fma-neg 26.5%

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

   3. Applied egg-rr 26.5%

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

      1. fma-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. *-commutative 26.5%

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

      4. fma-neg 26.5%

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

      5. remove-double-neg 26.5%

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

   5. Simplified 26.5%

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

   6. Taylor expanded in eps around inf 26.5%

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

      1. associate--r+ 79.6%

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

      2. *-commutative 79.6%

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

      3. *-rgt-identity 79.6%

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

      4. distribute-lft-out-- 79.5%

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

      5. sub-neg 79.5%

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

      6. metadata-eval 79.5%

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

      7. +-commutative 79.5%

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

      8. *-commutative 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in eps around 0 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016 \lor \neg \left(\varepsilon \leq 0.00018\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (<= eps -0.00016)
     (- (* (cos eps) (cos x)) (+ (cos x) t_0))
     (if (<= eps 0.00018)
       (- (* (cos x) (* -0.5 (pow eps 2.0))) t_0)
       (fma (cos x) (+ -1.0 (cos eps)) (* (sin eps) (- (sin x))))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.00016) {
		tmp = (cos(eps) * cos(x)) - (cos(x) + t_0);
	} else if (eps <= 0.00018) {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - t_0;
	} else {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.00016)
		tmp = Float64(Float64(cos(eps) * cos(x)) - Float64(cos(x) + t_0));
	elseif (eps <= 0.00018)
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - t_0);
	else
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00016], N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00018], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.60000000000000013e-4

   1. Initial program 48.7%

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

      1. expm1-log1p-u 29.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right) - \cos x\right)\right)} \]

   3. Applied egg-rr 29.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right) - \cos x\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 48.7%

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

      2. cos-sum 98.9%

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

      3. *-commutative 98.9%

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

      4. associate--l- 98.8%

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

   5. Applied egg-rr 98.8%

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

   if -1.60000000000000013e-4 < eps < 1.80000000000000011e-4

   1. Initial program 25.9%

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

      1. sub-neg 25.9%

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

      2. cos-sum 26.5%

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

      3. associate-+l- 26.5%

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

      4. fma-neg 26.5%

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

   3. Applied egg-rr 26.5%

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

      1. fma-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. *-commutative 26.5%

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

      4. fma-neg 26.5%

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

      5. remove-double-neg 26.5%

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

   5. Simplified 26.5%

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

   6. Taylor expanded in eps around inf 26.5%

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

      1. associate--r+ 79.6%

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

      2. *-commutative 79.6%

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

      3. *-rgt-identity 79.6%

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

      4. distribute-lft-out-- 79.5%

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

      5. sub-neg 79.5%

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

      6. metadata-eval 79.5%

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

      7. +-commutative 79.5%

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

      8. *-commutative 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in eps around 0 99.7%

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

   if 1.80000000000000011e-4 < eps

   1. Initial program 46.5%

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

      1. sub-neg 46.5%

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

      2. cos-sum 98.7%

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

      3. associate-+l- 98.7%

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

      4. fma-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. fma-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-neg 98.8%

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

      5. remove-double-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around inf 98.7%

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

      1. associate--r+ 98.8%

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

      2. *-commutative 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. distribute-lft-out-- 98.9%

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

      5. sub-neg 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

      8. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around inf 98.9%

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

       1. fma-neg 98.9%

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

       2. sub-neg 98.9%

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

       3. metadata-eval 98.9%

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

       4. distribute-rgt-neg-in 98.9%

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

   11. Simplified 98.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 6: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (<= eps -0.00016)
     (- (- (* (cos eps) (cos x)) t_0) (cos x))
     (if (<= eps 0.00018)
       (- (* (cos x) (* -0.5 (pow eps 2.0))) t_0)
       (fma (cos x) (+ -1.0 (cos eps)) (* (sin eps) (- (sin x))))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.00016) {
		tmp = ((cos(eps) * cos(x)) - t_0) - cos(x);
	} else if (eps <= 0.00018) {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - t_0;
	} else {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.00016)
		tmp = Float64(Float64(Float64(cos(eps) * cos(x)) - t_0) - cos(x));
	elseif (eps <= 0.00018)
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - t_0);
	else
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00016], N[(N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00018], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.60000000000000013e-4

   1. Initial program 48.7%

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

      1. cos-sum 98.9%

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

   3. Applied egg-rr 98.9%

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

   if -1.60000000000000013e-4 < eps < 1.80000000000000011e-4

   1. Initial program 25.9%

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

      1. sub-neg 25.9%

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

      2. cos-sum 26.5%

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

      3. associate-+l- 26.5%

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

      4. fma-neg 26.5%

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

   3. Applied egg-rr 26.5%

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

      1. fma-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. *-commutative 26.5%

