2cos (problem 3.3.5)

Percentage Accurate: 38.2% → 99.3%

Time: 20.8s

Alternatives: 17

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0027:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00265:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, \sin x \cdot {\varepsilon}^{3}, \cos x \cdot \mathsf{fma}\left(-0.5, \varepsilon \cdot \varepsilon, {\varepsilon}^{4} \cdot 0.041666666666666664\right) - \varepsilon \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \cos \varepsilon - \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0027)
   (fma (cos x) (+ -1.0 (cos eps)) (* (sin x) (- (sin eps))))
   (if (<= eps 0.00265)
     (fma
      0.16666666666666666
      (* (sin x) (pow eps 3.0))
      (-
       (*
        (cos x)
        (fma -0.5 (* eps eps) (* (pow eps 4.0) 0.041666666666666664)))
       (* eps (sin x))))
     (fma (sin eps) (- (sin x)) (- (* (cos x) (cos eps)) (cos x))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0027) {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(x) * -sin(eps)));
	} else if (eps <= 0.00265) {
		tmp = fma(0.16666666666666666, (sin(x) * pow(eps, 3.0)), ((cos(x) * fma(-0.5, (eps * eps), (pow(eps, 4.0) * 0.041666666666666664))) - (eps * sin(x))));
	} else {
		tmp = fma(sin(eps), -sin(x), ((cos(x) * cos(eps)) - cos(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0027)
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(x) * Float64(-sin(eps))));
	elseif (eps <= 0.00265)
		tmp = fma(0.16666666666666666, Float64(sin(x) * (eps ^ 3.0)), Float64(Float64(cos(x) * fma(-0.5, Float64(eps * eps), Float64((eps ^ 4.0) * 0.041666666666666664))) - Float64(eps * sin(x))));
	else
		tmp = fma(sin(eps), Float64(-sin(x)), Float64(Float64(cos(x) * cos(eps)) - cos(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0027], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00265], N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[(eps * eps), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0027000000000000001

   1. Initial program 48.1%

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

      1. cos-sum 98.7%

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

      2. sub-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. +-commutative 98.7%

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

      2. distribute-lft-neg-in 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-def 98.8%

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

      5. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around inf 98.7%

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

      1. associate--l+ 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 98.8%

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

      4. neg-mul-1 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-def 98.9%

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

      7. *-commutative 98.9%

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

      8. *-rgt-identity 98.9%

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

      9. distribute-lft-out-- 98.9%

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

      10. sub-neg 98.9%

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

      11. metadata-eval 98.9%

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

      12. +-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in eps around inf 98.9%

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

       1. +-commutative 98.9%

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

       2. sub-neg 98.9%

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

       3. metadata-eval 98.9%

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

       4. +-commutative 98.9%

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

       5. associate-*r* 98.9%

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

       6. neg-mul-1 98.9%

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

       7. *-commutative 98.9%

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

       8. fma-def 99.0%

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

   11. Simplified 99.0%

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

   if -0.0027000000000000001 < eps < 0.00265000000000000001

   1. Initial program 25.7%

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

      1. expm1-log1p-u 25.4%

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

   3. Applied egg-rr 25.4%

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

   4. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right)} + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      3. associate-+r+ 99.8%

         \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]

      4. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, {\varepsilon}^{3} \cdot \sin x, 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]

      5. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(0.16666666666666666, \color{blue}{\sin x \cdot {\varepsilon}^{3}}, 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) \]

      6. associate-+r+ 99.8%

         \[\leadsto \mathsf{fma}\left(0.16666666666666666, \sin x \cdot {\varepsilon}^{3}, \color{blue}{\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)}\right) \]

      7. neg-mul-1 99.8%

         \[\leadsto \mathsf{fma}\left(0.16666666666666666, \sin x \cdot {\varepsilon}^{3}, \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + \color{blue}{\left(-\varepsilon \cdot \sin x\right)}\right) \]

      8. unsub-neg 99.8%

         \[\leadsto \mathsf{fma}\left(0.16666666666666666, \sin x \cdot {\varepsilon}^{3}, \color{blue}{\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) - \varepsilon \cdot \sin x}\right) \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, \sin x \cdot {\varepsilon}^{3}, \cos x \cdot \mathsf{fma}\left(-0.5, \varepsilon \cdot \varepsilon, {\varepsilon}^{4} \cdot 0.041666666666666664\right) - \varepsilon \cdot \sin x\right)} \]

   if 0.00265000000000000001 < eps

   1. Initial program 56.7%

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

      1. cos-sum 98.4%

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

      2. sub-neg 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-commutative 98.4%

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

      2. distribute-lft-neg-in 98.4%

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

      3. *-commutative 98.4%

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

      4. fma-def 98.4%

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

      5. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in eps around inf 98.4%

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

      1. associate--l+ 98.6%

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

      2. *-commutative 98.6%

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

      3. associate-*r* 98.6%

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

      4. neg-mul-1 98.6%

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

      5. *-commutative 98.6%

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

      6. fma-def 98.7%

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

      7. *-commutative 98.7%

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

      8. *-rgt-identity 98.7%

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

      9. distribute-lft-out-- 98.6%

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

      10. sub-neg 98.6%

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

      11. metadata-eval 98.6%

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

      12. +-commutative 98.6%

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

   8. Simplified 98.6%

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

      1. distribute-lft-in 98.7%

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

   10. Applied egg-rr 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0027:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00265:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, \sin x \cdot {\varepsilon}^{3}, \cos x \cdot \mathsf{fma}\left(-0.5, \varepsilon \cdot \varepsilon, {\varepsilon}^{4} \cdot 0.041666666666666664\right) - \varepsilon \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \cos \varepsilon - \cos x\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin x\\ \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0048:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, t_0, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, t_0, \cos x \cdot \cos \varepsilon - \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin x))))
   (if (<= eps -0.0058)
     (fma (cos x) (+ -1.0 (cos eps)) (* (sin x) (- (sin eps))))
     (if (<= eps 0.0048)
       (fma
        (sin eps)
        t_0
        (*
         (cos x)
         (+ (* eps (* eps -0.5)) (* (pow eps 4.0) 0.041666666666666664))))
       (fma (sin eps) t_0 (- (* (cos x) (cos eps)) (cos x)))))))

C

double code(double x, double eps) {
	double t_0 = -sin(x);
	double tmp;
	if (eps <= -0.0058) {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(x) * -sin(eps)));
	} else if (eps <= 0.0048) {
		tmp = fma(sin(eps), t_0, (cos(x) * ((eps * (eps * -0.5)) + (pow(eps, 4.0) * 0.041666666666666664))));
	} else {
		tmp = fma(sin(eps), t_0, ((cos(x) * cos(eps)) - cos(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-sin(x))
	tmp = 0.0
	if (eps <= -0.0058)
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(x) * Float64(-sin(eps))));
	elseif (eps <= 0.0048)
		tmp = fma(sin(eps), t_0, Float64(cos(x) * Float64(Float64(eps * Float64(eps * -0.5)) + Float64((eps ^ 4.0) * 0.041666666666666664))));
	else
		tmp = fma(sin(eps), t_0, Float64(Float64(cos(x) * cos(eps)) - cos(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Sin[x], $MachinePrecision])}, If[LessEqual[eps, -0.0058], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0048], N[(N[Sin[eps], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] * t$95$0 + N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0058

   1. Initial program 48.1%

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

      1. cos-sum 98.7%

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

      2. sub-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. +-commutative 98.7%

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

      2. distribute-lft-neg-in 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-def 98.8%

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

      5. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around inf 98.7%

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

      1. associate--l+ 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 98.8%

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

      4. neg-mul-1 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-def 98.9%

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

      7. *-commutative 98.9%

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

      8. *-rgt-identity 98.9%

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

      9. distribute-lft-out-- 98.9%

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

      10. sub-neg 98.9%

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

      11. metadata-eval 98.9%

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

      12. +-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in eps around inf 98.9%

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

       1. +-commutative 98.9%

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

       2. sub-neg 98.9%

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

       3. metadata-eval 98.9%

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

       4. +-commutative 98.9%

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

       5. associate-*r* 98.9%

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

       6. neg-mul-1 98.9%

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

       7. *-commutative 98.9%

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

       8. fma-def 99.0%

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

   11. Simplified 99.0%

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

   if -0.0058 < eps < 0.00479999999999999958

   1. Initial program 25.7%

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

      1. cos-sum 27.0%

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

      2. sub-neg 27.0%

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

   3. Applied egg-rr 27.0%

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

      1. +-commutative 27.0%

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

      2. distribute-lft-neg-in 27.0%

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

      3. *-commutative 27.0%

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

      4. fma-def 27.0%

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

      5. *-commutative 27.0%

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

   5. Simplified 27.0%

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

   6. Taylor expanded in eps around inf 27.0%

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

      1. associate--l+ 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*r* 76.4%

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

      4. neg-mul-1 76.4%

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

      5. *-commutative 76.4%

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

      6. fma-def 76.4%

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

      7. *-commutative 76.4%

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

      8. *-rgt-identity 76.4%

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

      9. distribute-lft-out-- 76.4%

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

      10. sub-neg 76.4%

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

      11. metadata-eval 76.4%

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

      12. +-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in eps around 0 99.8%

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, -\sin x, \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)}\right) \]
   10. Step-by-step derivation

       1. +-commutative 99.8%

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

       2. associate-*r* 99.8%

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

       3. associate-*r* 99.8%

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

       4. distribute-rgt-out 99.8%

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

       5. *-commutative 99.8%

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

       6. unpow2 99.8%

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

       7. associate-*l* 99.8%

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

   11. Simplified 99.8%

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

   if 0.00479999999999999958 < eps

   1. Initial program 56.7%

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

      1. cos-sum 98.4%

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

      2. sub-neg 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-commutative 98.4%

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

      2. distribute-lft-neg-in 98.4%

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

      3. *-commutative 98.4%

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

      4. fma-def 98.4%

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

      5. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in eps around inf 98.4%

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

      1. associate--l+ 98.6%

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

      2. *-commutative 98.6%

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

      3. associate-*r* 98.6%

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

      4. neg-mul-1 98.6%

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

      5. *-commutative 98.6%

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

      6. fma-def 98.7%

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

      7. *-commutative 98.7%

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

      8. *-rgt-identity 98.7%

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

      9. distribute-lft-out-- 98.6%

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

      10. sub-neg 98.6%

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

      11. metadata-eval 98.6%

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

      12. +-commutative 98.6%

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

   8. Simplified 98.6%

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

      1. distribute-lft-in 98.7%

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

   10. Applied egg-rr 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0048:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \cos \varepsilon - \cos x\right)\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin x\\ t_1 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\mathsf{fma}\left(\cos x, t_1, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0033:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, t_0, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, t_0, \cos x \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin x))) (t_1 (+ -1.0 (cos eps))))
   (if (<= eps -0.0058)
     (fma (cos x) t_1 (* (sin x) (- (sin eps))))
     (if (<= eps 0.0033)
       (fma
        (sin eps)
        t_0
        (*
         (cos x)
         (+ (* eps (* eps -0.5)) (* (pow eps 4.0) 0.041666666666666664))))
       (fma (sin eps) t_0 (* (cos x) t_1))))))

C

double code(double x, double eps) {
	double t_0 = -sin(x);
	double t_1 = -1.0 + cos(eps);
	double tmp;
	if (eps <= -0.0058) {
		tmp = fma(cos(x), t_1, (sin(x) * -sin(eps)));
	} else if (eps <= 0.0033) {
		tmp = fma(sin(eps), t_0, (cos(x) * ((eps * (eps * -0.5)) + (pow(eps, 4.0) * 0.041666666666666664))));
	} else {
		tmp = fma(sin(eps), t_0, (cos(x) * t_1));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-sin(x))
	t_1 = Float64(-1.0 + cos(eps))
	tmp = 0.0
	if (eps <= -0.0058)
		tmp = fma(cos(x), t_1, Float64(sin(x) * Float64(-sin(eps))));
	elseif (eps <= 0.0033)
		tmp = fma(sin(eps), t_0, Float64(cos(x) * Float64(Float64(eps * Float64(eps * -0.5)) + Float64((eps ^ 4.0) * 0.041666666666666664))));
	else
		tmp = fma(sin(eps), t_0, Float64(cos(x) * t_1));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Sin[x], $MachinePrecision])}, Block[{t$95$1 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0058], N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0033], N[(N[Sin[eps], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0058