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

      4. fma-neg 26.5%

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

      5. remove-double-neg 26.5%

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

   5. Simplified 26.5%

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

   6. Taylor expanded in eps around inf 26.5%

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

      1. associate--r+ 79.6%

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

      2. *-commutative 79.6%

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

      3. *-rgt-identity 79.6%

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

      4. distribute-lft-out-- 79.5%

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

      5. sub-neg 79.5%

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

      6. metadata-eval 79.5%

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

      7. +-commutative 79.5%

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

      8. *-commutative 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in eps around 0 99.7%

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

   if 1.80000000000000011e-4 < eps

   1. Initial program 46.5%

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

      1. sub-neg 46.5%

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

      2. cos-sum 98.7%

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

      3. associate-+l- 98.7%

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

      4. fma-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. fma-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-neg 98.8%

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

      5. remove-double-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around inf 98.7%

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

      1. associate--r+ 98.8%

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

      2. *-commutative 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. distribute-lft-out-- 98.9%

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

      5. sub-neg 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

      8. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around inf 98.9%

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

       1. fma-neg 98.9%

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

       2. sub-neg 98.9%

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

       3. metadata-eval 98.9%

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

       4. distribute-rgt-neg-in 98.9%

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

   11. Simplified 98.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 7: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00016 \lor \neg \left(\varepsilon \leq 0.00018\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (or (<= eps -0.00016) (not (<= eps 0.00018)))
     (- (* (cos x) (+ -1.0 (cos eps))) t_0)
     (- (* (cos x) (* -0.5 (pow eps 2.0))) t_0))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if ((eps <= -0.00016) || !(eps <= 0.00018)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - t_0;
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - t_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) * sin(x)
    if ((eps <= (-0.00016d0)) .or. (.not. (eps <= 0.00018d0))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - t_0
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) * Math.sin(x);
	double tmp;
	if ((eps <= -0.00016) || !(eps <= 0.00018)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - t_0;
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) * math.sin(x)
	tmp = 0
	if (eps <= -0.00016) or not (eps <= 0.00018):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - t_0
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if ((eps <= -0.00016) || !(eps <= 0.00018))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - t_0);
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) * sin(x);
	tmp = 0.0;
	if ((eps <= -0.00016) || ~((eps <= 0.00018)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - t_0;
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -0.00016], N[Not[LessEqual[eps, 0.00018]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.60000000000000013e-4 or 1.80000000000000011e-4 < eps

   1. Initial program 47.6%

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

      1. sub-neg 47.6%

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

      2. cos-sum 98.8%

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

      3. associate-+l- 98.8%

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

      4. fma-neg 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. fma-neg 98.8%

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

      2. *-commutative 98.8%

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

      3. *-commutative 98.8%

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

      4. fma-neg 98.8%

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

      5. remove-double-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around inf 98.8%

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

      1. associate--r+ 98.8%

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

      2. *-commutative 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. distribute-lft-out-- 98.9%

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

      5. sub-neg 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

      8. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -1.60000000000000013e-4 < eps < 1.80000000000000011e-4

   1. Initial program 25.9%

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

      1. sub-neg 25.9%

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

      2. cos-sum 26.5%

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

      3. associate-+l- 26.5%

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

      4. fma-neg 26.5%

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

   3. Applied egg-rr 26.5%

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

      1. fma-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. *-commutative 26.5%

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

      4. fma-neg 26.5%

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

      5. remove-double-neg 26.5%

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

   5. Simplified 26.5%

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

   6. Taylor expanded in eps around inf 26.5%

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

      1. associate--r+ 79.6%

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

      2. *-commutative 79.6%

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

      3. *-rgt-identity 79.6%

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

      4. distribute-lft-out-- 79.5%

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

      5. sub-neg 79.5%

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

      6. metadata-eval 79.5%

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

      7. +-commutative 79.5%

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

      8. *-commutative 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in eps around 0 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016 \lor \neg \left(\varepsilon \leq 0.00018\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 8: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-13} \lor \neg \left(x \leq 3.8 \cdot 10^{-28}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.1e-13) (not (<= x 3.8e-28)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (* (sin (* 0.5 (fma 2.0 x eps))) (* -2.0 (sin (* eps 0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.1e-13) || !(x <= 3.8e-28)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = sin((0.5 * fma(2.0, x, eps))) * (-2.0 * sin((eps * 0.5)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.1e-13) || !(x <= 3.8e-28))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(sin(Float64(0.5 * fma(2.0, x, eps))) * Float64(-2.0 * sin(Float64(eps * 0.5))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.1e-13], N[Not[LessEqual[x, 3.8e-28]], $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[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999989e-13 or 3.80000000000000009e-28 < x

   1. Initial program 7.9%

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

      1. sub-neg 7.9%

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

      2. cos-sum 61.9%

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

      3. associate-+l- 61.9%

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

      4. fma-neg 61.9%

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

   3. Applied egg-rr 61.9%

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

      1. fma-neg 61.9%

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

      2. *-commutative 61.9%

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

      3. *-commutative 61.9%

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

      4. fma-neg 61.9%

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

      5. remove-double-neg 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in eps around inf 61.9%

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

      1. associate--r+ 98.9%

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

      2. *-commutative 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. distribute-lft-out-- 98.9%

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

      5. sub-neg 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

      8. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -2.09999999999999989e-13 < x < 3.80000000000000009e-28

   1. Initial program 71.3%

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

      1. diff-cos 91.0%

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

      2. div-inv 91.0%

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

      3. associate--l+ 91.0%

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

      4. metadata-eval 91.0%

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

      5. div-inv 91.0%

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

      6. +-commutative 91.0%

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

      7. associate-+l+ 91.0%

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

      8. metadata-eval 91.0%

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

   3. Applied egg-rr 91.0%

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

      1. associate-*r* 91.0%

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

      2. *-commutative 91.0%

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

      3. *-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. count-2 91.0%

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

      6. fma-def 91.0%

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

      7. sub-neg 91.0%

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

      8. mul-1-neg 91.0%

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

      9. +-commutative 91.0%

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

      10. associate-+r+ 99.6%

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

      11. mul-1-neg 99.6%

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

      12. sub-neg 99.6%

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

      13. +-inverses 99.6%

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

      14. remove-double-neg 99.6%

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

      15. mul-1-neg 99.6%

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

      16. sub-neg 99.6%

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

      17. neg-sub0 99.6%

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

      18. mul-1-neg 99.6%

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

      19. remove-double-neg 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-13} \lor \neg \left(x \leq 3.8 \cdot 10^{-28}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 9: 75.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

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 - x))))) * sin((0.5d0 * (x + (eps + x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.9%

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

   1. expm1-log1p-u 26.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right) - \cos x\right)\right)} \]

3. Applied egg-rr 26.1%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right) - \cos x\right)\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 37.9%

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

   2. diff-cos 46.7%

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

   3. associate-*r* 46.7%

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

   4. div-inv 46.7%

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

   5. +-commutative 46.7%

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

   6. associate--l+ 72.1%

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

   7. metadata-eval 72.1%

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

   8. div-inv 72.1%

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

   9. associate-+l+ 72.1%

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

   10. +-commutative 72.1%

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

   11. metadata-eval 72.1%

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

5. Applied egg-rr 72.1%

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

6. Final simplification 72.1%

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

Alternative 10: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := \sin x \cdot \left(-2 \cdot t_0\right)\\ \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-91}:\\ \;\;\;\;\left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))) (t_1 (* (sin x) (* -2.0 t_0))))
   (if (<= x -0.9)
     t_1
     (if (<= x -1.2e-91)
       (- (+ -1.0 (cos eps)) (* (sin eps) x))
       (if (<= x 7.5e-31) (* -2.0 (pow t_0 2.0)) t_1)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double t_1 = sin(x) * (-2.0 * t_0);
	double tmp;
	if (x <= -0.9) {
		tmp = t_1;
	} else if (x <= -1.2e-91) {
		tmp = (-1.0 + cos(eps)) - (sin(eps) * x);
	} else if (x <= 7.5e-31) {
		tmp = -2.0 * pow(t_0, 2.0);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    t_1 = sin(x) * ((-2.0d0) * t_0)
    if (x <= (-0.9d0)) then
        tmp = t_1
    else if (x <= (-1.2d-91)) then
        tmp = ((-1.0d0) + cos(eps)) - (sin(eps) * x)
    else if (x <= 7.5d-31) then
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double t_1 = Math.sin(x) * (-2.0 * t_0);
	double tmp;
	if (x <= -0.9) {
		tmp = t_1;
	} else if (x <= -1.2e-91) {
		tmp = (-1.0 + Math.cos(eps)) - (Math.sin(eps) * x);
	} else if (x <= 7.5e-31) {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	t_1 = math.sin(x) * (-2.0 * t_0)
	tmp = 0
	if x <= -0.9:
		tmp = t_1
	elif x <= -1.2e-91:
		tmp = (-1.0 + math.cos(eps)) - (math.sin(eps) * x)
	elif x <= 7.5e-31:
		tmp = -2.0 * math.pow(t_0, 2.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	t_1 = Float64(sin(x) * Float64(-2.0 * t_0))
	tmp = 0.0
	if (x <= -0.9)
		tmp = t_1;
	elseif (x <= -1.2e-91)
		tmp = Float64(Float64(-1.0 + cos(eps)) - Float64(sin(eps) * x));
	elseif (x <= 7.5e-31)
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	t_1 = sin(x) * (-2.0 * t_0);
	tmp = 0.0;
	if (x <= -0.9)
		tmp = t_1;
	elseif (x <= -1.2e-91)
		tmp = (-1.0 + cos(eps)) - (sin(eps) * x);
	elseif (x <= 7.5e-31)
		tmp = -2.0 * (t_0 ^ 2.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.9], t$95$1, If[LessEqual[x, -1.2e-91], N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e-31], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.900000000000000022 or 7.49999999999999975e-31 < x