   1. Initial program 48.1%

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

      1. cos-sum 98.7%

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

      2. sub-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. +-commutative 98.7%

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

      2. distribute-lft-neg-in 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-def 98.8%

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

      5. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around inf 98.7%

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

      1. associate--l+ 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 98.8%

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

      4. neg-mul-1 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-def 98.9%

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

      7. *-commutative 98.9%

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

      8. *-rgt-identity 98.9%

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

      9. distribute-lft-out-- 98.9%

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

      10. sub-neg 98.9%

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

      11. metadata-eval 98.9%

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

      12. +-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in eps around inf 98.9%

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

       1. +-commutative 98.9%

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

       2. sub-neg 98.9%

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

       3. metadata-eval 98.9%

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

       4. +-commutative 98.9%

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

       5. associate-*r* 98.9%

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

       6. neg-mul-1 98.9%

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

       7. *-commutative 98.9%

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

       8. fma-def 99.0%

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

   11. Simplified 99.0%

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

   if -0.0058 < eps < 0.0033

   1. Initial program 25.7%

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

      1. cos-sum 27.0%

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

      2. sub-neg 27.0%

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

   3. Applied egg-rr 27.0%

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

      1. +-commutative 27.0%

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

      2. distribute-lft-neg-in 27.0%

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

      3. *-commutative 27.0%

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

      4. fma-def 27.0%

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

      5. *-commutative 27.0%

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

   5. Simplified 27.0%

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

   6. Taylor expanded in eps around inf 27.0%

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

      1. associate--l+ 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*r* 76.4%

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

      4. neg-mul-1 76.4%

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

      5. *-commutative 76.4%

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

      6. fma-def 76.4%

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

      7. *-commutative 76.4%

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

      8. *-rgt-identity 76.4%

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

      9. distribute-lft-out-- 76.4%

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

      10. sub-neg 76.4%

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

      11. metadata-eval 76.4%

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

      12. +-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in eps around 0 99.8%

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, -\sin x, \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)}\right) \]
   10. Step-by-step derivation

       1. +-commutative 99.8%

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

       2. associate-*r* 99.8%

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

       3. associate-*r* 99.8%

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

       4. distribute-rgt-out 99.8%

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

       5. *-commutative 99.8%

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

       6. unpow2 99.8%

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

       7. associate-*l* 99.8%

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

   11. Simplified 99.8%

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

   if 0.0033 < eps

   1. Initial program 56.7%

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

      1. cos-sum 98.4%

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

      2. sub-neg 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-commutative 98.4%

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

      2. distribute-lft-neg-in 98.4%

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

      3. *-commutative 98.4%

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

      4. fma-def 98.4%

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

      5. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in eps around inf 98.4%

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

      1. associate--l+ 98.6%

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

      2. *-commutative 98.6%

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

      3. associate-*r* 98.6%

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

      4. neg-mul-1 98.6%

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

      5. *-commutative 98.6%

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

      6. fma-def 98.7%

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

      7. *-commutative 98.7%

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

      8. *-rgt-identity 98.7%

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

      9. distribute-lft-out-- 98.6%

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

      10. sub-neg 98.6%

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

      11. metadata-eval 98.6%

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

      12. +-commutative 98.6%

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

   8. Simplified 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0033:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 0.000135\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, -0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00015) (not (<= eps 0.000135)))
   (fma (cos x) (+ -1.0 (cos eps)) (* (sin x) (- (sin eps))))
   (fma (sin eps) (- (sin x)) (* -0.5 (* (cos x) (* eps eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00015) || !(eps <= 0.000135)) {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(x) * -sin(eps)));
	} else {
		tmp = fma(sin(eps), -sin(x), (-0.5 * (cos(x) * (eps * eps))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00015) || !(eps <= 0.000135))
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(x) * Float64(-sin(eps))));
	else
		tmp = fma(sin(eps), Float64(-sin(x)), Float64(-0.5 * Float64(cos(x) * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00015], N[Not[LessEqual[eps, 0.000135]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.49999999999999987e-4 or 1.35000000000000002e-4 < eps

   1. Initial program 52.4%

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

      1. cos-sum 98.6%

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

      2. sub-neg 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-lft-neg-in 98.6%

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

      3. *-commutative 98.6%

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

      4. fma-def 98.6%

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

      5. *-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in eps around inf 98.6%

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

      1. associate--l+ 98.7%

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

      2. *-commutative 98.7%

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

      3. associate-*r* 98.7%

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

      4. neg-mul-1 98.7%

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

      5. *-commutative 98.7%

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

      6. fma-def 98.8%

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

      7. *-commutative 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.7%

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

      10. sub-neg 98.7%

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

      11. metadata-eval 98.7%

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

      12. +-commutative 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in eps around inf 98.7%

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

       1. +-commutative 98.7%

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

       2. sub-neg 98.7%

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

       3. metadata-eval 98.7%

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

       4. +-commutative 98.7%

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

       5. associate-*r* 98.7%

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

       6. neg-mul-1 98.7%

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

       7. *-commutative 98.7%

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

       8. fma-def 98.7%

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

   11. Simplified 98.7%

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

   if -1.49999999999999987e-4 < eps < 1.35000000000000002e-4

   1. Initial program 25.7%

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

      1. cos-sum 27.0%

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

      2. sub-neg 27.0%

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

   3. Applied egg-rr 27.0%

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

      1. +-commutative 27.0%

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

      2. distribute-lft-neg-in 27.0%

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

      3. *-commutative 27.0%

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

      4. fma-def 27.0%

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

      5. *-commutative 27.0%

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

   5. Simplified 27.0%

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

   6. Taylor expanded in eps around inf 27.0%

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

      1. associate--l+ 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*r* 76.4%

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

      4. neg-mul-1 76.4%

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

      5. *-commutative 76.4%

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

      6. fma-def 76.4%

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

      7. *-commutative 76.4%

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

      8. *-rgt-identity 76.4%

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

      9. distribute-lft-out-- 76.4%

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

      10. sub-neg 76.4%

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

      11. metadata-eval 76.4%

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

      12. +-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in eps around 0 99.8%

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

       1. unpow2 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 0.000135\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, -0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin x\\ t_1 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00015:\\ \;\;\;\;\mathsf{fma}\left(\cos x, t_1, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000135:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, t_0, -0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, t_0, \cos x \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin x))) (t_1 (+ -1.0 (cos eps))))
   (if (<= eps -0.00015)
     (fma (cos x) t_1 (* (sin x) (- (sin eps))))
     (if (<= eps 0.000135)
       (fma (sin eps) t_0 (* -0.5 (* (cos x) (* eps eps))))
       (fma (sin eps) t_0 (* (cos x) t_1))))))