   1. Initial program 7.5%

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

      1. diff-cos 6.6%

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

      2. div-inv 6.6%

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

      3. associate--l+ 6.6%

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

      4. metadata-eval 6.6%

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

      5. div-inv 6.6%

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

      6. +-commutative 6.6%

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

      7. associate-+l+ 6.4%

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

      8. metadata-eval 6.4%

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

   3. Applied egg-rr 6.4%

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

      1. associate-*r* 6.4%

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

      2. *-commutative 6.4%

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

      3. *-commutative 6.4%

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

      4. +-commutative 6.4%

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

      5. count-2 6.4%

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

      6. fma-def 6.4%

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

      7. sub-neg 6.4%

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

      8. mul-1-neg 6.4%

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

      9. +-commutative 6.4%

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

      10. associate-+r+ 47.6%

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

      11. mul-1-neg 47.6%

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

      12. sub-neg 47.6%

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

      13. +-inverses 47.6%

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

      14. remove-double-neg 47.6%

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

      15. mul-1-neg 47.6%

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

      16. sub-neg 47.6%

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

      17. neg-sub0 47.6%

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

      18. mul-1-neg 47.6%

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

      19. remove-double-neg 47.6%

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

   5. Simplified 47.6%

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

   6. Taylor expanded in eps around 0 46.6%

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

   if -0.900000000000000022 < x < -1.20000000000000005e-91

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0 45.4%

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

      1. sub-neg 45.4%

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

      2. metadata-eval 45.4%

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

      3. +-commutative 45.4%

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

      4. associate-+r+ 81.8%

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

      5. +-commutative 81.8%

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

      6. +-commutative 81.8%

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

      7. mul-1-neg 81.8%

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

      8. unsub-neg 81.8%

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

      9. unpow2 81.8%

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

      10. associate-*l* 81.8%

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

      11. distribute-lft-out-- 81.8%

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

      12. +-commutative 81.8%

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

      13. *-commutative 81.8%

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

      14. fma-def 81.8%

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

   4. Simplified 81.8%

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

   5. Taylor expanded in x around 0 81.1%

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

      1. mul-1-neg 81.1%

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

      2. distribute-rgt-neg-out 81.1%

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

   7. Simplified 81.1%

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

   if -1.20000000000000005e-91 < x < 7.49999999999999975e-31

   1. Initial program 75.6%

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

      1. diff-cos 96.2%

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

      2. div-inv 96.2%

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

      3. associate--l+ 96.2%

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

      4. metadata-eval 96.2%

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

      5. div-inv 96.2%

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

      6. +-commutative 96.2%

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

      7. associate-+l+ 96.2%

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

      8. metadata-eval 96.2%

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

   3. Applied egg-rr 96.2%

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

      1. associate-*r* 96.2%

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

      2. *-commutative 96.2%

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

      3. *-commutative 96.2%

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

      4. +-commutative 96.2%

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

      5. count-2 96.2%

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

      6. fma-def 96.2%

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

      7. sub-neg 96.2%

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

      8. mul-1-neg 96.2%

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

      9. +-commutative 96.2%

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

      10. associate-+r+ 99.6%

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

      11. mul-1-neg 99.6%

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

      12. sub-neg 99.6%

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

      13. +-inverses 99.6%

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

      14. remove-double-neg 99.6%

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

      15. mul-1-neg 99.6%

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

      16. sub-neg 99.6%

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

      17. neg-sub0 99.6%

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

      18. mul-1-neg 99.6%

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

      19. remove-double-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 95.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\sin x \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-91}:\\ \;\;\;\;\left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 11: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.9%