C

double code(double x, double eps) {
	double t_0 = -sin(x);
	double t_1 = -1.0 + cos(eps);
	double tmp;
	if (eps <= -0.00015) {
		tmp = fma(cos(x), t_1, (sin(x) * -sin(eps)));
	} else if (eps <= 0.000135) {
		tmp = fma(sin(eps), t_0, (-0.5 * (cos(x) * (eps * eps))));
	} else {
		tmp = fma(sin(eps), t_0, (cos(x) * t_1));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-sin(x))
	t_1 = Float64(-1.0 + cos(eps))
	tmp = 0.0
	if (eps <= -0.00015)
		tmp = fma(cos(x), t_1, Float64(sin(x) * Float64(-sin(eps))));
	elseif (eps <= 0.000135)
		tmp = fma(sin(eps), t_0, Float64(-0.5 * Float64(cos(x) * Float64(eps * eps))));
	else
		tmp = fma(sin(eps), t_0, Float64(cos(x) * t_1));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Sin[x], $MachinePrecision])}, Block[{t$95$1 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00015], N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.000135], N[(N[Sin[eps], $MachinePrecision] * t$95$0 + N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.49999999999999987e-4

   1. Initial program 48.1%

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

      1. cos-sum 98.7%

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

      2. sub-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. +-commutative 98.7%

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

      2. distribute-lft-neg-in 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-def 98.8%

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

      5. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around inf 98.7%

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

      1. associate--l+ 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 98.8%

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

      4. neg-mul-1 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-def 98.9%

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

      7. *-commutative 98.9%

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

      8. *-rgt-identity 98.9%

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

      9. distribute-lft-out-- 98.9%

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

      10. sub-neg 98.9%

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

      11. metadata-eval 98.9%

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

      12. +-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in eps around inf 98.9%

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

       1. +-commutative 98.9%

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

       2. sub-neg 98.9%

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

       3. metadata-eval 98.9%

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

       4. +-commutative 98.9%

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

       5. associate-*r* 98.9%

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

       6. neg-mul-1 98.9%

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

       7. *-commutative 98.9%

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

       8. fma-def 99.0%

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

   11. Simplified 99.0%

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

   if -1.49999999999999987e-4 < eps < 1.35000000000000002e-4

   1. Initial program 25.7%

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

      1. cos-sum 27.0%

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

      2. sub-neg 27.0%

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

   3. Applied egg-rr 27.0%

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

      1. +-commutative 27.0%

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

      2. distribute-lft-neg-in 27.0%

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

      3. *-commutative 27.0%

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

      4. fma-def 27.0%

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

      5. *-commutative 27.0%

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

   5. Simplified 27.0%

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

   6. Taylor expanded in eps around inf 27.0%

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

      1. associate--l+ 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*r* 76.4%

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

      4. neg-mul-1 76.4%

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

      5. *-commutative 76.4%

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

      6. fma-def 76.4%

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

      7. *-commutative 76.4%

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

      8. *-rgt-identity 76.4%

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

      9. distribute-lft-out-- 76.4%

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

      10. sub-neg 76.4%

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

      11. metadata-eval 76.4%

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

      12. +-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in eps around 0 99.8%

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

       1. unpow2 99.8%

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

   11. Simplified 99.8%

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

   if 1.35000000000000002e-4 < eps

   1. Initial program 56.7%

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

      1. cos-sum 98.4%

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

      2. sub-neg 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-commutative 98.4%

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

      2. distribute-lft-neg-in 98.4%

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

      3. *-commutative 98.4%

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

      4. fma-def 98.4%

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

      5. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in eps around inf 98.4%

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

      1. associate--l+ 98.6%

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

      2. *-commutative 98.6%

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

      3. associate-*r* 98.6%

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

      4. neg-mul-1 98.6%

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

      5. *-commutative 98.6%

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

      6. fma-def 98.7%

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

      7. *-commutative 98.7%

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

      8. *-rgt-identity 98.7%

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

      9. distribute-lft-out-- 98.6%

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

      10. sub-neg 98.6%

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

      11. metadata-eval 98.6%

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

      12. +-commutative 98.6%

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

   8. Simplified 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000135:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, -0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \end{array} \]

Alternative 6: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 0.000135\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, -0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00015) (not (<= eps 0.000135)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (fma (sin eps) (- (sin x)) (* -0.5 (* (cos x) (* eps eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00015) || !(eps <= 0.000135)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = fma(sin(eps), -sin(x), (-0.5 * (cos(x) * (eps * eps))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00015) || !(eps <= 0.000135))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = fma(sin(eps), Float64(-sin(x)), Float64(-0.5 * Float64(cos(x) * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00015], N[Not[LessEqual[eps, 0.000135]], $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[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.49999999999999987e-4 or 1.35000000000000002e-4 < eps

   1. Initial program 52.4%

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

      1. cos-sum 98.6%

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

      2. sub-neg 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-lft-neg-in 98.6%

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

      3. *-commutative 98.6%

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

      4. fma-def 98.6%

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

      5. *-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in eps around inf 98.6%

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

      1. associate--l+ 98.7%

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

      2. *-commutative 98.7%

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

      3. associate-*r* 98.7%

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

      4. neg-mul-1 98.7%

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

      5. *-commutative 98.7%

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

      6. fma-def 98.8%

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

      7. *-commutative 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.7%

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

      10. sub-neg 98.7%

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

      11. metadata-eval 98.7%

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

      12. +-commutative 98.7%

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

   8. Simplified 98.7%

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

      1. fma-udef 98.7%

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

   10. Applied egg-rr 98.7%

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

   if -1.49999999999999987e-4 < eps < 1.35000000000000002e-4

   1. Initial program 25.7%

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

      1. cos-sum 27.0%

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

      2. sub-neg 27.0%

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

   3. Applied egg-rr 27.0%

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

      1. +-commutative 27.0%

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

      2. distribute-lft-neg-in 27.0%

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

      3. *-commutative 27.0%

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

      4. fma-def 27.0%

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

      5. *-commutative 27.0%

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

   5. Simplified 27.0%

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

   6. Taylor expanded in eps around inf 27.0%

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

      1. associate--l+ 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*r* 76.4%

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

      4. neg-mul-1 76.4%

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

      5. *-commutative 76.4%

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

      6. fma-def 76.4%

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

      7. *-commutative 76.4%

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

      8. *-rgt-identity 76.4%

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

      9. distribute-lft-out-- 76.4%

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

      10. sub-neg 76.4%

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

      11. metadata-eval 76.4%

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

      12. +-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in eps around 0 99.8%

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

       1. unpow2 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 0.000135\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, -0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]

Alternative 7: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 0.000135\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00015) (not (<= eps 0.000135)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (+
    (* -0.5 (* eps (* eps (cos x))))
    (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00015) || !(eps <= 0.000135)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - 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 <= (-0.00015d0)) .or. (.not. (eps <= 0.000135d0))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) + (sin(x) * ((0.16666666666666666d0 * (eps ** 3.0d0)) - eps))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00015) || !(eps <= 0.000135)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) + (Math.sin(x) * ((0.16666666666666666 * Math.pow(eps, 3.0)) - eps));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00015) or not (eps <= 0.000135):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) + (math.sin(x) * ((0.16666666666666666 * math.pow(eps, 3.0)) - eps))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00015) || !(eps <= 0.000135))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00015) || ~((eps <= 0.000135)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = (-0.5 * (eps * (eps * cos(x)))) + (sin(x) * ((0.16666666666666666 * (eps ^ 3.0)) - eps));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00015], N[Not[LessEqual[eps, 0.000135]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.49999999999999987e-4 or 1.35000000000000002e-4 < eps

   1. Initial program 52.4%

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

      1. cos-sum 98.6%

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

      2. sub-neg 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-lft-neg-in 98.6%

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

      3. *-commutative 98.6%

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

      4. fma-def 98.6%

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

      5. *-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in eps around inf 98.6%

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

      1. associate--l+ 98.7%

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

      2. *-commutative 98.7%

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

      3. associate-*r* 98.7%

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

      4. neg-mul-1 98.7%

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

      5. *-commutative 98.7%

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

      6. fma-def 98.8%

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

      7. *-commutative 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.7%

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

      10. sub-neg 98.7%

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

      11. metadata-eval 98.7%

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

      12. +-commutative 98.7%

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

   8. Simplified 98.7%

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

      1. fma-udef 98.7%

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

   10. Applied egg-rr 98.7%

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

   if -1.49999999999999987e-4 < eps < 1.35000000000000002e-4

   1. Initial program 25.7%

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

   2. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)} \]

      2. associate-+l+ 99.8%

         \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} \]

      3. unpow2 99.8%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      4. associate-*l* 99.8%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      5. associate-*r* 99.8%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      6. associate-*r* 99.8%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\left(-1 \cdot \varepsilon\right) \cdot \sin x + \color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x}\right) \]

      7. distribute-rgt-out 99.8%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \color{blue}{\sin x \cdot \left(-1 \cdot \varepsilon + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]

      8. mul-1-neg 99.8%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\color{blue}{\left(-\varepsilon\right)} + 0.16666666666666666 \cdot {\varepsilon}^{3}\right) \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\left(-\varepsilon\right) + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 0.000135\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \]

Alternative 8: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0145 \lor \neg \left(\varepsilon \leq 0.0031\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0145) (not (<= eps 0.0031)))
   (- (cos eps) (cos x))
   (- (* -0.5 (* eps (* eps (cos x)))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0145) || !(eps <= 0.0031)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0145d0)) .or. (.not. (eps <= 0.0031d0))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0145) || !(eps <= 0.0031)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0145) or not (eps <= 0.0031):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0145) || !(eps <= 0.0031))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0145) || ~((eps <= 0.0031)))
		tmp = cos(eps) - cos(x);
	else
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0145], N[Not[LessEqual[eps, 0.0031]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0145000000000000007 or 0.00309999999999999989 < eps

   1. Initial program 52.4%

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

   2. Taylor expanded in x around 0 55.2%

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

   if -0.0145000000000000007 < eps < 0.00309999999999999989

   1. Initial program 25.7%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

      3. unpow2 99.6%

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

      4. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0145 \lor \neg \left(\varepsilon \leq 0.0031\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 9: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(-\sin x\right)\\ t_1 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.00023:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -2.9 \cdot 10^{-41}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 5.4 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(-x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (- (sin x)))) (t_1 (- (cos eps) (cos x))))
   (if (<= eps -0.00023)
     t_1
     (if (<= eps -2.9e-41)
       (* eps (* eps -0.5))
       (if (<= eps 5.4e-120)
         t_0
         (if (<= eps 1.2e-61)
           (fma -0.5 (* eps eps) (* eps (- x)))
           (if (<= eps 3e-8) t_0 t_1)))))))

C

double code(double x, double eps) {
	double t_0 = eps * -sin(x);
	double t_1 = cos(eps) - cos(x);
	double tmp;
	if (eps <= -0.00023) {
		tmp = t_1;
	} else if (eps <= -2.9e-41) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 5.4e-120) {
		tmp = t_0;
	} else if (eps <= 1.2e-61) {
		tmp = fma(-0.5, (eps * eps), (eps * -x));
	} else if (eps <= 3e-8) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(-sin(x)))
	t_1 = Float64(cos(eps) - cos(x))
	tmp = 0.0
	if (eps <= -0.00023)
		tmp = t_1;
	elseif (eps <= -2.9e-41)
		tmp = Float64(eps * Float64(eps * -0.5));
	elseif (eps <= 5.4e-120)
		tmp = t_0;
	elseif (eps <= 1.2e-61)
		tmp = fma(-0.5, Float64(eps * eps), Float64(eps * Float64(-x)));
	elseif (eps <= 3e-8)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00023], t$95$1, If[LessEqual[eps, -2.9e-41], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.4e-120], t$95$0, If[LessEqual[eps, 1.2e-61], N[(-0.5 * N[(eps * eps), $MachinePrecision] + N[(eps * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3e-8], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < -2.3000000000000001e-4 or 2.99999999999999973e-8 < eps