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

   1. sub-neg 37.9%

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

   2. cos-sum 66.3%

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

   3. associate-+l- 66.3%

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

   4. fma-neg 66.3%

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

3. Applied egg-rr 66.3%

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

   1. fma-neg 66.3%

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

   2. *-commutative 66.3%

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

   3. *-commutative 66.3%

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

   4. fma-neg 66.3%

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

   5. remove-double-neg 66.3%

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

5. Simplified 66.3%

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

6. Taylor expanded in eps around inf 66.3%

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

   1. associate--r+ 90.2%

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

   2. *-commutative 90.2%

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

   3. *-rgt-identity 90.2%

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

   4. distribute-lft-out-- 90.2%

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

   5. sub-neg 90.2%

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

   6. metadata-eval 90.2%

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

   7. +-commutative 90.2%

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

   8. *-commutative 90.2%

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

8. Simplified 90.2%

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

   1. distribute-lft-in 90.2%

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

   2. associate--l+ 66.3%

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

   3. *-commutative 66.3%

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

   4. neg-mul-1 66.3%

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

   5. fma-neg 66.3%

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

   6. distribute-lft-neg-out 66.3%

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

   7. +-commutative 66.3%

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

   8. distribute-lft-neg-out 66.3%

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

   9. fma-neg 66.3%

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

   10. cos-sum 37.9%

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

   11. sub-neg 37.9%

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

   12. diff-cos 46.7%

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

10. Applied egg-rr 72.0%

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

    1. +-inverses 72.0%

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

    2. +-rgt-identity 72.0%

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

    3. *-commutative 72.0%

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

12. Simplified 72.0%

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

13. Final simplification 72.0%

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

Alternative 12: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-37} \lor \neg \left(x \leq 1.45 \cdot 10^{-28}\right):\\ \;\;\;\;\sin x \cdot \left(-2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -2.7e-37) (not (<= x 1.45e-28)))
     (* (sin x) (* -2.0 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 <= -2.7e-37) || !(x <= 1.45e-28)) {
		tmp = sin(x) * (-2.0 * 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 <= (-2.7d-37)) .or. (.not. (x <= 1.45d-28))) then
        tmp = sin(x) * ((-2.0d0) * 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 <= -2.7e-37) || !(x <= 1.45e-28)) {
		tmp = Math.sin(x) * (-2.0 * 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 <= -2.7e-37) or not (x <= 1.45e-28):
		tmp = math.sin(x) * (-2.0 * 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 <= -2.7e-37) || !(x <= 1.45e-28))
		tmp = Float64(sin(x) * Float64(-2.0 * 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 <= -2.7e-37) || ~((x <= 1.45e-28)))
		tmp = sin(x) * (-2.0 * 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, -2.7e-37], N[Not[LessEqual[x, 1.45e-28]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * N[(-2.0 * 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 < -2.70000000000000016e-37 or 1.45000000000000006e-28 < x