   1. Initial program 52.2%

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

   2. Taylor expanded in x around 0 55.0%

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

   if -2.3000000000000001e-4 < eps < -2.89999999999999977e-41

   1. Initial program 4.9%

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

   2. Taylor expanded in x around 0 5.6%

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

   3. Taylor expanded in eps around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow2 73.6%

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

      3. associate-*l* 73.6%

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

   5. Simplified 73.6%

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

   if -2.89999999999999977e-41 < eps < 5.3999999999999997e-120 or 1.2e-61 < eps < 2.99999999999999973e-8

   1. Initial program 31.9%

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

   2. Taylor expanded in eps around 0 89.3%

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

      1. associate-*r* 89.3%

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

      2. mul-1-neg 89.3%

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

   4. Simplified 89.3%

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

   if 5.3999999999999997e-120 < eps < 1.2e-61

   1. Initial program 4.8%

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

   2. Taylor expanded in eps around 0 4.8%

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

      1. associate-+r+ 4.8%

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

      2. mul-1-neg 4.8%

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

      3. unsub-neg 4.8%

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

      4. associate-*r* 4.8%

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

      5. distribute-rgt1-in 4.8%

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

      6. unpow2 4.8%

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

      7. associate-*r* 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in x around 0 79.3%

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

      1. +-commutative 79.3%

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

      2. fma-def 79.3%

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

      3. unpow2 79.3%

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

      4. mul-1-neg 79.3%

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

      5. distribute-rgt-neg-in 79.3%

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

   7. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(-x\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00023:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -2.9 \cdot 10^{-41}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 5.4 \cdot 10^{-120}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(-x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 10: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. metadata-eval 50.1%

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

   4. div-inv 50.1%

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

   5. +-commutative 50.1%

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

   6. metadata-eval 50.1%

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

3. Applied egg-rr 50.1%

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

   1. *-commutative 50.1%

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

   2. +-commutative 50.1%

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

   3. associate--l+ 75.3%

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

   4. +-inverses 75.3%

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

   5. distribute-lft-in 75.3%

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

   6. metadata-eval 75.3%

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

   7. *-commutative 75.3%

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

   8. +-commutative 75.3%

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

5. Simplified 75.3%

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

   1. expm1-log1p-u 75.3%

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

   2. *-commutative 75.3%

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

   3. +-commutative 75.3%

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

   4. +-rgt-identity 75.3%

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

7. Applied egg-rr 75.3%

   \[\leadsto -2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 75.3%

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

9. Applied egg-rr 75.3%

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

10. Final simplification 75.3%

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

Alternative 11: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-14} \lor \neg \left(x \leq 3.3 \cdot 10^{-25}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -5.5e-14) (not (<= x 3.3e-25)))
     (* -2.0 (* (sin x) t_0))
     (* -2.0 (pow t_0 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -5.5e-14) || !(x <= 3.3e-25)) {
		tmp = -2.0 * (sin(x) * t_0);
	} else {
		tmp = -2.0 * pow(t_0, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((x <= (-5.5d-14)) .or. (.not. (x <= 3.3d-25))) then
        tmp = (-2.0d0) * (sin(x) * t_0)
    else
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((x <= -5.5e-14) || !(x <= 3.3e-25)) {
		tmp = -2.0 * (Math.sin(x) * t_0);
	} else {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -5.5e-14) or not (x <= 3.3e-25):
		tmp = -2.0 * (math.sin(x) * t_0)
	else:
		tmp = -2.0 * math.pow(t_0, 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -5.5e-14) || !(x <= 3.3e-25))
		tmp = Float64(-2.0 * Float64(sin(x) * t_0));
	else
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((x <= -5.5e-14) || ~((x <= 3.3e-25)))
		tmp = -2.0 * (sin(x) * t_0);
	else
		tmp = -2.0 * (t_0 ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -5.5e-14], N[Not[LessEqual[x, 3.3e-25]], $MachinePrecision]], N[(-2.0 * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.49999999999999991e-14 or 3.2999999999999998e-25 < x

   1. Initial program 8.3%

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

      1. diff-cos 7.2%

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

      2. div-inv 7.2%

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

      3. metadata-eval 7.2%

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

      4. div-inv 7.2%

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

      5. +-commutative 7.2%

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

      6. metadata-eval 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. *-commutative 7.2%

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

      2. +-commutative 7.2%

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

      3. associate--l+ 52.2%

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

      4. +-inverses 52.2%

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

      5. distribute-lft-in 52.2%

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

      6. metadata-eval 52.2%

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

      7. *-commutative 52.2%

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

      8. +-commutative 52.2%

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

   5. Simplified 52.2%

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

   6. Taylor expanded in eps around 0 51.2%

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

   if -5.49999999999999991e-14 < x < 3.2999999999999998e-25

   1. Initial program 72.3%

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

      1. diff-cos 95.1%

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

      2. div-inv 95.1%

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

      3. metadata-eval 95.1%

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

      4. div-inv 95.1%

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

      5. +-commutative 95.1%

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

      6. metadata-eval 95.1%

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

   3. Applied egg-rr 95.1%

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

      1. *-commutative 95.1%

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

      2. +-commutative 95.1%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 94.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-14} \lor \neg \left(x \leq 3.3 \cdot 10^{-25}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 12: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-14} \lor \neg \left(x \leq 1.8 \cdot 10^{-25}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.6e-14) (not (<= x 1.8e-25)))
   (* eps (- (sin x)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.6e-14) || !(x <= 1.8e-25)) {
		tmp = eps * -sin(x);
	} else {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-3.6d-14)) .or. (.not. (x <= 1.8d-25))) then
        tmp = eps * -sin(x)
    else
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -3.6e-14) || !(x <= 1.8e-25)) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.6e-14) or not (x <= 1.8e-25):
		tmp = eps * -math.sin(x)
	else:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.6e-14) || !(x <= 1.8e-25))
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.6e-14) || ~((x <= 1.8e-25)))
		tmp = eps * -sin(x);
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.6e-14], N[Not[LessEqual[x, 1.8e-25]], $MachinePrecision]], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5999999999999998e-14 or 1.8e-25 < x

   1. Initial program 8.3%

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

   2. Taylor expanded in eps around 0 47.8%

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

      1. associate-*r* 47.8%

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

      2. mul-1-neg 47.8%

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

   4. Simplified 47.8%

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

   if -3.5999999999999998e-14 < x < 1.8e-25

   1. Initial program 72.3%

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

      1. diff-cos 95.1%

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

      2. div-inv 95.1%

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

      3. metadata-eval 95.1%

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

      4. div-inv 95.1%

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

      5. +-commutative 95.1%

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

      6. metadata-eval 95.1%

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

   3. Applied egg-rr 95.1%

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

      1. *-commutative 95.1%

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

      2. +-commutative 95.1%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 94.6%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-14} \lor \neg \left(x \leq 1.8 \cdot 10^{-25}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 13: 66.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(-\sin x\right)\\ t_1 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -2.6 \cdot 10^{-36}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.02 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(-x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (- (sin x)))) (t_1 (+ -1.0 (cos eps))))
   (if (<= eps -1.1e-5)
     t_1
     (if (<= eps -2.6e-36)
       (* eps (* eps -0.5))
       (if (<= eps 1.4e-124)
         t_0
         (if (<= eps 1.02e-58)
           (fma -0.5 (* eps eps) (* eps (- x)))
           (if (<= eps 3e-8) t_0 t_1)))))))

C

double code(double x, double eps) {
	double t_0 = eps * -sin(x);
	double t_1 = -1.0 + cos(eps);
	double tmp;
	if (eps <= -1.1e-5) {
		tmp = t_1;
	} else if (eps <= -2.6e-36) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 1.4e-124) {
		tmp = t_0;
	} else if (eps <= 1.02e-58) {
		tmp = fma(-0.5, (eps * eps), (eps * -x));
	} else if (eps <= 3e-8) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(-sin(x)))
	t_1 = Float64(-1.0 + cos(eps))
	tmp = 0.0
	if (eps <= -1.1e-5)
		tmp = t_1;
	elseif (eps <= -2.6e-36)
		tmp = Float64(eps * Float64(eps * -0.5));
	elseif (eps <= 1.4e-124)
		tmp = t_0;
	elseif (eps <= 1.02e-58)
		tmp = fma(-0.5, Float64(eps * eps), Float64(eps * Float64(-x)));
	elseif (eps <= 3e-8)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.1e-5], t$95$1, If[LessEqual[eps, -2.6e-36], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.4e-124], t$95$0, If[LessEqual[eps, 1.02e-58], N[(-0.5 * N[(eps * eps), $MachinePrecision] + N[(eps * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3e-8], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < -1.1e-5 or 2.99999999999999973e-8 < eps

   1. Initial program 51.8%

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

   2. Taylor expanded in x around 0 53.7%

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

   if -1.1e-5 < eps < -2.6e-36

   1. Initial program 5.0%

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

   2. Taylor expanded in x around 0 5.0%

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

   3. Taylor expanded in eps around 0 78.3%

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

      1. *-commutative 78.3%

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

      2. unpow2 78.3%

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

      3. associate-*l* 78.3%

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

   5. Simplified 78.3%

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

   if -2.6e-36 < eps < 1.39999999999999999e-124 or 1.0199999999999999e-58 < eps < 2.99999999999999973e-8

   1. Initial program 31.9%

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

   2. Taylor expanded in eps around 0 89.3%

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

      1. associate-*r* 89.3%

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

      2. mul-1-neg 89.3%

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

   4. Simplified 89.3%

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

   if 1.39999999999999999e-124 < eps < 1.0199999999999999e-58

   1. Initial program 4.8%

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

   2. Taylor expanded in eps around 0 4.8%

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

      1. associate-+r+ 4.8%

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

      2. mul-1-neg 4.8%

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

      3. unsub-neg 4.8%

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

      4. associate-*r* 4.8%

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

      5. distribute-rgt1-in 4.8%

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

      6. unpow2 4.8%

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

      7. associate-*r* 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in x around 0 79.3%

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

      1. +-commutative 79.3%

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

      2. fma-def 79.3%

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

      3. unpow2 79.3%

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

      4. mul-1-neg 79.3%

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

      5. distribute-rgt-neg-in 79.3%

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

   7. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(-x\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq -2.6 \cdot 10^{-36}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-124}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.02 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(-x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 14: 49.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -5.5 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 3.8 \cdot 10^{-175}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000135:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* eps (* eps -0.5))))
   (if (<= eps -1.1e-5)
     t_0
     (if (<= eps -5.5e-118)
       t_1
       (if (<= eps 3.8e-175) (* eps (- x)) (if (<= eps 0.000135) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double t_1 = eps * (eps * -0.5);
	double tmp;
	if (eps <= -1.1e-5) {
		tmp = t_0;
	} else if (eps <= -5.5e-118) {
		tmp = t_1;
	} else if (eps <= 3.8e-175) {
		tmp = eps * -x;
	} else if (eps <= 0.000135) {
		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 = eps * (eps * (-0.5d0))
    if (eps <= (-1.1d-5)) then
        tmp = t_0
    else if (eps <= (-5.5d-118)) then
        tmp = t_1
    else if (eps <= 3.8d-175) then
        tmp = eps * -x
    else if (eps <= 0.000135d0) 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 = eps * (eps * -0.5);
	double tmp;
	if (eps <= -1.1e-5) {
		tmp = t_0;
	} else if (eps <= -5.5e-118) {
		tmp = t_1;
	} else if (eps <= 3.8e-175) {
		tmp = eps * -x;
	} else if (eps <= 0.000135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	t_1 = eps * (eps * -0.5)
	tmp = 0
	if eps <= -1.1e-5:
		tmp = t_0
	elif eps <= -5.5e-118:
		tmp = t_1
	elif eps <= 3.8e-175:
		tmp = eps * -x
	elif eps <= 0.000135:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	t_1 = Float64(eps * Float64(eps * -0.5))
	tmp = 0.0
	if (eps <= -1.1e-5)
		tmp = t_0;
	elseif (eps <= -5.5e-118)
		tmp = t_1;
	elseif (eps <= 3.8e-175)
		tmp = Float64(eps * Float64(-x));
	elseif (eps <= 0.000135)
		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 = eps * (eps * -0.5);
	tmp = 0.0;
	if (eps <= -1.1e-5)
		tmp = t_0;
	elseif (eps <= -5.5e-118)
		tmp = t_1;
	elseif (eps <= 3.8e-175)
		tmp = eps * -x;
	elseif (eps <= 0.000135)
		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[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.1e-5], t$95$0, If[LessEqual[eps, -5.5e-118], t$95$1, If[LessEqual[eps, 3.8e-175], N[(eps * (-x)), $MachinePrecision], If[LessEqual[eps, 0.000135], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.1e-5 or 1.35000000000000002e-4 < eps