   1. Initial program 8.5%

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

      1. diff-cos 7.7%

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

      2. div-inv 7.7%

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

      3. associate--l+ 7.7%

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

      4. metadata-eval 7.7%

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

      5. div-inv 7.7%

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

      6. +-commutative 7.7%

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

      7. associate-+l+ 7.4%

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

      8. metadata-eval 7.4%

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

   3. Applied egg-rr 7.4%

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

      1. associate-*r* 7.4%

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

      2. *-commutative 7.4%

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

      3. *-commutative 7.4%

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

      4. +-commutative 7.4%

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

      5. count-2 7.4%

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

      6. fma-def 7.4%

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

      7. sub-neg 7.4%

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

      8. mul-1-neg 7.4%

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

      9. +-commutative 7.4%

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

      10. associate-+r+ 48.6%

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

      11. mul-1-neg 48.6%

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

      12. sub-neg 48.6%

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

      13. +-inverses 48.6%

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

      14. remove-double-neg 48.6%

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

      15. mul-1-neg 48.6%

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

      16. sub-neg 48.6%

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

      17. neg-sub0 48.6%

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

      18. mul-1-neg 48.6%

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

      19. remove-double-neg 48.6%

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

   5. Simplified 48.6%

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

   6. Taylor expanded in eps around 0 46.6%

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

   if -2.70000000000000016e-37 < x < 1.45000000000000006e-28

   1. Initial program 72.7%

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

      1. diff-cos 93.1%

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

      2. div-inv 93.1%

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

      3. associate--l+ 93.1%

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

      4. metadata-eval 93.1%

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

      5. div-inv 93.1%

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

      6. +-commutative 93.1%

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

      7. associate-+l+ 93.1%

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

      8. metadata-eval 93.1%

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

   3. Applied egg-rr 93.1%

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

      1. associate-*r* 93.1%

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

      2. *-commutative 93.1%

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

      3. *-commutative 93.1%

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

      4. +-commutative 93.1%

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

      5. count-2 93.1%

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

      6. fma-def 93.1%

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

      7. sub-neg 93.1%

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

      8. mul-1-neg 93.1%

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

      9. +-commutative 93.1%

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

      10. associate-+r+ 99.6%

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

      11. mul-1-neg 99.6%

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

      12. sub-neg 99.6%

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

      13. +-inverses 99.6%

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

      14. remove-double-neg 99.6%

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

      15. mul-1-neg 99.6%

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

      16. sub-neg 99.6%

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

      17. neg-sub0 99.6%

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

      18. mul-1-neg 99.6%

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

      19. remove-double-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 91.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-37} \lor \neg \left(x \leq 1.45 \cdot 10^{-28}\right):\\ \;\;\;\;\sin x \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 13: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-65} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-67}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.1e-65) (not (<= eps 3.8e-67)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.1e-65) || !(eps <= 3.8e-67)) {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.1d-65)) .or. (.not. (eps <= 3.8d-67))) then
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.1e-65) || !(eps <= 3.8e-67)) {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.1e-65) or not (eps <= 3.8e-67):
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.1e-65) || !(eps <= 3.8e-67))
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.1e-65) || ~((eps <= 3.8e-67)))
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.1e-65], N[Not[LessEqual[eps, 3.8e-67]], $MachinePrecision]], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.10000000000000016e-65 or 3.79999999999999988e-67 < eps

   1. Initial program 41.8%

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

      1. diff-cos 50.1%

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

      2. div-inv 50.1%

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

      3. associate--l+ 50.1%

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

      4. metadata-eval 50.1%

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

      5. div-inv 50.1%

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

      6. +-commutative 50.1%

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

      7. associate-+l+ 49.9%

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

      8. metadata-eval 49.9%

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

   3. Applied egg-rr 49.9%

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

      1. associate-*r* 49.9%

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

      2. *-commutative 49.9%

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

      3. *-commutative 49.9%

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

      4. +-commutative 49.9%

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

      5. count-2 49.9%

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

      6. fma-def 49.9%

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

      7. sub-neg 49.9%

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

      8. mul-1-neg 49.9%

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

      9. +-commutative 49.9%

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

      10. associate-+r+ 56.2%

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

      11. mul-1-neg 56.2%

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

      12. sub-neg 56.2%

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

      13. +-inverses 56.2%

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

      14. remove-double-neg 56.2%

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

      15. mul-1-neg 56.2%

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

      16. sub-neg 56.2%

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

      17. neg-sub0 56.2%

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

      18. mul-1-neg 56.2%

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

      19. remove-double-neg 56.2%

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0 51.2%

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

   if -3.10000000000000016e-65 < eps < 3.79999999999999988e-67

   1. Initial program 30.8%

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

   2. Taylor expanded in eps around 0 90.5%

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

      1. mul-1-neg 90.5%

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

      2. *-commutative 90.5%

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

      3. distribute-rgt-neg-in 90.5%

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

   4. Simplified 90.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-65} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-67}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 14: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0003:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -3.6 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 17:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos eps) (cos x))))
   (if (<= eps -0.0003)
     t_0
     (if (<= eps -3.6e-65)
       (* -0.5 (pow eps 2.0))
       (if (<= eps 17.0) (* (sin x) (- eps)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) - cos(x);
	double tmp;
	if (eps <= -0.0003) {
		tmp = t_0;
	} else if (eps <= -3.6e-65) {
		tmp = -0.5 * pow(eps, 2.0);
	} else if (eps <= 17.0) {
		tmp = sin(x) * -eps;
	} else {
		tmp = t_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 = cos(eps) - cos(x)
    if (eps <= (-0.0003d0)) then
        tmp = t_0
    else if (eps <= (-3.6d-65)) then
        tmp = (-0.5d0) * (eps ** 2.0d0)
    else if (eps <= 17.0d0) then
        tmp = sin(x) * -eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) - Math.cos(x);
	double tmp;
	if (eps <= -0.0003) {
		tmp = t_0;
	} else if (eps <= -3.6e-65) {
		tmp = -0.5 * Math.pow(eps, 2.0);
	} else if (eps <= 17.0) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) - math.cos(x)
	tmp = 0
	if eps <= -0.0003:
		tmp = t_0
	elif eps <= -3.6e-65:
		tmp = -0.5 * math.pow(eps, 2.0)
	elif eps <= 17.0:
		tmp = math.sin(x) * -eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) - cos(x))
	tmp = 0.0
	if (eps <= -0.0003)
		tmp = t_0;
	elseif (eps <= -3.6e-65)
		tmp = Float64(-0.5 * (eps ^ 2.0));
	elseif (eps <= 17.0)
		tmp = Float64(sin(x) * Float64(-eps));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) - cos(x);
	tmp = 0.0;
	if (eps <= -0.0003)
		tmp = t_0;
	elseif (eps <= -3.6e-65)
		tmp = -0.5 * (eps ^ 2.0);
	elseif (eps <= 17.0)
		tmp = sin(x) * -eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0003], t$95$0, If[LessEqual[eps, -3.6e-65], N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 17.0], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.99999999999999974e-4 or 17 < eps

   1. Initial program 48.2%

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

   2. Taylor expanded in x around 0 51.3%

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

   if -2.99999999999999974e-4 < eps < -3.5999999999999998e-65

   1. Initial program 3.8%

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

   2. Taylor expanded in x around 0 4.5%

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

   3. Taylor expanded in eps around 0 67.3%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]

   if -3.5999999999999998e-65 < eps < 17

   1. Initial program 28.5%

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

   2. Taylor expanded in eps around 0 84.7%

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

      1. mul-1-neg 84.7%

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

      2. *-commutative 84.7%

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

      3. distribute-rgt-neg-in 84.7%

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

   4. Simplified 84.7%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0003:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -3.6 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 17:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 15: 48.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.0003:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.7 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* -0.5 (pow eps 2.0))))
   (if (<= eps -0.0003)
     t_0
     (if (<= eps -2.7e-160)
       t_1
       (if (<= eps 2.4e-71) (* x (- eps)) (if (<= eps 0.00018) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double t_1 = -0.5 * pow(eps, 2.0);
	double tmp;
	if (eps <= -0.0003) {
		tmp = t_0;
	} else if (eps <= -2.7e-160) {
		tmp = t_1;
	} else if (eps <= 2.4e-71) {
		tmp = x * -eps;
	} else if (eps <= 0.00018) {
		tmp = t_1;
	} else {
		tmp = t_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) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) + cos(eps)
    t_1 = (-0.5d0) * (eps ** 2.0d0)
    if (eps <= (-0.0003d0)) then
        tmp = t_0
    else if (eps <= (-2.7d-160)) then
        tmp = t_1
    else if (eps <= 2.4d-71) then
        tmp = x * -eps
    else if (eps <= 0.00018d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = -1.0 + Math.cos(eps);
	double t_1 = -0.5 * Math.pow(eps, 2.0);
	double tmp;
	if (eps <= -0.0003) {
		tmp = t_0;
	} else if (eps <= -2.7e-160) {
		tmp = t_1;
	} else if (eps <= 2.4e-71) {
		tmp = x * -eps;
	} else if (eps <= 0.00018) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	t_1 = -0.5 * math.pow(eps, 2.0)
	tmp = 0
	if eps <= -0.0003:
		tmp = t_0
	elif eps <= -2.7e-160:
		tmp = t_1
	elif eps <= 2.4e-71:
		tmp = x * -eps
	elif eps <= 0.00018:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	t_1 = Float64(-0.5 * (eps ^ 2.0))
	tmp = 0.0
	if (eps <= -0.0003)
		tmp = t_0;
	elseif (eps <= -2.7e-160)
		tmp = t_1;
	elseif (eps <= 2.4e-71)
		tmp = Float64(x * Float64(-eps));
	elseif (eps <= 0.00018)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -1.0 + cos(eps);
	t_1 = -0.5 * (eps ^ 2.0);
	tmp = 0.0;
	if (eps <= -0.0003)
		tmp = t_0;
	elseif (eps <= -2.7e-160)
		tmp = t_1;
	elseif (eps <= 2.4e-71)
		tmp = x * -eps;
	elseif (eps <= 0.00018)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0003], t$95$0, If[LessEqual[eps, -2.7e-160], t$95$1, If[LessEqual[eps, 2.4e-71], N[(x * (-eps)), $MachinePrecision], If[LessEqual[eps, 0.00018], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.99999999999999974e-4 or 1.80000000000000011e-4 < eps

   1. Initial program 47.9%

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

   2. Taylor expanded in x around 0 48.7%

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

   if -2.99999999999999974e-4 < eps < -2.7000000000000001e-160 or 2.4e-71 < eps < 1.80000000000000011e-4