   1. Initial program 52.0%

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

   2. Taylor expanded in x around 0 53.9%

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

   if -1.1e-5 < eps < -5.5000000000000003e-118 or 3.8e-175 < eps < 1.35000000000000002e-4

   1. Initial program 8.1%

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

   2. Taylor expanded in x around 0 8.1%

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

   3. Taylor expanded in eps around 0 54.4%

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

      1. *-commutative 54.4%

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

      2. unpow2 54.4%

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

      3. associate-*l* 54.4%

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

   5. Simplified 54.4%

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

   if -5.5000000000000003e-118 < eps < 3.8e-175

   1. Initial program 40.9%

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

   2. Taylor expanded in eps around 0 97.0%

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

      1. associate-*r* 97.0%

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

      2. mul-1-neg 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in x around 0 50.4%

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

      1. associate-*r* 50.4%

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

      2. neg-mul-1 50.4%

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

   7. Simplified 50.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq -5.5 \cdot 10^{-118}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 3.8 \cdot 10^{-175}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000135:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 15: 66.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -3 \cdot 10^{-37}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))))
   (if (<= eps -1.1e-5)
     t_0
     (if (<= eps -3e-37)
       (* eps (* eps -0.5))
       (if (<= eps 3e-8) (* eps (- (sin x))) t_0)))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double tmp;
	if (eps <= -1.1e-5) {
		tmp = t_0;
	} else if (eps <= -3e-37) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 3e-8) {
		tmp = eps * -sin(x);
	} 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 <= (-1.1d-5)) then
        tmp = t_0
    else if (eps <= (-3d-37)) then
        tmp = eps * (eps * (-0.5d0))
    else if (eps <= 3d-8) then
        tmp = eps * -sin(x)
    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 <= -1.1e-5) {
		tmp = t_0;
	} else if (eps <= -3e-37) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 3e-8) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	tmp = 0
	if eps <= -1.1e-5:
		tmp = t_0
	elif eps <= -3e-37:
		tmp = eps * (eps * -0.5)
	elif eps <= 3e-8:
		tmp = eps * -math.sin(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	tmp = 0.0
	if (eps <= -1.1e-5)
		tmp = t_0;
	elseif (eps <= -3e-37)
		tmp = Float64(eps * Float64(eps * -0.5));
	elseif (eps <= 3e-8)
		tmp = Float64(eps * Float64(-sin(x)));
	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 <= -1.1e-5)
		tmp = t_0;
	elseif (eps <= -3e-37)
		tmp = eps * (eps * -0.5);
	elseif (eps <= 3e-8)
		tmp = eps * -sin(x);
	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, -1.1e-5], t$95$0, If[LessEqual[eps, -3e-37], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3e-8], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.1e-5 or 2.99999999999999973e-8 < eps

   1. Initial program 51.8%

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

   2. Taylor expanded in x around 0 53.7%

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

   if -1.1e-5 < eps < -3e-37

   1. Initial program 5.0%

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

   2. Taylor expanded in x around 0 5.0%

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

   3. Taylor expanded in eps around 0 78.3%

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

      1. *-commutative 78.3%

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

      2. unpow2 78.3%

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

      3. associate-*l* 78.3%

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

   5. Simplified 78.3%

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

   if -3e-37 < eps < 2.99999999999999973e-8

   1. Initial program 28.4%

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

   2. Taylor expanded in eps around 0 82.9%

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

      1. associate-*r* 82.9%

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

      2. mul-1-neg 82.9%

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

   4. Simplified 82.9%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq -3 \cdot 10^{-37}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 16: 23.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-72}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e-72) (* eps (- x)) (* eps (* eps -0.5))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-72) {
		tmp = eps * -x;
	} else {
		tmp = eps * (eps * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.4d-72)) then
        tmp = eps * -x
    else
        tmp = eps * (eps * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-72) {
		tmp = eps * -x;
	} else {
		tmp = eps * (eps * -0.5);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.4e-72:
		tmp = eps * -x
	else:
		tmp = eps * (eps * -0.5)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e-72)
		tmp = Float64(eps * Float64(-x));
	else
		tmp = Float64(eps * Float64(eps * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.4e-72)
		tmp = eps * -x;
	else
		tmp = eps * (eps * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e-72], N[(eps * (-x)), $MachinePrecision], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e-72

   1. Initial program 20.6%

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

   2. Taylor expanded in eps around 0 49.1%

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

      1. associate-*r* 49.1%

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

      2. mul-1-neg 49.1%

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

   4. Simplified 49.1%

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

   5. Taylor expanded in x around 0 9.5%

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

      1. associate-*r* 9.5%

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

      2. neg-mul-1 9.5%

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

   7. Simplified 9.5%

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

   if -2.4e-72 < x

   1. Initial program 48.2%

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

   2. Taylor expanded in x around 0 49.3%

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

   3. Taylor expanded in eps around 0 34.9%

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

      1. *-commutative 34.9%

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

      2. unpow2 34.9%

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

      3. associate-*l* 34.9%

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

   5. Simplified 34.9%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-72}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 17: 18.3% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

2. Taylor expanded in eps around 0 38.6%

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

   1. associate-*r* 38.6%

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

   2. mul-1-neg 38.6%

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

4. Simplified 38.6%

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

5. Taylor expanded in x around 0 17.6%

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

   1. associate-*r* 17.6%

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

   2. neg-mul-1 17.6%

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

7. Simplified 17.6%

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

8. Final simplification 17.6%

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

Reproduce

herbie shell --seed 2023200 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))