   1. Initial program 5.4%

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

   2. Taylor expanded in x around 0 5.8%

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

   3. Taylor expanded in eps around 0 48.5%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]

   if -2.7000000000000001e-160 < eps < 2.4e-71

   1. Initial program 39.1%

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

   2. Taylor expanded in eps around 0 96.0%

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

      1. mul-1-neg 96.0%

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

      2. *-commutative 96.0%

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

      3. distribute-rgt-neg-in 96.0%

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

   4. Simplified 96.0%

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

   5. Taylor expanded in x around 0 54.1%

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

      1. mul-1-neg 54.1%

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

      2. distribute-rgt-neg-in 54.1%

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

   7. Simplified 54.1%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0003:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq -2.7 \cdot 10^{-160}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 16: 66.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0003:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.7 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 4000:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))))
   (if (<= eps -0.0003)
     t_0
     (if (<= eps -1.7e-65)
       (* -0.5 (pow eps 2.0))
       (if (<= eps 4000.0) (* (sin x) (- eps)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double tmp;
	if (eps <= -0.0003) {
		tmp = t_0;
	} else if (eps <= -1.7e-65) {
		tmp = -0.5 * pow(eps, 2.0);
	} else if (eps <= 4000.0) {
		tmp = sin(x) * -eps;
	} else {
		tmp = t_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 = (-1.0d0) + cos(eps)
    if (eps <= (-0.0003d0)) then
        tmp = t_0
    else if (eps <= (-1.7d-65)) then
        tmp = (-0.5d0) * (eps ** 2.0d0)
    else if (eps <= 4000.0d0) then
        tmp = sin(x) * -eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = -1.0 + Math.cos(eps);
	double tmp;
	if (eps <= -0.0003) {
		tmp = t_0;
	} else if (eps <= -1.7e-65) {
		tmp = -0.5 * Math.pow(eps, 2.0);
	} else if (eps <= 4000.0) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	tmp = 0
	if eps <= -0.0003:
		tmp = t_0
	elif eps <= -1.7e-65:
		tmp = -0.5 * math.pow(eps, 2.0)
	elif eps <= 4000.0:
		tmp = math.sin(x) * -eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	tmp = 0.0
	if (eps <= -0.0003)
		tmp = t_0;
	elseif (eps <= -1.7e-65)
		tmp = Float64(-0.5 * (eps ^ 2.0));
	elseif (eps <= 4000.0)
		tmp = Float64(sin(x) * Float64(-eps));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -1.0 + cos(eps);
	tmp = 0.0;
	if (eps <= -0.0003)
		tmp = t_0;
	elseif (eps <= -1.7e-65)
		tmp = -0.5 * (eps ^ 2.0);
	elseif (eps <= 4000.0)
		tmp = sin(x) * -eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0003], t$95$0, If[LessEqual[eps, -1.7e-65], N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4000.0], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.99999999999999974e-4 or 4e3 < eps

   1. Initial program 48.9%

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

   2. Taylor expanded in x around 0 49.7%

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

   if -2.99999999999999974e-4 < eps < -1.69999999999999993e-65

   1. Initial program 3.8%

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

   2. Taylor expanded in x around 0 4.5%

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

   3. Taylor expanded in eps around 0 67.3%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]

   if -1.69999999999999993e-65 < eps < 4e3

   1. Initial program 28.0%

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

   2. Taylor expanded in eps around 0 83.2%

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

      1. mul-1-neg 83.2%

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

      2. *-commutative 83.2%

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

      3. distribute-rgt-neg-in 83.2%

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

   4. Simplified 83.2%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0003:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq -1.7 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 4000:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 17: 42.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -48000000000 \lor \neg \left(\varepsilon \leq 4.6 \cdot 10^{-10}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -48000000000.0) (not (<= eps 4.6e-10)))
   (+ -1.0 (cos eps))
   (* x (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -48000000000.0) || !(eps <= 4.6e-10)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = x * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-48000000000.0d0)) .or. (.not. (eps <= 4.6d-10))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = x * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -48000000000.0) || !(eps <= 4.6e-10)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = x * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -48000000000.0) or not (eps <= 4.6e-10):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = x * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -48000000000.0) || !(eps <= 4.6e-10))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(x * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -48000000000.0) || ~((eps <= 4.6e-10)))
		tmp = -1.0 + cos(eps);
	else
		tmp = x * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -48000000000.0], N[Not[LessEqual[eps, 4.6e-10]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(x * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.8e10 or 4.60000000000000014e-10 < eps

   1. Initial program 47.6%

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

   2. Taylor expanded in x around 0 48.5%

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

   if -4.8e10 < eps < 4.60000000000000014e-10

   1. Initial program 25.7%

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

   2. Taylor expanded in eps around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. *-commutative 79.5%

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

      3. distribute-rgt-neg-in 79.5%

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

   4. Simplified 79.5%

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

   5. Taylor expanded in x around 0 37.1%

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

      1. mul-1-neg 37.1%

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

      2. distribute-rgt-neg-in 37.1%

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

   7. Simplified 37.1%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -48000000000 \lor \neg \left(\varepsilon \leq 4.6 \cdot 10^{-10}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 18: 17.4% 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 37.9%

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

2. Taylor expanded in eps around 0 37.9%

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

   1. mul-1-neg 37.9%

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

   2. *-commutative 37.9%

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

   3. distribute-rgt-neg-in 37.9%

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

4. Simplified 37.9%

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

5. Taylor expanded in x around 0 18.4%

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

   1. mul-1-neg 18.4%

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

   2. distribute-rgt-neg-in 18.4%

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

7. Simplified 18.4%

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

8. Final simplification 18.4%

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

Reproduce

herbie shell --seed 2023320 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))